Step | Hyp | Ref
| Expression |
1 | | hspmbllem2.i |
. . 3
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ) |
2 | | hspmbllem2.f |
. . 3
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∈ ℝ) |
3 | 1, 2 | readdcld 9948 |
. 2
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ∈ ℝ) |
4 | | hspmbllem2.r |
. . . 4
⊢ (𝜑 → ((voln*‘𝑋)‘𝐴) ∈ ℝ) |
5 | | hspmbllem2.e |
. . . . 5
⊢ (𝜑 → 𝐸 ∈
ℝ+) |
6 | 5 | rpred 11748 |
. . . 4
⊢ (𝜑 → 𝐸 ∈ ℝ) |
7 | 4, 6 | readdcld 9948 |
. . 3
⊢ (𝜑 → (((voln*‘𝑋)‘𝐴) + 𝐸) ∈ ℝ) |
8 | | nfv 1830 |
. . . 4
⊢
Ⅎ𝑗𝜑 |
9 | | nnex 10903 |
. . . . 5
⊢ ℕ
∈ V |
10 | 9 | a1i 11 |
. . . 4
⊢ (𝜑 → ℕ ∈
V) |
11 | | icossicc 12131 |
. . . . 5
⊢
(0[,)+∞) ⊆ (0[,]+∞) |
12 | | hspmbllem2.l |
. . . . . 6
⊢ 𝐿 = (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
13 | | hspmbllem2.x |
. . . . . . 7
⊢ (𝜑 → 𝑋 ∈ Fin) |
14 | 13 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ∈ Fin) |
15 | | hspmbllem2.c |
. . . . . . . 8
⊢ (𝜑 → 𝐶:ℕ⟶(ℝ
↑𝑚 𝑋)) |
16 | 15 | ffvelrnda 6267 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗) ∈ (ℝ ↑𝑚
𝑋)) |
17 | | elmapi 7765 |
. . . . . . 7
⊢ ((𝐶‘𝑗) ∈ (ℝ ↑𝑚
𝑋) → (𝐶‘𝑗):𝑋⟶ℝ) |
18 | 16, 17 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):𝑋⟶ℝ) |
19 | | hspmbllem2.d |
. . . . . . . 8
⊢ (𝜑 → 𝐷:ℕ⟶(ℝ
↑𝑚 𝑋)) |
20 | 19 | ffvelrnda 6267 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗) ∈ (ℝ ↑𝑚
𝑋)) |
21 | | elmapi 7765 |
. . . . . . 7
⊢ ((𝐷‘𝑗) ∈ (ℝ ↑𝑚
𝑋) → (𝐷‘𝑗):𝑋⟶ℝ) |
22 | 20, 21 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ) |
23 | 12, 14, 18, 22 | hoidmvcl 39472 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞)) |
24 | 11, 23 | sseldi 3566 |
. . . 4
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞)) |
25 | 8, 10, 24 | sge0clmpt 39318 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ (0[,]+∞)) |
26 | | hspmbllem2.k |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ 𝑋) |
27 | | ne0i 3880 |
. . . . . . . . 9
⊢ (𝐾 ∈ 𝑋 → 𝑋 ≠ ∅) |
28 | 26, 27 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑋 ≠ ∅) |
29 | 28 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑋 ≠ ∅) |
30 | 12, 14, 29, 18, 22 | hoidmvn0val 39474 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
31 | 30 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) = (𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))) |
32 | 31 | fveq2d 6107 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))))) |
33 | | hspmbllem2.g |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ∏𝑘 ∈ 𝑋 (vol‘(((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸)) |
34 | 32, 33 | eqbrtrd 4605 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸)) |
35 | 7, 25, 34 | ge0lere 38606 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ) |
36 | | hspmbllem2.t |
. . . . . . . . 9
⊢ 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) |
37 | | hspmbllem2.y |
. . . . . . . . . 10
⊢ (𝜑 → 𝑌 ∈ ℝ) |
38 | 37 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝑌 ∈ ℝ) |
39 | 36, 38, 14, 22 | hsphoif 39466 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑇‘𝑌)‘(𝐷‘𝑗)):𝑋⟶ℝ) |
40 | 12, 14, 18, 39 | hoidmvcl 39472 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) ∈ (0[,)+∞)) |
41 | 11, 40 | sseldi 3566 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) ∈ (0[,]+∞)) |
42 | 8, 10, 41 | sge0clmpt 39318 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) ∈ (0[,]+∞)) |
43 | | oveq2 6557 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → (ℝ ↑𝑚
𝑥) = (ℝ
↑𝑚 𝑦)) |
44 | | eqeq1 2614 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → (𝑥 = ∅ ↔ 𝑦 = ∅)) |
45 | | prodeq1 14478 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑦 → ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))) = ∏𝑘 ∈ 𝑦 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) |
46 | 44, 45 | ifbieq2d 4061 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑦 → if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))) = if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) |
47 | 43, 43, 46 | mpt2eq123dv 6615 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑦 → (𝑎 ∈ (ℝ ↑𝑚
𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘))))) = (𝑎 ∈ (ℝ ↑𝑚
𝑦), 𝑏 ∈ (ℝ ↑𝑚
𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
48 | 47 | cbvmptv 4678 |
. . . . . . . . 9
⊢ (𝑥 ∈ Fin ↦ (𝑎 ∈ (ℝ
↑𝑚 𝑥), 𝑏 ∈ (ℝ ↑𝑚
𝑥) ↦ if(𝑥 = ∅, 0, ∏𝑘 ∈ 𝑥 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) = (𝑦 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑦), 𝑏 ∈ (ℝ ↑𝑚
𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
49 | 12, 48 | eqtri 2632 |
. . . . . . . 8
⊢ 𝐿 = (𝑦 ∈ Fin ↦ (𝑎 ∈ (ℝ ↑𝑚
𝑦), 𝑏 ∈ (ℝ ↑𝑚
𝑦) ↦ if(𝑦 = ∅, 0, ∏𝑘 ∈ 𝑦 (vol‘((𝑎‘𝑘)[,)(𝑏‘𝑘)))))) |
50 | | diffi 8077 |
. . . . . . . . . . 11
⊢ (𝑋 ∈ Fin → (𝑋 ∖ {𝐾}) ∈ Fin) |
51 | 13, 50 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → (𝑋 ∖ {𝐾}) ∈ Fin) |
52 | | snfi 7923 |
. . . . . . . . . . 11
⊢ {𝐾} ∈ Fin |
53 | 52 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → {𝐾} ∈ Fin) |
54 | | unfi 8112 |
. . . . . . . . . 10
⊢ (((𝑋 ∖ {𝐾}) ∈ Fin ∧ {𝐾} ∈ Fin) → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ∈ Fin) |
55 | 51, 53, 54 | syl2anc 691 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ∈ Fin) |
56 | 55 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ∈ Fin) |
57 | | snidg 4153 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ 𝑋 → 𝐾 ∈ {𝐾}) |
