Step | Hyp | Ref
| Expression |
1 | | 1nn 10908 |
. . . . 5
⊢ 1 ∈
ℕ |
2 | 1 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = ∅) → 1 ∈
ℕ) |
3 | | 0le0 10987 |
. . . . . 6
⊢ 0 ≤
0 |
4 | 3 | a1i 11 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 ≤
0) |
5 | | hoidmvlelem3.g |
. . . . . . . 8
⊢ 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) |
6 | 5 | a1i 11 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))) |
7 | | fveq2 6103 |
. . . . . . . . 9
⊢ (𝑌 = ∅ → (𝐿‘𝑌) = (𝐿‘∅)) |
8 | | reseq2 5312 |
. . . . . . . . . 10
⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = (𝐴 ↾ ∅)) |
9 | | res0 5321 |
. . . . . . . . . . 11
⊢ (𝐴 ↾ ∅) =
∅ |
10 | 9 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑌 = ∅ → (𝐴 ↾ ∅) =
∅) |
11 | 8, 10 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑌 = ∅ → (𝐴 ↾ 𝑌) = ∅) |
12 | | reseq2 5312 |
. . . . . . . . . 10
⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = (𝐵 ↾ ∅)) |
13 | | res0 5321 |
. . . . . . . . . . 11
⊢ (𝐵 ↾ ∅) =
∅ |
14 | 13 | a1i 11 |
. . . . . . . . . 10
⊢ (𝑌 = ∅ → (𝐵 ↾ ∅) =
∅) |
15 | 12, 14 | eqtrd 2644 |
. . . . . . . . 9
⊢ (𝑌 = ∅ → (𝐵 ↾ 𝑌) = ∅) |
16 | 7, 11, 15 | oveq123d 6570 |
. . . . . . . 8
⊢ (𝑌 = ∅ → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅)) |
17 | 16 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = (∅(𝐿‘∅)∅)) |
18 | | hoidmvlelem3.l |
. . . . . . . 8
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
19 | | f0 5999 |
. . . . . . . . 9
⊢
∅:∅⟶ℝ |
20 | 19 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = ∅) →
∅:∅⟶ℝ) |
21 | 18, 20, 20 | hoidmv0val 39473 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → (∅(𝐿‘∅)∅) =
0) |
22 | 6, 17, 21 | 3eqtrd 2648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 = 0) |
23 | | nfcvd 2752 |
. . . . . . . . . . 11
⊢ (𝜑 → Ⅎ𝑗(𝑃‘1)) |
24 | | nfv 1830 |
. . . . . . . . . . 11
⊢
Ⅎ𝑗𝜑 |
25 | | simpr 476 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 = 1) → 𝑗 = 1) |
26 | 25 | fveq2d 6107 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 = 1) → (𝑃‘𝑗) = (𝑃‘1)) |
27 | | 1red 9934 |
. . . . . . . . . . 11
⊢ (𝜑 → 1 ∈
ℝ) |
28 | | rge0ssre 12151 |
. . . . . . . . . . . . 13
⊢
(0[,)+∞) ⊆ ℝ |
29 | | id 22 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝜑) |
30 | 1 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 1 ∈
ℕ) |
31 | 1 | elexi 3186 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
32 | | eleq1 2676 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 1 → (𝑗 ∈ ℕ ↔ 1 ∈
ℕ)) |
33 | 32 | anbi2d 736 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 1 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 1 ∈ ℕ))) |
34 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 1 → (𝑃‘𝑗) = (𝑃‘1)) |
35 | 34 | eleq1d 2672 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 1 → ((𝑃‘𝑗) ∈ (0[,)+∞) ↔ (𝑃‘1) ∈
(0[,)+∞))) |
36 | 33, 35 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) ↔ ((𝜑 ∧ 1 ∈ ℕ) →
(𝑃‘1) ∈
(0[,)+∞)))) |
37 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℕ) |
38 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) |
40 | | hoidmvlelem3.p |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝑃 = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
41 | 40 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ ℕ ∧ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ V) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
42 | 37, 39, 41 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
43 | 42 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
44 | | hoidmvlelem3.x |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑋 ∈ Fin) |
45 | | hoidmvlelem3.w |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 𝑊 = (𝑌 ∪ {𝑍}) |
46 | 45 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → 𝑊 = (𝑌 ∪ {𝑍})) |
47 | | hoidmvlelem3.y |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
48 | | hoidmvlelem3.z |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝑍 ∈ (𝑋 ∖ 𝑌)) |
49 | 48 | eldifad 3552 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝑍 ∈ 𝑋) |
50 | | snssi 4280 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑍 ∈ 𝑋 → {𝑍} ⊆ 𝑋) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → {𝑍} ⊆ 𝑋) |
52 | 47, 51 | unssd 3751 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑌 ∪ {𝑍}) ⊆ 𝑋) |
53 | 46, 52 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝑊 ⊆ 𝑋) |
54 | 44, 53 | ssfid 8068 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑊 ∈ Fin) |
55 | | ssun1 3738 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ 𝑌 ⊆ (𝑌 ∪ {𝑍}) |
56 | 45 | eqcomi 2619 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑌 ∪ {𝑍}) = 𝑊 |
57 | 55, 56 | sseqtri 3600 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑌 ⊆ 𝑊 |
58 | 57 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝑌 ⊆ 𝑊) |
59 | 54, 58 | ssfid 8068 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝑌 ∈ Fin) |
60 | 59 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ Fin) |
61 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌)) |
62 | 61 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌)) |
63 | | hoidmvlelem3.c |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝐶:ℕ⟶(ℝ
↑𝑚 𝑊)) |
64 | 63 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚
𝑊)) |
65 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐶‘𝑗) ∈ (ℝ ↑𝑚
𝑊) → (𝐶‘𝑗):𝑊⟶ℝ) |
66 | 64, 65 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑊⟶ℝ) |
67 | 55, 45 | sseqtr4i 3601 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑌 ⊆ 𝑊 |
68 | 67 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ⊆ 𝑊) |
69 | 66, 68 | fssresd 5984 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ) |
70 | | reex 9906 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ℝ
∈ V |
71 | 70 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ℝ ∈
V) |
72 | 54, 58 | ssexd 4733 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → 𝑌 ∈ V) |
73 | 72 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ V) |
74 | 71, 73 | elmapd 7758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)) |
75 | 69, 74 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
76 | 75 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
77 | 62, 76 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
78 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹) |
79 | 78 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = 𝐹) |
80 | | 0red 9920 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑦 ∈ 𝑌) → 0 ∈ ℝ) |
81 | | hoidmvlelem3.f |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ 𝐹 = (𝑦 ∈ 𝑌 ↦ 0) |
82 | 80, 81 | fmptd 6292 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → 𝐹:𝑌⟶ℝ) |
83 | 70 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ℝ ∈
V) |
84 | 83, 59 | elmapd 7758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝜑 → (𝐹 ∈ (ℝ ↑𝑚
𝑌) ↔ 𝐹:𝑌⟶ℝ)) |
85 | 82, 84 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐹 ∈ (ℝ ↑𝑚
𝑌)) |
86 | 85 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ (ℝ ↑𝑚
𝑌)) |
87 | 79, 86 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
88 | 77, 87 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
89 | | hoidmvlelem3.j |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐽 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) |
90 | 88, 89 | fmptd 6292 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐽:ℕ⟶(ℝ
↑𝑚 𝑌)) |
91 | 90 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) ∈ (ℝ ↑𝑚
𝑌)) |
92 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐽‘𝑗) ∈ (ℝ ↑𝑚
𝑌) → (𝐽‘𝑗):𝑌⟶ℝ) |
93 | 91, 92 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) |
94 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌)) |
95 | 94 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌)) |
96 | | hoidmvlelem3.d |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝐷:ℕ⟶(ℝ
↑𝑚 𝑊)) |
97 | 96 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚
𝑊)) |
98 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝐷‘𝑗) ∈ (ℝ ↑𝑚
𝑊) → (𝐷‘𝑗):𝑊⟶ℝ) |
99 | 97, 98 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑊⟶ℝ) |
100 | 99, 68 | fssresd 5984 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ) |
101 | 71, 73 | elmapd 7758 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ↔ ((𝐷‘𝑗) ↾ 𝑌):𝑌⟶ℝ)) |
102 | 100, 101 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
103 | 102 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐷‘𝑗) ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
104 | 95, 103 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
105 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹) |
106 | 105 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = 𝐹) |
107 | 106, 86 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
108 | 104, 107 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ (ℝ ↑𝑚
𝑌)) |
109 | | hoidmvlelem3.k |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝐾 = (𝑗 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) |
110 | 108, 109 | fmptd 6292 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐾:ℕ⟶(ℝ
↑𝑚 𝑌)) |
111 | 110 | ffvelrnda 6267 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) ∈ (ℝ ↑𝑚
𝑌)) |