58 | 26, 57 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐾 ∈ {𝐾}) |
59 | | elun2 3743 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ {𝐾} → 𝐾 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})) |
60 | 58, 59 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐾 ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾})) |
61 | | neldifsnd 4263 |
. . . . . . . . . 10
⊢ (𝜑 → ¬ 𝐾 ∈ (𝑋 ∖ {𝐾})) |
62 | 60, 61 | eldifd 3551 |
. . . . . . . . 9
⊢ (𝜑 → 𝐾 ∈ (((𝑋 ∖ {𝐾}) ∪ {𝐾}) ∖ (𝑋 ∖ {𝐾}))) |
63 | 62 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐾 ∈ (((𝑋 ∖ {𝐾}) ∪ {𝐾}) ∖ (𝑋 ∖ {𝐾}))) |
64 | | eqid 2610 |
. . . . . . . 8
⊢ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = ((𝑋 ∖ {𝐾}) ∪ {𝐾}) |
65 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ
↑𝑚 ((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) |
66 | | uncom 3719 |
. . . . . . . . . . . . 13
⊢ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = ({𝐾} ∪ (𝑋 ∖ {𝐾})) |
67 | 66 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = ({𝐾} ∪ (𝑋 ∖ {𝐾}))) |
68 | 26 | snssd 4281 |
. . . . . . . . . . . . 13
⊢ (𝜑 → {𝐾} ⊆ 𝑋) |
69 | | undif 4001 |
. . . . . . . . . . . . 13
⊢ ({𝐾} ⊆ 𝑋 ↔ ({𝐾} ∪ (𝑋 ∖ {𝐾})) = 𝑋) |
70 | 68, 69 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝜑 → ({𝐾} ∪ (𝑋 ∖ {𝐾})) = 𝑋) |
71 | 67, 70 | eqtrd 2644 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = 𝑋) |
72 | 71 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑋 ∖ {𝐾}) ∪ {𝐾}) = 𝑋) |
73 | 72 | feq2d 5944 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗):((𝑋 ∖ {𝐾}) ∪ {𝐾})⟶ℝ ↔ (𝐶‘𝑗):𝑋⟶ℝ)) |
74 | 18, 73 | mpbird 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐶‘𝑗):((𝑋 ∖ {𝐾}) ∪ {𝐾})⟶ℝ) |
75 | 72 | feq2d 5944 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐷‘𝑗):((𝑋 ∖ {𝐾}) ∪ {𝐾})⟶ℝ ↔ (𝐷‘𝑗):𝑋⟶ℝ)) |
76 | 22, 75 | mpbird 246 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):((𝑋 ∖ {𝐾}) ∪ {𝐾})⟶ℝ) |
77 | 49, 56, 63, 64, 38, 65, 74, 76 | hsphoidmvle 39476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(𝐷‘𝑗))) |
78 | 71 | fveq2d 6107 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾})) = (𝐿‘𝑋)) |
79 | | eqidd 2611 |
. . . . . . . . . 10
⊢ (𝜑 → (𝐶‘𝑗) = (𝐶‘𝑗)) |
80 | 36 | a1i 11 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑇 = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))) |
81 | 71 | oveq2d 6565 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ℝ
↑𝑚 ((𝑋 ∖ {𝐾}) ∪ {𝐾})) = (ℝ ↑𝑚
𝑋)) |
82 | 71 | mpteq1d 4666 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))) |
83 | 81, 82 | mpteq12dv 4663 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))) = (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) |
84 | 83 | eqcomd 2616 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))) = (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) |
85 | 84 | mpteq2dv 4673 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) = (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))) |
86 | 80, 85 | eqtr2d 2645 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))))) = 𝑇) |
87 | 86 | fveq1d 6105 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌) = (𝑇‘𝑌)) |
88 | 87 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝜑 → (((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌)‘(𝐷‘𝑗)) = ((𝑇‘𝑌)‘(𝐷‘𝑗))) |
89 | 78, 79, 88 | oveq123d 6570 |
. . . . . . . . 9
⊢ (𝜑 → ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))) |
90 | 89 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌)‘(𝐷‘𝑗))) = ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))) |
91 | 78 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾})) = (𝐿‘𝑋)) |
92 | 91 | oveqd 6566 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(𝐷‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) |
93 | 90, 92 | breq12d 4596 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(((𝑦 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
((𝑋 ∖ {𝐾}) ∪ {𝐾})) ↦ (ℎ ∈ ((𝑋 ∖ {𝐾}) ∪ {𝐾}) ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))))‘𝑌)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘((𝑋 ∖ {𝐾}) ∪ {𝐾}))(𝐷‘𝑗)) ↔ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) |
94 | 77, 93 | mpbid 221 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) ≤ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) |
95 | 8, 10, 41, 24, 94 | sge0lempt 39303 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
96 | 35, 42, 95 | ge0lere 38606 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) ∈ ℝ) |
97 | | hspmbllem2.s |
. . . . . . . . . 10
⊢ 𝑆 = (𝑥 ∈ ℝ ↦ (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ = 𝐾, if(𝑥 ≤ (𝑐‘ℎ), (𝑐‘ℎ), 𝑥), (𝑐‘ℎ))))) |
98 | 97, 38, 14, 18 | hoidifhspf 39508 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑆‘𝑌)‘(𝐶‘𝑗)):𝑋⟶ℝ) |
99 | 12, 14, 98, 22 | hoidmvcl 39472 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,)+∞)) |
100 | | eqid 2610 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))) = (𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))) |
101 | 99, 100 | fmptd 6292 |
. . . . . . 7
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))):ℕ⟶(0[,)+∞)) |
102 | 11 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → (0[,)+∞) ⊆
(0[,]+∞)) |
103 | 101, 102 | fssd 5970 |
. . . . . 6
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))):ℕ⟶(0[,]+∞)) |