112 | | elmapi 7765 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾‘𝑗) ∈ (ℝ ↑𝑚
𝑌) → (𝐾‘𝑗):𝑌⟶ℝ) |
113 | 111, 112 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) |
114 | 18, 60, 93, 113 | hoidmvcl 39472 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ∈ (0[,)+∞)) |
115 | 43, 114 | eqeltrd 2688 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) |
116 | 31, 36, 115 | vtocl 3232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 ∈ ℕ) →
(𝑃‘1) ∈
(0[,)+∞)) |
117 | 29, 30, 116 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑃‘1) ∈
(0[,)+∞)) |
118 | 28, 117 | sseldi 3566 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑃‘1) ∈ ℝ) |
119 | 118 | recnd 9947 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑃‘1) ∈ ℂ) |
120 | 23, 24, 26, 27, 119 | sumsnd 38208 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1)) |
121 | 120 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = (𝑃‘1)) |
122 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 1 → (𝐽‘𝑗) = (𝐽‘1)) |
123 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 1 → (𝐾‘𝑗) = (𝐾‘1)) |
124 | 122, 123 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝑗 = 1 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))) |
125 | | ovex 6577 |
. . . . . . . . . . . 12
⊢ ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) ∈ V |
126 | 124, 40, 125 | fvmpt 6191 |
. . . . . . . . . . 11
⊢ (1 ∈
ℕ → (𝑃‘1)
= ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))) |
127 | 1, 126 | ax-mp 5 |
. . . . . . . . . 10
⊢ (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) |
128 | 127 | a1i 11 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝑃‘1) = ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1))) |
129 | 7 | oveqd 6566 |
. . . . . . . . . . 11
⊢ (𝑌 = ∅ → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1))) |
130 | 129 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = ((𝐽‘1)(𝐿‘∅)(𝐾‘1))) |
131 | 122 | feq1d 5943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 1 → ((𝐽‘𝑗):𝑌⟶ℝ ↔ (𝐽‘1):𝑌⟶ℝ)) |
132 | 33, 131 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐽‘1):𝑌⟶ℝ))) |
133 | 69 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ) |
134 | 62 | feq1d 5943 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ ((𝐶‘𝑗) ↾ 𝑌):𝑌⟶ℝ)) |
135 | 133, 134 | mpbird 246 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ) |
136 | 82 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹:𝑌⟶ℝ) |
137 | 79 | feq1d 5943 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ ↔ 𝐹:𝑌⟶ℝ)) |
138 | 136, 137 | mpbird 246 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ) |
139 | 135, 138 | pm2.61dan 828 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ) |
140 | | simpr 476 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
141 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐶‘𝑗) ∈ V |
142 | 141 | resex 5363 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐶‘𝑗) ↾ 𝑌) ∈ V |
143 | 62, 142 | syl6eqel 2696 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) |
144 | 85 | elexd 3187 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝜑 → 𝐹 ∈ V) |
145 | 144 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐹 ∈ V) |
146 | 145 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → 𝐹 ∈ V) |
147 | 79, 146 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ ¬ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) |
148 | 143, 147 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) |
149 | 89 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) |
150 | 140, 148,
149 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) |
151 | 150 | feq1d 5943 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐽‘𝑗):𝑌⟶ℝ ↔ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹):𝑌⟶ℝ)) |
152 | 139, 151 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗):𝑌⟶ℝ) |
153 | 31, 132, 152 | vtocl 3232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 ∈ ℕ) →
(𝐽‘1):𝑌⟶ℝ) |
154 | 29, 30, 153 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐽‘1):𝑌⟶ℝ) |
155 | 154 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):𝑌⟶ℝ) |
156 | | id 22 |
. . . . . . . . . . . . . . 15
⊢ (𝑌 = ∅ → 𝑌 = ∅) |
157 | 156 | eqcomd 2616 |
. . . . . . . . . . . . . 14
⊢ (𝑌 = ∅ → ∅ =
𝑌) |
158 | 157 | feq2d 5944 |
. . . . . . . . . . . . 13
⊢ (𝑌 = ∅ → ((𝐽‘1):∅⟶ℝ
↔ (𝐽‘1):𝑌⟶ℝ)) |
159 | 158 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1):∅⟶ℝ ↔
(𝐽‘1):𝑌⟶ℝ)) |
160 | 155, 159 | mpbird 246 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐽‘1):∅⟶ℝ) |
161 | 123 | feq1d 5943 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 1 → ((𝐾‘𝑗):𝑌⟶ℝ ↔ (𝐾‘1):𝑌⟶ℝ)) |
162 | 33, 161 | imbi12d 333 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 1 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗):𝑌⟶ℝ) ↔ ((𝜑 ∧ 1 ∈ ℕ) → (𝐾‘1):𝑌⟶ℝ))) |
163 | 31, 162, 113 | vtocl 3232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 1 ∈ ℕ) →
(𝐾‘1):𝑌⟶ℝ) |
164 | 29, 30, 163 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝐾‘1):𝑌⟶ℝ) |
165 | 164 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):𝑌⟶ℝ) |
166 | 157 | feq2d 5944 |
. . . . . . . . . . . . 13
⊢ (𝑌 = ∅ → ((𝐾‘1):∅⟶ℝ
↔ (𝐾‘1):𝑌⟶ℝ)) |
167 | 166 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐾‘1):∅⟶ℝ ↔
(𝐾‘1):𝑌⟶ℝ)) |
168 | 165, 167 | mpbird 246 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐾‘1):∅⟶ℝ) |
169 | 18, 160, 168 | hoidmv0val 39473 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘∅)(𝐾‘1)) = 0) |
170 | 130, 169 | eqtrd 2644 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((𝐽‘1)(𝐿‘𝑌)(𝐾‘1)) = 0) |
171 | 121, 128,
170 | 3eqtrd 2648 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑌 = ∅) → Σ𝑗 ∈ {1} (𝑃‘𝑗) = 0) |
172 | 171 | oveq2d 6565 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = ((1 + 𝐸) · 0)) |
173 | | hoidmvlelem3.e |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
174 | 173 | rpred 11748 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐸 ∈ ℝ) |
175 | 27, 174 | readdcld 9948 |
. . . . . . . . . 10
⊢ (𝜑 → (1 + 𝐸) ∈ ℝ) |
176 | 175 | recnd 9947 |
. . . . . . . . 9
⊢ (𝜑 → (1 + 𝐸) ∈ ℂ) |
177 | 176 | mul01d 10114 |
. . . . . . . 8
⊢ (𝜑 → ((1 + 𝐸) · 0) = 0) |
178 | 177 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · 0) = 0) |
179 | | eqidd 2611 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑌 = ∅) → 0 = 0) |
180 | 172, 178,
179 | 3eqtrd 2648 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 = ∅) → ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) = 0) |
181 | 22, 180 | breq12d 4596 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 = ∅) → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)) ↔ 0 ≤ 0)) |
182 | 4, 181 | mpbird 246 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 = ∅) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))) |
183 | | oveq2 6557 |
. . . . . . . . 9
⊢ (𝑚 = 1 → (1...𝑚) = (1...1)) |
184 | 1 | nnzi 11278 |
. . . . . . . . . . 11
⊢ 1 ∈
ℤ |
185 | | fzsn 12254 |
. . . . . . . . . . 11
⊢ (1 ∈
ℤ → (1...1) = {1}) |
186 | 184, 185 | ax-mp 5 |
. . . . . . . . . 10
⊢ (1...1) =
{1} |
187 | 186 | a1i 11 |
. . . . . . . . 9
⊢ (𝑚 = 1 → (1...1) =
{1}) |
188 | 183, 187 | eqtrd 2644 |
. . . . . . . 8
⊢ (𝑚 = 1 → (1...𝑚) = {1}) |
189 | 188 | sumeq1d 14279 |
. . . . . . 7
⊢ (𝑚 = 1 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) = Σ𝑗 ∈ {1} (𝑃‘𝑗)) |
190 | 189 | oveq2d 6565 |
. . . . . 6
⊢ (𝑚 = 1 → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗))) |
191 | 190 | breq2d 4595 |
. . . . 5
⊢ (𝑚 = 1 → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ {1} (𝑃‘𝑗)))) |
192 | 191 | rspcev 3282 |
. . . 4
⊢ ((1
∈ ℕ ∧ 𝐺 ≤
((1 + 𝐸) ·
Σ𝑗 ∈ {1} (𝑃‘𝑗))) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
193 | 2, 182, 192 | syl2anc 691 |
. . 3
⊢ ((𝜑 ∧ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
194 | | simpl 472 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝜑) |
195 | | neqne 2790 |
. . . . 5
⊢ (¬
𝑌 = ∅ → 𝑌 ≠ ∅) |
196 | 195 | adantl 481 |
. . . 4
⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → 𝑌 ≠ ∅) |
197 | | nfv 1830 |
. . . . . 6
⊢
Ⅎ𝑗(𝜑 ∧ 𝑌 ≠ ∅) |
198 | 184 | a1i 11 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 ∈
ℤ) |
199 | | nnuz 11599 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
200 | 115 | adantlr 747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,)+∞)) |
201 | | hoidmvlelem3.a |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴:𝑊⟶ℝ) |
202 | 67 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝑌 ⊆ 𝑊) |
203 | 201, 202 | fssresd 5984 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐴 ↾ 𝑌):𝑌⟶ℝ) |
204 | | hoidmvlelem3.b |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐵:𝑊⟶ℝ) |
205 | 204, 202 | fssresd 5984 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝐵 ↾ 𝑌):𝑌⟶ℝ) |
206 | 18, 59, 203, 205 | hoidmvcl 39472 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ (0[,)+∞)) |
207 | 28, 206 | sseldi 3566 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ∈ ℝ) |
208 | 5, 207 | syl5eqel 2692 |
. . . . . . . 8
⊢ (𝜑 → 𝐺 ∈ ℝ) |
209 | | 0red 9920 |
. . . . . . . . 9
⊢ (𝜑 → 0 ∈
ℝ) |
210 | | 1rp 11712 |
. . . . . . . . . . . . 13
⊢ 1 ∈
ℝ+ |
211 | 210 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℝ+) |
212 | 211, 173 | jca 553 |
. . . . . . . . . . 11
⊢ (𝜑 → (1 ∈
ℝ+ ∧ 𝐸
∈ ℝ+)) |
213 | | rpaddcl 11730 |
. . . . . . . . . . 11
⊢ ((1
∈ ℝ+ ∧ 𝐸 ∈ ℝ+) → (1 +
𝐸) ∈
ℝ+) |
214 | 212, 213 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (1 + 𝐸) ∈
ℝ+) |
215 | | rpgt0 11720 |
. . . . . . . . . 10
⊢ ((1 +
𝐸) ∈
ℝ+ → 0 < (1 + 𝐸)) |
216 | 214, 215 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 0 < (1 + 𝐸)) |
217 | 209, 216 | gtned 10051 |
. . . . . . . 8
⊢ (𝜑 → (1 + 𝐸) ≠ 0) |
218 | 208, 175,
217 | redivcld 10732 |
. . . . . . 7
⊢ (𝜑 → (𝐺 / (1 + 𝐸)) ∈ ℝ) |
219 | 218 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) ∈ ℝ) |
220 | 218 | ltpnfd 11831 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐺 / (1 + 𝐸)) < +∞) |
221 | 220 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) < +∞) |
222 | | id 22 |
. . . . . . . . . . 11
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) |
223 | 222 | eqcomd 2616 |
. . . . . . . . . 10
⊢
((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞ → +∞ =
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
224 | 223 | adantl 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → +∞ =
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
225 | 221, 224 | breqtrd 4609 |
. . . . . . . 8
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) <
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
226 | 225 | adantlr 747 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) <
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
227 | | simpl 472 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝜑 ∧ 𝑌 ≠ ∅)) |
228 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) |
229 | | nnex 10903 |
. . . . . . . . . . . 12
⊢ ℕ
∈ V |
230 | 229 | a1i 11 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → ℕ ∈
V) |
231 | | icossicc 12131 |
. . . . . . . . . . . . . 14
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
232 | 231, 115 | sseldi 3566 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ (0[,]+∞)) |
233 | | eqid 2610 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) |
234 | 232, 233 | fmptd 6292 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞)) |
235 | 234 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)):ℕ⟶(0[,]+∞)) |
236 | 230, 235 | sge0repnf 39279 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) →
((Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ ↔ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞)) |
237 | 228, 236 | mpbird 246 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) |
238 | 237 | adantlr 747 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) |
239 | 219 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) ∈ ℝ) |
240 | 208 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ) |
241 | 240 | adantlr 747 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ∈ ℝ) |
242 | | simpr 476 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) |
243 | 27, 173 | ltaddrpd 11781 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 < (1 + 𝐸)) |
244 | 243 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 1 < (1 + 𝐸)) |
245 | 59 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ∈ Fin) |
246 | | simpr 476 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝑌 ≠ ∅) |
247 | 203 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐴 ↾ 𝑌):𝑌⟶ℝ) |
248 | 205 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐵 ↾ 𝑌):𝑌⟶ℝ) |
249 | 18, 245, 246, 247, 248 | hoidmvn0val 39474 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)))) |
250 | 5 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))) |
251 | | fvres 6117 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ 𝑌 → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘)) |
252 | | fvres 6117 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ 𝑌 → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘)) |
253 | 251, 252 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 ∈ 𝑌 → (((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
254 | 253 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 ∈ 𝑌 → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
255 | 254 | adantl 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
256 | 201 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐴:𝑊⟶ℝ) |
257 | | elun1 3742 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ (𝑌 ∪ {𝑍})) |
258 | 257, 45 | syl6eleqr 2699 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑘 ∈ 𝑌 → 𝑘 ∈ 𝑊) |
259 | 258 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊) |
260 | 256, 259 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) ∈ ℝ) |
261 | 204 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → 𝐵:𝑊⟶ℝ) |
262 | 261, 259 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐵‘𝑘) ∈ ℝ) |
263 | | volico 38876 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0)) |
264 | 260, 262,
263 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘((𝐴‘𝑘)[,)(𝐵‘𝑘))) = if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0)) |
265 | | hoidmvlelem3.lt |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
266 | 259, 265 | syldan 486 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (𝐴‘𝑘) < (𝐵‘𝑘)) |
267 | 266 | iftrued 4044 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if((𝐴‘𝑘) < (𝐵‘𝑘), ((𝐵‘𝑘) − (𝐴‘𝑘)), 0) = ((𝐵‘𝑘) − (𝐴‘𝑘))) |
268 | 255, 264,
267 | 3eqtrd 2648 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ((𝐵‘𝑘) − (𝐴‘𝑘))) |
269 | 268 | prodeq2dv 14492 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘))) = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
270 | 269 | eqcomd 2616 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)))) |
271 | 270 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) = ∏𝑘 ∈ 𝑌 (vol‘(((𝐴 ↾ 𝑌)‘𝑘)[,)((𝐵 ↾ 𝑌)‘𝑘)))) |
272 | 249, 250,
271 | 3eqtr4d 2654 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 = ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘))) |
273 | | difrp 11744 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐴‘𝑘) ∈ ℝ ∧ (𝐵‘𝑘) ∈ ℝ) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
274 | 260, 262,
273 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘) < (𝐵‘𝑘) ↔ ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+)) |
275 | 266, 274 | mpbid 221 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+) |
276 | 59, 275 | fprodrpcl 14525 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+) |
277 | 276 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∏𝑘 ∈ 𝑌 ((𝐵‘𝑘) − (𝐴‘𝑘)) ∈
ℝ+) |
278 | 272, 277 | eqeltrd 2688 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ∈
ℝ+) |
279 | 214 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 + 𝐸) ∈
ℝ+) |
280 | 278, 279 | ltdivgt1 38513 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (1 < (1 + 𝐸) ↔ (𝐺 / (1 + 𝐸)) < 𝐺)) |
281 | 244, 280 | mpbid 221 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) < 𝐺) |
282 | 281 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) < 𝐺) |
283 | | hoidmvlelem3.i2 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → X𝑘 ∈
𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
284 | 283 | adantr 480 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
285 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑥‘𝑘) ∈ V |
286 | 285 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → (𝑥‘𝑘) ∈ V) |
287 | | hoidmvlelem3.s |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → 𝑆 ∈ 𝑈) |
288 | 287 | elexd 3187 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝜑 → 𝑆 ∈ V) |
289 | 286, 288 | ifcld 4081 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) |
290 | 289 | ralrimivw 2950 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝜑 → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) |
291 | 290 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) |
292 | | eqid 2610 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
293 | 292 | fnmpt 5933 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(∀𝑘 ∈
𝑊 if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊) |
294 | 291, 293 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊) |
295 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
296 | | mptexg 6389 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑊 ∈ Fin → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) |
297 | 54, 296 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) |
298 | 297 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) |
299 | | hoidmvlelem3.o |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑂 = (𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↦ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))) |
300 | 299 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) ∈ V) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))) |
301 | 295, 298,
300 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))) |
302 | 301 | fneq1d 5895 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ↔ (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) Fn 𝑊)) |
303 | 294, 302 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) Fn 𝑊) |