104 | 10, 103 | sge0cl 39274 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ (0[,]+∞)) |
105 | 11, 99 | sseldi 3566 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)) ∈ (0[,]+∞)) |
106 | 26 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → 𝐾 ∈ 𝑋) |
107 | 12, 14, 18, 22, 106, 97, 38 | hoidifhspdmvle 39510 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)) ≤ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) |
108 | 8, 10, 105, 24, 107 | sge0lempt 39303 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
109 | 35, 104, 108 | ge0lere 38606 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))) ∈ ℝ) |
110 | 37 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → 𝑌 ∈ ℝ) |
111 | 13 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → 𝑋 ∈ Fin) |
112 | | eleq1 2676 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → (𝑗 ∈ ℕ ↔ 𝑙 ∈ ℕ)) |
113 | 112 | anbi2d 736 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → ((𝜑 ∧ 𝑗 ∈ ℕ) ↔ (𝜑 ∧ 𝑙 ∈ ℕ))) |
114 | | fveq2 6103 |
. . . . . . . . . . . 12
⊢ (𝑗 = 𝑙 → (𝐷‘𝑗) = (𝐷‘𝑙)) |
115 | 114 | feq1d 5943 |
. . . . . . . . . . 11
⊢ (𝑗 = 𝑙 → ((𝐷‘𝑗):𝑋⟶ℝ ↔ (𝐷‘𝑙):𝑋⟶ℝ)) |
116 | 113, 115 | imbi12d 333 |
. . . . . . . . . 10
⊢ (𝑗 = 𝑙 → (((𝜑 ∧ 𝑗 ∈ ℕ) → (𝐷‘𝑗):𝑋⟶ℝ) ↔ ((𝜑 ∧ 𝑙 ∈ ℕ) → (𝐷‘𝑙):𝑋⟶ℝ))) |
117 | 116, 22 | chvarv 2251 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (𝐷‘𝑙):𝑋⟶ℝ) |
118 | 36, 110, 111, 117 | hsphoif 39466 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑇‘𝑌)‘(𝐷‘𝑙)):𝑋⟶ℝ) |
119 | | reex 9906 |
. . . . . . . . . . . 12
⊢ ℝ
∈ V |
120 | 119 | a1i 11 |
. . . . . . . . . . 11
⊢ (𝜑 → ℝ ∈
V) |
121 | 120, 13 | jca 553 |
. . . . . . . . . 10
⊢ (𝜑 → (ℝ ∈ V ∧
𝑋 ∈
Fin)) |
122 | 121 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (ℝ ∈ V
∧ 𝑋 ∈
Fin)) |
123 | | elmapg 7757 |
. . . . . . . . 9
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → (((𝑇‘𝑌)‘(𝐷‘𝑙)) ∈ (ℝ ↑𝑚
𝑋) ↔ ((𝑇‘𝑌)‘(𝐷‘𝑙)):𝑋⟶ℝ)) |
124 | 122, 123 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (((𝑇‘𝑌)‘(𝐷‘𝑙)) ∈ (ℝ ↑𝑚
𝑋) ↔ ((𝑇‘𝑌)‘(𝐷‘𝑙)):𝑋⟶ℝ)) |
125 | 118, 124 | mpbird 246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑇‘𝑌)‘(𝐷‘𝑙)) ∈ (ℝ ↑𝑚
𝑋)) |
126 | | eqid 2610 |
. . . . . . 7
⊢ (𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙))) = (𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙))) |
127 | 125, 126 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙))):ℕ⟶(ℝ
↑𝑚 𝑋)) |
128 | | simpl 472 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → 𝜑) |
129 | | elinel1 3761 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) → 𝑓 ∈ 𝐴) |
130 | 129 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 ∈ 𝐴) |
131 | | hspmbllem2.a |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐴 ⊆ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
132 | 131 | sselda 3568 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
133 | | eliun 4460 |
. . . . . . . . . . . . 13
⊢ (𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
134 | 132, 133 | sylib 207 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
135 | 128, 130,
134 | syl2anc 691 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
136 | 128 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) → 𝜑) |
137 | | elinel2 3762 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) → 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
138 | 137 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
139 | 138 | adantr 480 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) → 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
140 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
141 | | ixpfn 7800 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑓 Fn 𝑋) |
142 | 141 | adantl 481 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 Fn 𝑋) |
143 | | nfv 1830 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) |
144 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘𝑓 |
145 | | nfixp1 7814 |
. . . . . . . . . . . . . . . . . . 19
⊢
Ⅎ𝑘X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) |
146 | 144, 145 | nfel 2763 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) |
147 | 143, 146 | nfan 1816 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
148 | 18 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (𝐶‘𝑗):𝑋⟶ℝ) |
149 | | simp3 1056 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
150 | 148, 149 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ) |
151 | 150 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈
ℝ*) |
152 | 151 | ad5ant135 1306 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈
ℝ*) |
153 | 39 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑌)‘(𝐷‘𝑗)):𝑋⟶ℝ) |
154 | 153, 149 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) ∈ ℝ) |
155 | 154 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) ∈
ℝ*) |
156 | 155 | ad5ant135 1306 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) ∈
ℝ*) |
157 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌)) |
158 | | ioossre 12106 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(-∞(,)𝑌)
⊆ ℝ |
159 | 158 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑘 = 𝐾 → (-∞(,)𝑌) ⊆ ℝ) |
160 | 157, 159 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ⊆
ℝ) |
161 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑘 = 𝐾 → if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = ℝ) |