304 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘𝜑 |
305 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘𝑥 |
306 | | nfixp1 7814 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
307 | 305, 306 | nfel 2763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) |
308 | 304, 307 | nfan 1816 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
309 | 301 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘)) |
310 | 309 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘)) |
311 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → 𝑘 ∈ 𝑊) |
312 | 289 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) |
313 | 292 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 ∈ 𝑊 ∧ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
314 | 311, 312,
313 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
315 | 314 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
316 | 310, 315 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
317 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘)) |
318 | 317 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘)) |
319 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ 𝑥 ∈ V |
320 | 319 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
321 | 320 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
322 | 321 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
323 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑌) |
324 | | rspa 2914 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
((∀𝑘 ∈
𝑌 (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
325 | 322, 323,
324 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
326 | 325 | ad4ant24 1290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
327 | 318, 326 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
328 | | snidg 4153 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑍 ∈ (𝑋 ∖ 𝑌) → 𝑍 ∈ {𝑍}) |
329 | 48, 328 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → 𝑍 ∈ {𝑍}) |
330 | | elun2 3743 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑍 ∈ {𝑍} → 𝑍 ∈ (𝑌 ∪ {𝑍})) |
331 | 329, 330 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → 𝑍 ∈ (𝑌 ∪ {𝑍})) |
332 | 56 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → (𝑌 ∪ {𝑍}) = 𝑊) |
333 | 331, 332 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝜑 → 𝑍 ∈ 𝑊) |
334 | 201, 333 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → (𝐴‘𝑍) ∈ ℝ) |
335 | 334 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → (𝐴‘𝑍) ∈
ℝ*) |
336 | 204, 333 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → (𝐵‘𝑍) ∈ ℝ) |
337 | 336 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → (𝐵‘𝑍) ∈
ℝ*) |
338 | | iccssxr 12127 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ⊆
ℝ* |
339 | | hoidmvlelem3.u |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} |
340 | | ssrab2 3650 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)) |
341 | 339, 340 | eqsstri 3598 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ 𝑈 ⊆ ((𝐴‘𝑍)[,](𝐵‘𝑍)) |
342 | 341, 287 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) |
343 | 338, 342 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝑆 ∈
ℝ*) |
344 | | iccgelb 12101 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝐴‘𝑍) ∈ ℝ* ∧ (𝐵‘𝑍) ∈ ℝ* ∧ 𝑆 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍))) → (𝐴‘𝑍) ≤ 𝑆) |
345 | 335, 337,
342, 344 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → (𝐴‘𝑍) ≤ 𝑆) |
346 | | hoidmvlelem3.sb |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝑆 < (𝐵‘𝑍)) |
347 | 335, 337,
343, 345, 346 | elicod 12095 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝜑 → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))) |
348 | 347 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍))) |
349 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (¬
𝑘 ∈ 𝑌 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆) |
350 | 349 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = 𝑆) |
351 | 45 | eleq2i 2680 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑘 ∈ 𝑊 ↔ 𝑘 ∈ (𝑌 ∪ {𝑍})) |
352 | 351 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 ∈ 𝑊 → 𝑘 ∈ (𝑌 ∪ {𝑍})) |
353 | 352 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ (𝑌 ∪ {𝑍})) |
354 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ¬ 𝑘 ∈ 𝑌) |
355 | | elunnel1 3716 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑘 ∈ (𝑌 ∪ {𝑍}) ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍}) |
356 | 353, 354,
355 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 ∈ {𝑍}) |
357 | | elsni 4142 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 ∈ {𝑍} → 𝑘 = 𝑍) |
358 | 356, 357 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → 𝑘 = 𝑍) |
359 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝑍 → (𝐴‘𝑘) = (𝐴‘𝑍)) |
360 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝑍 → (𝐵‘𝑘) = (𝐵‘𝑍)) |
361 | 359, 360 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑘 = 𝑍 → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍))) |
362 | 358, 361 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑘 ∈ 𝑊 ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍))) |
363 | 362 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝐵‘𝑘)) = ((𝐴‘𝑍)[,)(𝐵‘𝑍))) |
364 | 350, 363 | eleq12d 2682 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → (if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ 𝑆 ∈ ((𝐴‘𝑍)[,)(𝐵‘𝑍)))) |
365 | 348, 364 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
366 | 365 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) ∧ ¬ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
367 | 327, 366 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
368 | 316, 367 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
369 | 368 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑘 ∈ 𝑊 → ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
370 | 308, 369 | ralrimi 2940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
371 | 303, 370 | jca 553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
372 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑂‘𝑥) ∈ V |
373 | 372 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ ((𝐴‘𝑘)[,)(𝐵‘𝑘)))) |
374 | 371, 373 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
375 | 284, 374 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (𝑂‘𝑥) ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
376 | | eliun 4460 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑂‘𝑥) ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
377 | 375, 376 | sylib 207 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
378 | | ixpfn 7800 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) → 𝑥 Fn 𝑌) |
379 | 378 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 Fn 𝑌) |
380 | 379 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 Fn 𝑌) |
381 | | nfv 1830 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘 𝑗 ∈ ℕ |
382 | 308, 381 | nfan 1816 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) |
383 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘(𝑂‘𝑥) |
384 | | nfixp1 7814 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
Ⅎ𝑘X𝑘 ∈
𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) |
385 | 383, 384 | nfel 2763 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
Ⅎ𝑘(𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) |
386 | 382, 385 | nfan 1816 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
387 | 309 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘)) |
388 | 289 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) ∈ V) |
389 | 259, 388,
313 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
390 | 389 | 3adant2 1073 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑘) = if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) |
391 | 317 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = (𝑥‘𝑘)) |
392 | 387, 390,
391 | 3eqtrrd 2649 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘)) |
393 | 392 | ad5ant125 1304 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) = ((𝑂‘𝑥)‘𝑘)) |
394 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
395 | 372 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
396 | 394, 395 | sylib 207 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥) Fn 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
397 | 396 | simprd 478 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
398 | 258 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → 𝑘 ∈ 𝑊) |
399 | | rspa 2914 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
((∀𝑘 ∈
𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑊) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
400 | 397, 398,
399 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
401 | 400 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
402 | 393, 401 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
403 | 29 | ad3antrrr 762 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝜑) |
404 | 37 | ad2antlr 759 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑗 ∈ ℕ) |
405 | 301 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑂‘𝑥)‘𝑍) = ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍)) |
406 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝜑 → (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆)) = (𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))) |
407 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑘 = 𝑍 → (𝑘 ∈ 𝑌 ↔ 𝑍 ∈ 𝑌)) |
408 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑘 = 𝑍 → (𝑥‘𝑘) = (𝑥‘𝑍)) |
409 | 407, 408 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑘 = 𝑍 → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆)) |
410 | 409 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ 𝑘 = 𝑍) → if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆)) |
411 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑥‘𝑍) ∈ V |
412 | 411 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝜑 → (𝑥‘𝑍) ∈ V) |
413 | 412, 288 | ifcld 4081 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) ∈ V) |
414 | 406, 410,
333, 413 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆)) |
415 | 414 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ((𝑘 ∈ 𝑊 ↦ if(𝑘 ∈ 𝑌, (𝑥‘𝑘), 𝑆))‘𝑍) = if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆)) |
416 | 48 | eldifbd 3553 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝜑 → ¬ 𝑍 ∈ 𝑌) |
417 | 416 | iffalsed 4047 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆) |
418 | 417 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → if(𝑍 ∈ 𝑌, (𝑥‘𝑍), 𝑆) = 𝑆) |
419 | 405, 415,
418 | 3eqtrrd 2649 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍)) |
420 | 419 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 = ((𝑂‘𝑥)‘𝑍)) |
421 | 403, 333 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑍 ∈ 𝑊) |
422 | 395 | simprbi 479 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
423 | 422 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
424 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 = 𝑍 → ((𝑂‘𝑥)‘𝑘) = ((𝑂‘𝑥)‘𝑍)) |
425 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑘 = 𝑍 → ((𝐶‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑍)) |
426 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑘 = 𝑍 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑍)) |
427 | 425, 426 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 = 𝑍 → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) |
428 | 424, 427 | eleq12d 2682 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝑍 → (((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)))) |
429 | 428 | rspcva 3280 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑍 ∈ 𝑊 ∧ ∀𝑘 ∈ 𝑊 ((𝑂‘𝑥)‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) |
430 | 421, 423,
429 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝑂‘𝑥)‘𝑍) ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) |
431 | 420, 430 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) |
432 | 150 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) |
433 | 61 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = ((𝐶‘𝑗) ↾ 𝑌)) |
434 | 432, 433 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐽‘𝑗) = ((𝐶‘𝑗) ↾ 𝑌)) |
435 | 434 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘)) |
436 | 403, 404,
431, 435 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘)) |
437 | 436 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = (((𝐶‘𝑗) ↾ 𝑌)‘𝑘)) |
438 | | fvres 6117 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 ∈ 𝑌 → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘)) |
439 | 438 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐶‘𝑗)‘𝑘)) |
440 | 437, 439 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐽‘𝑗)‘𝑘) = ((𝐶‘𝑗)‘𝑘)) |
441 | 108 | elexd 3187 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) |
442 | 109 | fvmpt2 6200 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑗 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ∈ V) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) |
443 | 140, 441,
442 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) |
444 | 443 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) |
445 | 94 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = ((𝐷‘𝑗) ↾ 𝑌)) |
446 | 444, 445 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → (𝐾‘𝑗) = ((𝐷‘𝑗) ↾ 𝑌)) |
447 | 446 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘)) |
448 | 403, 404,
431, 447 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘)) |
449 | 448 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = (((𝐷‘𝑗) ↾ 𝑌)‘𝑘)) |
450 | | fvres 6117 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 ∈ 𝑌 → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘)) |
451 | 450 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐷‘𝑗) ↾ 𝑌)‘𝑘) = ((𝐷‘𝑗)‘𝑘)) |
452 | 449, 451 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → ((𝐾‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝑘)) |
453 | 440, 452 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
454 | 453 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
455 | 402, 454 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
(((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑌) → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
456 | 455 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑌 → (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
457 | 386, 456 | ralrimi 2940 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
458 | 380, 457 | jca 553 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
459 | 319 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 ∈ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ (𝑥 Fn 𝑌 ∧ ∀𝑘 ∈ 𝑌 (𝑥‘𝑘) ∈ (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
460 | 458, 459 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) ∧ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
461 | 460 | ex 449 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) ∧ 𝑗 ∈ ℕ) → ((𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
462 | 461 | reximdva 3000 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → (∃𝑗 ∈ ℕ (𝑂‘𝑥) ∈ X𝑘 ∈ 𝑊 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
463 | 377, 462 | mpd 15 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
464 | | eliun 4460 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑥 ∈ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
465 | 463, 464 | sylibr 223 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) → 𝑥 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
466 | 465 | ralrimiva 2949 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
467 | | dfss3 3558 |
. . . . . . . . . . . . . . 15
⊢ (X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) ↔ ∀𝑥 ∈ X 𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))𝑥 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
468 | 466, 467 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
469 | | ovex 6577 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℝ
↑𝑚 𝑌) ∈ V |
470 | 469 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (ℝ
↑𝑚 𝑌) ∈ V) |
471 | 229 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ℕ ∈
V) |
472 | 470, 471 | elmapd 7758 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (𝐾 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ) ↔ 𝐾:ℕ⟶(ℝ
↑𝑚 𝑌))) |
473 | 110, 472 | mpbird 246 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)) |
474 | 470, 471 | elmapd 7758 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐽 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ) ↔ 𝐽:ℕ⟶(ℝ
↑𝑚 𝑌))) |
475 | 90, 474 | mpbird 246 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐽 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)) |
476 | 83, 72 | elmapd 7758 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ↔ (𝐵 ↾ 𝑌):𝑌⟶ℝ)) |
477 | 205, 476 | mpbird 246 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
478 | 83, 72 | elmapd 7758 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ↔ (𝐴 ↾ 𝑌):𝑌⟶ℝ)) |
479 | 203, 478 | mpbird 246 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌)) |
480 | | hoidmvlelem3.i |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → ∀𝑒 ∈ (ℝ ↑𝑚
𝑌)∀𝑓 ∈ (ℝ
↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
481 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘)) |
482 | 481 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = ((𝐴 ↾ 𝑌)‘𝑘)) |
483 | 251 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴 ↾ 𝑌)‘𝑘) = (𝐴‘𝑘)) |
484 | 482, 483 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑒‘𝑘) = (𝐴‘𝑘)) |
485 | 484 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑒 = (𝐴 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝑓‘𝑘))) |
486 | 485 | ixpeq2dva 7809 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑒 = (𝐴 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘))) |
487 | 486 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))) |
488 | | oveq1 6556 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (𝑒(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓)) |
489 | 488 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑒 = (𝐴 ↾ 𝑌) → ((𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
490 | 487, 489 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑒 = (𝐴 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