162 | | ssid 3587 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ℝ
⊆ ℝ |
163 | 162 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (¬
𝑘 = 𝐾 → ℝ ⊆
ℝ) |
164 | 161, 163 | eqsstrd 3602 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑘 = 𝐾 → if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ⊆
ℝ) |
165 | 160, 164 | pm2.61i 175 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ⊆ ℝ |
166 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) → 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
167 | | hspmbllem2.h |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ 𝐻 = (𝑥 ∈ Fin ↦ (𝑙 ∈ 𝑥, 𝑦 ∈ ℝ ↦ X𝑘 ∈
𝑥 if(𝑘 = 𝑙, (-∞(,)𝑦), ℝ))) |
168 | 167, 13, 26, 37 | hspval 39499 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝜑 → (𝐾(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
169 | 168 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) → (𝐾(𝐻‘𝑋)𝑌) = X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
170 | 166, 169 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) → 𝑓 ∈ X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
171 | 170 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋) → 𝑓 ∈ X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
172 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
173 | | vex 3176 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ 𝑓 ∈ V |
174 | 173 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ))) |
175 | 174 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ))) |
176 | 175 | simprd 478 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
177 | 176 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ∧ 𝑘 ∈ 𝑋) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
178 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
179 | | rspa 2914 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((∀𝑘 ∈
𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
180 | 177, 178,
179 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
181 | 171, 172,
180 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
182 | 165, 181 | sseldi 3566 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ ℝ) |
183 | 182 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈
ℝ*) |
184 | 183 | ad4ant14 1285 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈
ℝ*) |
185 | 151 | ad4ant124 1287 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈
ℝ*) |
186 | 22 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (𝐷‘𝑗):𝑋⟶ℝ) |
187 | 186, 149 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈ ℝ) |
188 | 187 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈
ℝ*) |
189 | 188 | ad4ant124 1287 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈
ℝ*) |
190 | 173 | elixp 7801 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
191 | 190 | biimpi 205 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
192 | 191 | simprd 478 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
193 | 192 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑋) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
194 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
195 | | rspa 2914 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((∀𝑘 ∈
𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
196 | 193, 194,
195 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
197 | 196 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
198 | | icogelb 12096 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ* ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ* ∧ (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
199 | 185, 189,
197, 198 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
200 | 199 | ad5ant1345 1308 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
201 | | icoltub 38579 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ* ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ* ∧ (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
202 | 185, 189,
197, 201 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
203 | 202 | ad5ant1345 1308 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
204 | 203 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
205 | | simpll 786 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) → 𝜑) |
206 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) → 𝑗 ∈ ℕ) |
207 | 205, 206 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) → (𝜑 ∧ 𝑗 ∈ ℕ)) |
208 | 207 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝜑 ∧ 𝑗 ∈ ℕ)) |
209 | | simp2 1055 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → 𝑘 = 𝐾) |
210 | | simp3 1056 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) |
211 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑘 = 𝐾 → ((𝐷‘𝑗)‘𝑘) = ((𝐷‘𝑗)‘𝐾)) |
212 | 211 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝐾 → (((𝐷‘𝑗)‘𝑘) ≤ 𝑌 ↔ ((𝐷‘𝑗)‘𝐾) ≤ 𝑌)) |
213 | 212 | biimpa 500 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝐾) ≤ 𝑌) |
214 | 213 | iftrued 4044 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = ((𝐷‘𝑗)‘𝐾)) |
215 | 211 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑘 = 𝐾 → ((𝐷‘𝑗)‘𝐾) = ((𝐷‘𝑗)‘𝑘)) |
216 | 215 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝐾) = ((𝐷‘𝑗)‘𝑘)) |
217 | 214, 216 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = ((𝐷‘𝑗)‘𝑘)) |