491 | 490 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
492 | 491 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
493 | 492 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑒 = (𝐴 ↾ 𝑌) → (∀𝑓 ∈ (ℝ ↑𝑚
𝑌)∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑓 ∈ (ℝ ↑𝑚
𝑌)∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
494 | 493 | rspcva 3280 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐴 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ∧ ∀𝑒 ∈ (ℝ
↑𝑚 𝑌)∀𝑓 ∈ (ℝ ↑𝑚
𝑌)∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝑒‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → (𝑒(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑓 ∈ (ℝ ↑𝑚
𝑌)∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
495 | 479, 480,
494 | syl2anc 691 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ∀𝑓 ∈ (ℝ ↑𝑚
𝑌)∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
496 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓 = (𝐵 ↾ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘)) |
497 | 496 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = ((𝐵 ↾ 𝑌)‘𝑘)) |
498 | 252 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐵 ↾ 𝑌)‘𝑘) = (𝐵‘𝑘)) |
499 | 497, 498 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → (𝑓‘𝑘) = (𝐵‘𝑘)) |
500 | 499 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓 = (𝐵 ↾ 𝑌) ∧ 𝑘 ∈ 𝑌) → ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
501 | 500 | ixpeq2dva 7809 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 = (𝐵 ↾ 𝑌) → X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) = X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘))) |
502 | 501 | sseq1d 3595 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 = (𝐵 ↾ 𝑌) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))) |
503 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) = ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌))) |
504 | 503 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓 = (𝐵 ↾ 𝑌) → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
505 | 502, 504 | imbi12d 333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 = (𝐵 ↾ 𝑌) → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
506 | 505 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑓 = (𝐵 ↾ 𝑌) → (∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
507 | 506 | ralbidv 2969 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝐵 ↾ 𝑌) → (∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
508 | 507 | rspcva 3280 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐵 ↾ 𝑌) ∈ (ℝ ↑𝑚
𝑌) ∧ ∀𝑓 ∈ (ℝ
↑𝑚 𝑌)∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝑓‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)𝑓) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
509 | 477, 495,
508 | syl2anc 691 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ∀𝑔 ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
510 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑔 = 𝐽 → (𝑔‘𝑗) = (𝐽‘𝑗)) |
511 | 510 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)‘𝑘) = ((𝐽‘𝑗)‘𝑘)) |
512 | 511 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔 = 𝐽 → (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
513 | 512 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = 𝐽 → X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
514 | 513 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝐽 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘))) |
515 | 514 | sseq2d 3596 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = 𝐽 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)))) |
516 | 510 | oveq1d 6564 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔 = 𝐽 → ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) |
517 | 516 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑔 = 𝐽 → (𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) |
518 | 517 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑔 = 𝐽 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) |
519 | 518 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑔 = 𝐽 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
520 | 515, 519 | imbi12d 333 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑔 = 𝐽 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
521 | 520 | ralbidv 2969 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑔 = 𝐽 → (∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ ∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))))) |
522 | 521 | rspcva 3280 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚 ℕ) ∧
∀𝑔 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚
ℕ)∀ℎ ∈
((ℝ ↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝑔‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝑔‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → ∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
523 | 475, 509,
522 | syl2anc 691 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → ∀ℎ ∈ ((ℝ ↑𝑚
𝑌)
↑𝑚 ℕ)(X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) |
524 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (ℎ = 𝐾 → (ℎ‘𝑗) = (𝐾‘𝑗)) |
525 | 524 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (ℎ = 𝐾 → ((ℎ‘𝑗)‘𝑘) = ((𝐾‘𝑗)‘𝑘)) |
526 | 525 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = 𝐾 → (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
527 | 526 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = 𝐾 → X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
528 | 527 | iuneq2d 4483 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = 𝐾 → ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘))) |
529 | 528 | sseq2d 3596 |
. . . . . . . . . . . . . . . . 17
⊢ (ℎ = 𝐾 → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) ↔ X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)))) |
530 | 524 | oveq2d 6565 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (ℎ = 𝐾 → ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
531 | 530 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . 19
⊢ (ℎ = 𝐾 → (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))) |
532 | 531 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . 18
⊢ (ℎ = 𝐾 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))) |
533 | 532 | breq2d 4595 |
. . . . . . . . . . . . . . . . 17
⊢ (ℎ = 𝐾 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))) |
534 | 529, 533 | imbi12d 333 |
. . . . . . . . . . . . . . . 16
⊢ (ℎ = 𝐾 → ((X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗))))) ↔ (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))))) |
535 | 534 | rspcva 3280 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚 ℕ) ∧
∀ℎ ∈ ((ℝ
↑𝑚 𝑌) ↑𝑚 ℕ)(X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((ℎ‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(ℎ‘𝑗)))))) → (X𝑘 ∈ 𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))) |
536 | 473, 523,
535 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (X𝑘 ∈
𝑌 ((𝐴‘𝑘)[,)(𝐵‘𝑘)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑌 (((𝐽‘𝑗)‘𝑘)[,)((𝐾‘𝑗)‘𝑘)) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))) |
537 | 468, 536 | mpd 15 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))) |
538 | | idd 24 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))) |
539 | 537, 538 | mpd 15 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))) |
540 | 539 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))) |
541 | 42 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) |
542 | 541 | mpteq2dva 4672 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝑗 ∈ ℕ ↦ (𝑃‘𝑗)) = (𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))) |
543 | 542 | fveq2d 6107 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) →
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))))) |
544 | 250, 543 | breq12d 4596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ↔ ((𝐴 ↾ 𝑌)(𝐿‘𝑌)(𝐵 ↾ 𝑌)) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)))))) |
545 | 540, 544 | mpbird 246 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → 𝐺 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
546 | 545 | adantr 480 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → 𝐺 ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
547 | 239, 241,
242, 282, 546 | ltletrd 10076 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) ∈ ℝ) → (𝐺 / (1 + 𝐸)) <
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
548 | 227, 238,
547 | syl2anc 691 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ ¬
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗))) = +∞) → (𝐺 / (1 + 𝐸)) <
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
549 | 226, 548 | pm2.61dan 828 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (𝐺 / (1 + 𝐸)) <
(Σ^‘(𝑗 ∈ ℕ ↦ (𝑃‘𝑗)))) |
550 | 197, 198,
199, 200, 219, 549 | sge0uzfsumgt 39337 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) |
551 | 218 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ∈ ℝ) |
552 | | fzfid 12634 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1...𝑚) ∈ Fin) |
553 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝜑) |
554 | | elfznn 12241 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ (1...𝑚) → 𝑗 ∈ ℕ) |
555 | 554 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → 𝑗 ∈ ℕ) |
556 | 28, 115 | sseldi 3566 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑃‘𝑗) ∈ ℝ) |
557 | 553, 555,
556 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ (1...𝑚)) → (𝑃‘𝑗) ∈ ℝ) |
558 | 552, 557 | fsumrecl 14312 |
. . . . . . . . . . . 12
⊢ (𝜑 → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ) |
559 | 558 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ∈ ℝ) |
560 | | simpr 476 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) |
561 | 551, 559,
560 | ltled 10064 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) |
562 | 208 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ∈ ℝ) |
563 | 214 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → (1 + 𝐸) ∈
ℝ+) |
564 | 562, 559,
563 | ledivmuld 11801 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((𝐺 / (1 + 𝐸)) ≤ Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) ↔ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))) |
565 | 561, 564 | mpbid 221 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
566 | 565 | ex 449 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))) |
567 | 566 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))) |
568 | 567 | adantlr 747 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑌 ≠ ∅) ∧ 𝑚 ∈ ℕ) → ((𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))) |
569 | 568 | reximdva 3000 |
. . . . 5
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → (∃𝑚 ∈ ℕ (𝐺 / (1 + 𝐸)) < Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)))) |
570 | 550, 569 | mpd 15 |
. . . 4
⊢ ((𝜑 ∧ 𝑌 ≠ ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
571 | 194, 196,
570 | syl2anc 691 |
. . 3
⊢ ((𝜑 ∧ ¬ 𝑌 = ∅) → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
572 | 193, 571 | pm2.61dan 828 |
. 2
⊢ (𝜑 → ∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
573 | 44 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑋 ∈ Fin) |
574 | 47 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑌 ⊆ 𝑋) |
575 | 48 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑍 ∈ (𝑋 ∖ 𝑌)) |
576 | 201 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐴:𝑊⟶ℝ) |
577 | 204 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐵:𝑊⟶ℝ) |
578 | 63 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐶:ℕ⟶(ℝ
↑𝑚 𝑊)) |
579 | | eqid 2610 |
. . . . 5
⊢ (𝑦 ∈ 𝑌 ↦ 0) = (𝑦 ∈ 𝑌 ↦ 0) |
580 | | eqid 2610 |
. . . . 5
⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
581 | 96 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐷:ℕ⟶(ℝ
↑𝑚 𝑊)) |
582 | | eqid 2610 |
. . . . 5
⊢ (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) = (𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
583 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝐶‘𝑖) = (𝐶‘𝑗)) |
584 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → (𝐷‘𝑖) = (𝐷‘𝑗)) |
585 | 583, 584 | oveq12d 6567 |
. . . . . . . . 9
⊢ (𝑖 = 𝑗 → ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)) = ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))) |
586 | 585 | cbvmptv 4678 |
. . . . . . . 8
⊢ (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗))) |
587 | 586 | fveq2i 6106 |
. . . . . . 7
⊢
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) |
588 | | hoidmvlelem3.r |
. . . . . . 7
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)(𝐷‘𝑗)))) ∈ ℝ) |
589 | 587, 588 | syl5eqel 2692 |
. . . . . 6
⊢ (𝜑 →
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) |
590 | 589 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) →
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)(𝐷‘𝑖)))) ∈ ℝ) |
591 | | hoidmvlelem3.h |
. . . . . 6
⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) |
592 | | eleq1 2676 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑗 ∈ 𝑌 ↔ 𝑖 ∈ 𝑌)) |
593 | | fveq2 6103 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → (𝑐‘𝑗) = (𝑐‘𝑖)) |
594 | 593 | breq1d 4593 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑖 → ((𝑐‘𝑗) ≤ 𝑥 ↔ (𝑐‘𝑖) ≤ 𝑥)) |
595 | 594, 593 | ifbieq1d 4059 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑖 → if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥) = if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)) |
596 | 592, 593,
595 | ifbieq12d 4063 |
. . . . . . . . 9
⊢ (𝑗 = 𝑖 → if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)) = if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) |
597 | 596 | cbvmptv 4678 |
. . . . . . . 8
⊢ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))) = (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))) |
598 | 597 | mpteq2i 4669 |
. . . . . . 7
⊢ (𝑐 ∈ (ℝ
↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥)))) = (𝑐 ∈ (ℝ ↑𝑚
𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥)))) |
599 | 598 | mpteq2i 4669 |
. . . . . 6
⊢ (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ
↑𝑚 𝑊) ↦ (𝑗 ∈ 𝑊 ↦ if(𝑗 ∈ 𝑌, (𝑐‘𝑗), if((𝑐‘𝑗) ≤ 𝑥, (𝑐‘𝑗), 𝑥))))) = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))))) |
600 | 591, 599 | eqtri 2632 |
. . . . 5
⊢ 𝐻 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑊) ↦ (𝑖 ∈ 𝑊 ↦ if(𝑖 ∈ 𝑌, (𝑐‘𝑖), if((𝑐‘𝑖) ≤ 𝑥, (𝑐‘𝑖), 𝑥))))) |
601 | 173 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐸 ∈
ℝ+) |
602 | | fveq2 6103 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → (𝐶‘𝑗) = (𝐶‘𝑖)) |
603 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑖 → (𝐷‘𝑗) = (𝐷‘𝑖)) |
604 | 603 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑗 = 𝑖 → ((𝐻‘𝑧)‘(𝐷‘𝑗)) = ((𝐻‘𝑧)‘(𝐷‘𝑖))) |
605 | 602, 604 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))) = ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))) |
606 | 605 | cbvmptv 4678 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))) = (𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))) |
607 | 606 | fveq2i 6106 |
. . . . . . . . . 10
⊢
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))) =
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))) |
608 | 607 | oveq2i 6560 |
. . . . . . . . 9
⊢ ((1 +
𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) = ((1 + 𝐸) ·
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))) |
609 | 608 | breq2i 4591 |
. . . . . . . 8
⊢ ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))) |
610 | 609 | a1i 11 |
. . . . . . 7
⊢ (𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) → ((𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗)))))) ↔ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖)))))))) |
611 | 610 | rabbiia 3161 |
. . . . . 6
⊢ {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑗))))))} = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))} |
612 | 339, 611 | eqtri 2632 |
. . . . 5
⊢ 𝑈 = {𝑧 ∈ ((𝐴‘𝑍)[,](𝐵‘𝑍)) ∣ (𝐺 · (𝑧 − (𝐴‘𝑍))) ≤ ((1 + 𝐸) ·
(Σ^‘(𝑖 ∈ ℕ ↦ ((𝐶‘𝑖)(𝐿‘𝑊)((𝐻‘𝑧)‘(𝐷‘𝑖))))))} |
613 | 287 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 ∈ 𝑈) |
614 | 346 | 3ad2ant1 1075 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑆 < (𝐵‘𝑍)) |
615 | | eqid 2610 |
. . . . 5
⊢ (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) = (𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) |
616 | | simp2 1055 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝑚 ∈ ℕ) |
617 | | id 22 |
. . . . . . . 8
⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) |
618 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑖 → (𝑃‘𝑗) = (𝑃‘𝑖)) |
619 | 618 | cbvsumv 14274 |
. . . . . . . . . 10
⊢
Σ𝑗 ∈
(1...𝑚)(𝑃‘𝑗) = Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖) |
620 | 619 | oveq2i 6560 |
. . . . . . . . 9
⊢ ((1 +
𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)) |
621 | 620 | a1i 11 |
. . . . . . . 8
⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))) |
622 | 617, 621 | breqtrd 4609 |
. . . . . . 7
⊢ (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))) |
623 | 622 | 3ad2ant3 1077 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖))) |
624 | | simpl 472 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝜑) |
625 | | elfznn 12241 |
. . . . . . . . . . 11
⊢ (𝑖 ∈ (1...𝑚) → 𝑖 ∈ ℕ) |
626 | 625 | adantl 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → 𝑖 ∈ ℕ) |
627 | | eleq1 2676 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑖 → (𝑗 ∈ ℕ ↔ 𝑖 ∈ ℕ)) |
628 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → (𝐽‘𝑗) = (𝐽‘𝑖)) |
629 | | fveq2 6103 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → (𝐾‘𝑗) = (𝐾‘𝑖)) |
630 | 628, 629 | oveq12d 6567 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑖 → ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))) |
631 | 618, 630 | eqeq12d 2625 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑖 → ((𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗)) ↔ (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)))) |
632 | 627, 631 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑖 → ((𝑗 ∈ ℕ → (𝑃‘𝑗) = ((𝐽‘𝑗)(𝐿‘𝑌)(𝐾‘𝑗))) ↔ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))))) |
633 | 632, 42 | chvarv 2251 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ ℕ → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))) |
634 | 633 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖))) |