218 | 217 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = ((𝐷‘𝑗)‘𝑘)) |
219 | | breq2 4587 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑦 = 𝑌 → ((𝑐‘ℎ) ≤ 𝑦 ↔ (𝑐‘ℎ) ≤ 𝑌)) |
220 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑦 = 𝑌 → 𝑦 = 𝑌) |
221 | 219, 220 | ifbieq2d 4061 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑦 = 𝑌 → if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦) = if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌)) |
222 | 221 | ifeq2d 4055 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑦 = 𝑌 → if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)) = if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))) |
223 | 222 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑦 = 𝑌 → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦))) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌)))) |
224 | 223 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑦 = 𝑌 → (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))) = (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))))) |
225 | 224 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ ((𝜑 ∧ 𝑦 = 𝑌) → (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑦, (𝑐‘ℎ), 𝑦)))) = (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))))) |
226 | | ovex 6577 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (ℝ
↑𝑚 𝑋) ∈ V |
227 | 226 | mptex 6390 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 ∈ (ℝ
↑𝑚 𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌)))) ∈ V |
228 | 227 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌)))) ∈ V) |
229 | 80, 225, 37, 228 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝜑 → (𝑇‘𝑌) = (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))))) |
230 | 229 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝑇‘𝑌) = (𝑐 ∈ (ℝ ↑𝑚
𝑋) ↦ (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))))) |
231 | | fveq1 6102 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = (𝐷‘𝑗) → (𝑐‘ℎ) = ((𝐷‘𝑗)‘ℎ)) |
232 | 231 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = (𝐷‘𝑗) → ((𝑐‘ℎ) ≤ 𝑌 ↔ ((𝐷‘𝑗)‘ℎ) ≤ 𝑌)) |
233 | 232, 231 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = (𝐷‘𝑗) → if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌) = if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)) |
234 | 231, 233 | ifeq12d 4056 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑐 = (𝐷‘𝑗) → if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌)) = if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌))) |
235 | 234 | mpteq2dv 4673 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑐 = (𝐷‘𝑗) → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))) |
236 | 235 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑐 = (𝐷‘𝑗)) → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), (𝑐‘ℎ), if((𝑐‘ℎ) ≤ 𝑌, (𝑐‘ℎ), 𝑌))) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))) |
237 | | mptexg 6389 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑋 ∈ Fin → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌))) ∈ V) |
238 | 13, 237 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝜑 → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌))) ∈ V) |
239 | 238 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌))) ∈ V) |
240 | 230, 236,
20, 239 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝑇‘𝑌)‘(𝐷‘𝑗)) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))) |
241 | 240 | fveq1d 6105 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘)) |
242 | 241 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 = 𝐾) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘)) |
243 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → 𝜑) |
244 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → 𝑘 = 𝐾) |
245 | 243, 26 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → 𝐾 ∈ 𝑋) |
246 | 244, 245 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → 𝑘 ∈ 𝑋) |
247 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌))) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))) |
248 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (ℎ = 𝑘 → (ℎ ∈ (𝑋 ∖ {𝐾}) ↔ 𝑘 ∈ (𝑋 ∖ {𝐾}))) |
249 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (ℎ = 𝑘 → ((𝐷‘𝑗)‘ℎ) = ((𝐷‘𝑗)‘𝑘)) |
250 | 249 | breq1d 4593 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (ℎ = 𝑘 → (((𝐷‘𝑗)‘ℎ) ≤ 𝑌 ↔ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌)) |
251 | 250, 249 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (ℎ = 𝑘 → if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌) = if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) |
252 | 248, 249,
251 | ifbieq12d 4063 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (ℎ = 𝑘 → if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
253 | 252 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ ℎ = 𝑘) → if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
254 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → 𝑘 ∈ 𝑋) |
255 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝐷‘𝑗)‘𝑘) ∈ V |
256 | 255 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → ((𝐷‘𝑗)‘𝑘) ∈ V) |
257 | | ifexg 4107 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((((𝐷‘𝑗)‘𝑘) ∈ V ∧ 𝑌 ∈ ℝ) → if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌) ∈ V) |
258 | 256, 37, 257 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝜑 → if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌) ∈ V) |
259 | | ifexg 4107 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝐷‘𝑗)‘𝑘) ∈ V ∧ if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌) ∈ V) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) ∈ V) |
260 | 256, 258,
259 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝜑 → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) ∈ V) |
261 | 260 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) ∈ V) |
262 | 247, 253,
254, 261 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
263 | 243, 246,
262 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
264 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝐾 → (𝑘 ∈ (𝑋 ∖ {𝐾}) ↔ 𝐾 ∈ (𝑋 ∖ {𝐾}))) |
265 | 212, 211 | ifbieq1d 4059 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝐾 → if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌) = if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) |
266 | 264, 211,
265 | ifbieq12d 4063 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (𝑘 = 𝐾 → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) = if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌))) |
267 | 266 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) = if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌))) |
268 | 263, 267 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ 𝑘 = 𝐾) → ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘) = if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌))) |
269 | 268 | 3adant2 1073 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 = 𝐾) → ((ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))‘𝑘) = if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌))) |
270 | | neldifsnd 4263 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑘 = 𝐾 → ¬ 𝐾 ∈ (𝑋 ∖ {𝐾})) |
271 | 270 | iffalsed 4047 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝐾 → if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) = if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) |
272 | 271 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 = 𝐾) → if(𝐾 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝐾), if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) = if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) |
273 | 242, 269,
272 | 3eqtrrd 2649 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 = 𝐾) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
274 | 273 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
275 | 274 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
276 | 218, 275 | eqtr3d 2646 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝑘) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
277 | 208, 209,
210, 276 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 = 𝐾 ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝑘) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
278 | 277 | ad5ant145 1307 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ((𝐷‘𝑗)‘𝑘) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
279 | 204, 278 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
280 | | mnfxr 9975 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ -∞
∈ ℝ* |
281 | 280 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → -∞ ∈
ℝ*) |
282 | 37 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → 𝑌 ∈
ℝ*) |
283 | 282 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) → 𝑌 ∈
ℝ*) |
284 | 283 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → 𝑌 ∈
ℝ*) |
285 | 181 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
286 | 157 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = (-∞(,)𝑌)) |
287 | 285, 286 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ (-∞(,)𝑌)) |
288 | | iooltub 38582 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
((-∞ ∈ ℝ* ∧ 𝑌 ∈ ℝ* ∧ (𝑓‘𝑘) ∈ (-∞(,)𝑌)) → (𝑓‘𝑘) < 𝑌) |
289 | 281, 284,
287, 288 | syl3anc 1318 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) < 𝑌) |
290 | 289 | 3adant1r 1311 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) < 𝑌) |
291 | 290 | ad4ant123 1286 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝑓‘𝑘) < 𝑌) |
292 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) |
293 | 212 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑘 = 𝐾 → (¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌 ↔ ¬ ((𝐷‘𝑗)‘𝐾) ≤ 𝑌)) |
294 | 293 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌 ↔ ¬ ((𝐷‘𝑗)‘𝐾) ≤ 𝑌)) |
295 | 292, 294 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → ¬ ((𝐷‘𝑗)‘𝐾) ≤ 𝑌) |
296 | 295 | iffalsed 4047 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = 𝑌) |
297 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → 𝑌 = 𝑌) |
298 | 296, 297 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑘 = 𝐾 ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → 𝑌 = if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) |
299 | 298 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → 𝑌 = if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌)) |
300 | 274 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
301 | 300 | adantlllr 38222 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
302 | 301 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → if(((𝐷‘𝑗)‘𝐾) ≤ 𝑌, ((𝐷‘𝑗)‘𝐾), 𝑌) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
303 | 299, 302 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → 𝑌 = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
304 | 291, 303 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
305 | 304 | ad5ant1345 1308 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ ((𝐷‘𝑗)‘𝑘) ≤ 𝑌) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
306 | 279, 305 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
307 | 203 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
308 | 240 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝑇‘𝑌)‘(𝐷‘𝑗)) = (ℎ ∈ 𝑋 ↦ if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)))) |
309 | 252 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) ∧ ℎ = 𝑘) → if(ℎ ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘ℎ), if(((𝐷‘𝑗)‘ℎ) ≤ 𝑌, ((𝐷‘𝑗)‘ℎ), 𝑌)) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
310 | 261 | 3adant2 1073 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) ∈ V) |
311 | 308, 309,
149, 310 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
312 | 311 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
313 | 312 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
314 | 313 | ad4ant13 1284 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘) = if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌))) |
315 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑘 ∈ 𝑋 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ 𝑋) |
316 | | neqne 2790 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (¬
𝑘 = 𝐾 → 𝑘 ≠ 𝐾) |
317 | | nelsn 4159 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑘 ≠ 𝐾 → ¬ 𝑘 ∈ {𝐾}) |
318 | 316, 317 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (¬
𝑘 = 𝐾 → ¬ 𝑘 ∈ {𝐾}) |
319 | 318 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑘 ∈ 𝑋 ∧ ¬ 𝑘 = 𝐾) → ¬ 𝑘 ∈ {𝐾}) |
320 | 315, 319 | eldifd 3551 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑘 ∈ 𝑋 ∧ ¬ 𝑘 = 𝐾) → 𝑘 ∈ (𝑋 ∖ {𝐾})) |
321 | 320 | iftrued 4044 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑘 ∈ 𝑋 ∧ ¬ 𝑘 = 𝐾) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) = ((𝐷‘𝑗)‘𝑘)) |
322 | 321 | adantll 746 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → if(𝑘 ∈ (𝑋 ∖ {𝐾}), ((𝐷‘𝑗)‘𝑘), if(((𝐷‘𝑗)‘𝑘) ≤ 𝑌, ((𝐷‘𝑗)‘𝑘), 𝑌)) = ((𝐷‘𝑗)‘𝑘)) |
323 | 314, 322 | eqtr2d 2645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → ((𝐷‘𝑗)‘𝑘) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
324 | 307, 323 | breqtrd 4609 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
325 | 306, 324 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) < (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
326 | 152, 156,
184, 200, 325 | elicod 12095 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
327 | 326 | ex 449 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑋 → (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
328 | 147, 327 | ralrimi 2940 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
329 | 142, 328 | jca 553 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
330 | 173 | elixp 7801 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
331 | 329, 330 | sylibr 223 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
332 | 331 | ex 449 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) ∧ 𝑗 ∈ ℕ) → (𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
333 | 136, 139,
140, 332 | syl21anc 1317 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) → (𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
334 | 333 | reximdva 3000 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → (∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)))) |
335 | 135, 334 | mpd 15 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
336 | | eliun 4460 |
. . . . . . . . . 10
⊢ (𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
337 | 335, 336 | sylibr 223 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
338 | 337 | ralrimiva 2949 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
339 | | dfss3 3558 |
. . . . . . . 8
⊢ ((𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) ↔ ∀𝑓 ∈ (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
340 | 338, 339 | sylibr 223 |
. . . . . . 7
⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
341 | | eqidd 2611 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → (𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙))) = (𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))) |
342 | | fveq2 6103 |
. . . . . . . . . . . . . 14
⊢ (𝑙 = 𝑗 → (𝐷‘𝑙) = (𝐷‘𝑗)) |
343 | 342 | fveq2d 6107 |
. . . . . . . . . . . . 13
⊢ (𝑙 = 𝑗 → ((𝑇‘𝑌)‘(𝐷‘𝑙)) = ((𝑇‘𝑌)‘(𝐷‘𝑗))) |
344 | 343 | adantl 481 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℕ ∧ 𝑙 = 𝑗) → ((𝑇‘𝑌)‘(𝐷‘𝑙)) = ((𝑇‘𝑌)‘(𝐷‘𝑗))) |
345 | | id 22 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℕ) |
346 | | fvex 6113 |
. . . . . . . . . . . . 13
⊢ ((𝑇‘𝑌)‘(𝐷‘𝑗)) ∈ V |
347 | 346 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → ((𝑇‘𝑌)‘(𝐷‘𝑗)) ∈ V) |
348 | 341, 344,
345, 347 | fvmptd 6197 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ ℕ → ((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗) = ((𝑇‘𝑌)‘(𝐷‘𝑗))) |
349 | 348 | fveq1d 6105 |
. . . . . . . . . 10