635 | 627 | anbi2d 736 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑖 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑖 ∈ ℕ))) |
636 | 602 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗)‘𝑍) = ((𝐶‘𝑖)‘𝑍)) |
637 | 603 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘𝑖)‘𝑍)) |
638 | 636, 637 | oveq12d 6567 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 = 𝑖 → (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))) |
639 | 638 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → (𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))) |
640 | 602 | reseq1d 5316 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → ((𝐶‘𝑗) ↾ 𝑌) = ((𝐶‘𝑖) ↾ 𝑌)) |
641 | 639, 640 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)) |
642 | 628, 641 | eqeq12d 2625 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑖 → ((𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹))) |
643 | 635, 642 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐽‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐶‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)))) |
644 | 643, 150 | chvarv 2251 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐽‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)) |
645 | 603 | reseq1d 5316 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 = 𝑖 → ((𝐷‘𝑗) ↾ 𝑌) = ((𝐷‘𝑖) ↾ 𝑌)) |
646 | 639, 645 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 = 𝑖 → if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)) |
647 | 629, 646 | eqeq12d 2625 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 = 𝑖 → ((𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹) ↔ (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))) |
648 | 635, 647 | imbi12d 333 |
. . . . . . . . . . . . . 14
⊢ (𝑗 = 𝑖 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐾‘𝑗) = if(𝑆 ∈ (((𝐶‘𝑗)‘𝑍)[,)((𝐷‘𝑗)‘𝑍)), ((𝐷‘𝑗) ↾ 𝑌), 𝐹)) ↔ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)))) |
649 | 648, 443 | chvarv 2251 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝐾‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)) |
650 | 644, 649 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝐽‘𝑖)(𝐿‘𝑌)(𝐾‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))) |
651 | 634, 650 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))) |
652 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) |
653 | | ovex 6577 |
. . . . . . . . . . . . . 14
⊢ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V |
654 | 653 | a1i 11 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V) |
655 | 615 | fvmpt2 6200 |
. . . . . . . . . . . . 13
⊢ ((𝑖 ∈ ℕ ∧ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) ∈ V) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) |
656 | 652, 654,
655 | syl2anc 691 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖))) |
657 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐶‘𝑖) ∈ V |
658 | 657 | resex 5363 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐶‘𝑖) ↾ 𝑌) ∈ V |
659 | 658 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐶‘𝑖) ↾ 𝑌) ∈ V) |
660 | 81, 144 | syl5eqelr 2693 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → (𝑦 ∈ 𝑌 ↦ 0) ∈ V) |
661 | 659, 660 | ifcld 4081 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) |
662 | 661 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) |
663 | 580 | fvmpt2 6200 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
664 | 652, 662,
663 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
665 | 81 | eqcomi 2619 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ 𝑌 ↦ 0) = 𝐹 |
666 | | ifeq2 4041 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ 𝑌 ↦ 0) = 𝐹 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)) |
667 | 665, 666 | ax-mp 5 |
. . . . . . . . . . . . . . 15
⊢ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹) |
668 | 667 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)) |
669 | 664, 668 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)) |
670 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐷‘𝑖) ∈ V |
671 | 670 | resex 5363 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐷‘𝑖) ↾ 𝑌) ∈ V |
672 | 671 | a1i 11 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ((𝐷‘𝑖) ↾ 𝑌) ∈ V) |
673 | 672, 660 | ifcld 4081 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) |
674 | 673 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) |
675 | 582 | fvmpt2 6200 |
. . . . . . . . . . . . . . 15
⊢ ((𝑖 ∈ ℕ ∧ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) ∈ V) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
676 | 652, 674,
675 | syl2anc 691 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0))) |
677 | | biid 250 |
. . . . . . . . . . . . . . . 16
⊢ (𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))) |
678 | 677, 665 | ifbieq2i 4060 |
. . . . . . . . . . . . . . 15
⊢ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹) |
679 | 678 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)) |
680 | 676, 679 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖) = if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹)) |
681 | 669, 680 | oveq12d 6567 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))) |
682 | 656, 681 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖) = (if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), 𝐹)(𝐿‘𝑌)if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), 𝐹))) |
683 | 651, 682 | eqtr4d 2647 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ ℕ) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)) |
684 | 624, 626,
683 | syl2anc 691 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)) |
685 | 684 | 3ad2antl1 1216 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) ∧ 𝑖 ∈ (1...𝑚)) → (𝑃‘𝑖) = ((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)) |
686 | 685 | sumeq2dv 14281 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖) = Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖)) |
687 | 686 | oveq2d 6565 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)(𝑃‘𝑖)) = ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))) |
688 | 623, 687 | breqtrd 4609 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → 𝐺 ≤ ((1 + 𝐸) · Σ𝑖 ∈ (1...𝑚)((𝑖 ∈ ℕ ↦ (((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐶‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)(𝐿‘𝑌)((𝑖 ∈ ℕ ↦ if(𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)), ((𝐷‘𝑖) ↾ 𝑌), (𝑦 ∈ 𝑌 ↦ 0)))‘𝑖)))‘𝑖))) |
689 | | fveq2 6103 |
. . . . . . . 8
⊢ (𝑗 = ℎ → (𝐷‘𝑗) = (𝐷‘ℎ)) |
690 | 689 | fveq1d 6105 |
. . . . . . 7
⊢ (𝑗 = ℎ → ((𝐷‘𝑗)‘𝑍) = ((𝐷‘ℎ)‘𝑍)) |
691 | 690 | cbvmptv 4678 |
. . . . . 6
⊢ (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍)) |
692 | 691 | rneqi 5273 |
. . . . 5
⊢ ran
(𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (ℎ ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘ℎ)‘𝑍)) |
693 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑖 → (𝐶‘ℎ) = (𝐶‘𝑖)) |
694 | 693 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑖 → ((𝐶‘ℎ)‘𝑍) = ((𝐶‘𝑖)‘𝑍)) |
695 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (ℎ = 𝑖 → (𝐷‘ℎ) = (𝐷‘𝑖)) |
696 | 695 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (ℎ = 𝑖 → ((𝐷‘ℎ)‘𝑍) = ((𝐷‘𝑖)‘𝑍)) |
697 | 694, 696 | oveq12d 6567 |
. . . . . . . . . 10
⊢ (ℎ = 𝑖 → (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) = (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))) |
698 | 697 | eleq2d 2673 |
. . . . . . . . 9
⊢ (ℎ = 𝑖 → (𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍)) ↔ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍)))) |
699 | 698 | cbvrabv 3172 |
. . . . . . . 8
⊢ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} = {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} |
700 | 699 | mpteq1i 4667 |
. . . . . . 7
⊢ (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) |
701 | 700 | rneqi 5273 |
. . . . . 6
⊢ ran
(𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) = ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍)) |
702 | 701 | uneq2i 3726 |
. . . . 5
⊢ ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))) = ({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {𝑖 ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘𝑖)‘𝑍)[,)((𝐷‘𝑖)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))) |
703 | | eqid 2610 |
. . . . 5
⊢
inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < ) = inf(({(𝐵‘𝑍)} ∪ ran (𝑗 ∈ {ℎ ∈ (1...𝑚) ∣ 𝑆 ∈ (((𝐶‘ℎ)‘𝑍)[,)((𝐷‘ℎ)‘𝑍))} ↦ ((𝐷‘𝑗)‘𝑍))), ℝ, < ) |
704 | 18, 573, 574, 575, 45, 576, 577, 578, 579, 580, 581, 582, 590, 600, 5, 601, 612, 613, 614, 615, 616, 688, 692, 702, 703 | hoidmvlelem2 39486 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ ∧ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗))) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |
705 | 704 | 3exp 1256 |
. . 3
⊢ (𝜑 → (𝑚 ∈ ℕ → (𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢))) |
706 | 705 | rexlimdv 3012 |
. 2
⊢ (𝜑 → (∃𝑚 ∈ ℕ 𝐺 ≤ ((1 + 𝐸) · Σ𝑗 ∈ (1...𝑚)(𝑃‘𝑗)) → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢)) |
707 | 572, 706 | mpd 15 |
1
⊢ (𝜑 → ∃𝑢 ∈ 𝑈 𝑆 < 𝑢) |