⊢ (𝑗 ∈ ℕ → (((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)‘𝑘) = (((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
350 | 349 | oveq2d 6565 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℕ → (((𝐶‘𝑗)‘𝑘)[,)(((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)‘𝑘)) = (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
351 | 350 | ixpeq2dv 7810 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ → X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘))) |
352 | 351 | iuneq2i 4475 |
. . . . . . 7
⊢ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)‘𝑘)) = ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑇‘𝑌)‘(𝐷‘𝑗))‘𝑘)) |
353 | 340, 352 | syl6sseqr 3615 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)(((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)‘𝑘))) |
354 | 13, 15, 127, 353, 12 | ovnlecvr2 39500 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗))))) |
355 | 348 | oveq2d 6565 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ → ((𝐶‘𝑗)(𝐿‘𝑋)((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)) = ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))) |
356 | 355 | mpteq2ia 4668 |
. . . . . . 7
⊢ (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗))) = (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))) |
357 | 356 | fveq2i 6106 |
. . . . . 6
⊢
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) |
358 | 357 | a1i 11 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑙 ∈ ℕ ↦ ((𝑇‘𝑌)‘(𝐷‘𝑙)))‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))))) |
359 | 354, 358 | breqtrd 4609 |
. . . 4
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗)))))) |
360 | 15 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (𝐶‘𝑙) ∈ (ℝ ↑𝑚
𝑋)) |
361 | | elmapi 7765 |
. . . . . . . . . 10
⊢ ((𝐶‘𝑙) ∈ (ℝ ↑𝑚
𝑋) → (𝐶‘𝑙):𝑋⟶ℝ) |
362 | 360, 361 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (𝐶‘𝑙):𝑋⟶ℝ) |
363 | 97, 110, 111, 362 | hoidifhspf 39508 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑆‘𝑌)‘(𝐶‘𝑙)):𝑋⟶ℝ) |
364 | | elmapg 7757 |
. . . . . . . . . 10
⊢ ((ℝ
∈ V ∧ 𝑋 ∈
Fin) → (((𝑆‘𝑌)‘(𝐶‘𝑙)) ∈ (ℝ ↑𝑚
𝑋) ↔ ((𝑆‘𝑌)‘(𝐶‘𝑙)):𝑋⟶ℝ)) |
365 | 121, 364 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑆‘𝑌)‘(𝐶‘𝑙)) ∈ (ℝ ↑𝑚
𝑋) ↔ ((𝑆‘𝑌)‘(𝐶‘𝑙)):𝑋⟶ℝ)) |
366 | 365 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → (((𝑆‘𝑌)‘(𝐶‘𝑙)) ∈ (ℝ ↑𝑚
𝑋) ↔ ((𝑆‘𝑌)‘(𝐶‘𝑙)):𝑋⟶ℝ)) |
367 | 363, 366 | mpbird 246 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑙 ∈ ℕ) → ((𝑆‘𝑌)‘(𝐶‘𝑙)) ∈ (ℝ ↑𝑚
𝑋)) |
368 | | eqid 2610 |
. . . . . . 7
⊢ (𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙))) = (𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙))) |
369 | 367, 368 | fmptd 6292 |
. . . . . 6
⊢ (𝜑 → (𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙))):ℕ⟶(ℝ
↑𝑚 𝑋)) |
370 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → 𝜑) |
371 | | eldifi 3694 |
. . . . . . . . . . . 12
⊢ (𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) → 𝑓 ∈ 𝐴) |
372 | 371 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 ∈ 𝐴) |
373 | 370, 372,
134 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
374 | 141 | adantl 481 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 Fn 𝑋) |
375 | | nfv 1830 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) |
376 | 375, 146 | nfan 1816 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘(((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
377 | 98 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → ((𝑆‘𝑌)‘(𝐶‘𝑗)):𝑋⟶ℝ) |
378 | 377, 149 | ffvelrnd 6268 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ∈ ℝ) |
379 | 378 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ∈
ℝ*) |
380 | 379 | ad5ant135 1306 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ∈
ℝ*) |
381 | 189 | adantl3r 782 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈
ℝ*) |
382 | 150 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ∈ ℝ) |
383 | 188 | 3expa 1257 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → ((𝐷‘𝑗)‘𝑘) ∈
ℝ*) |
384 | | icossre 12125 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝐶‘𝑗)‘𝑘) ∈ ℝ ∧ ((𝐷‘𝑗)‘𝑘) ∈ ℝ*) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ) |
385 | 382, 383,
384 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ) |
386 | 385 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ⊆ ℝ) |
387 | 386, 197 | sseldd 3569 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ ℝ) |
388 | 387 | rexrd 9968 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈
ℝ*) |
389 | 388 | ad5ant1345 1308 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈
ℝ*) |
390 | 38 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → 𝑌 ∈ ℝ) |
391 | 14 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → 𝑋 ∈ Fin) |
392 | 97, 390, 391, 148, 149 | hoidifhspval3 39509 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘))) |
393 | 392 | ad5ant134 1305 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘))) |
394 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑘 = 𝐾 → if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘)) = if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌)) |
395 | 394 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘)) = if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌)) |
396 | 393, 395 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌)) |
397 | 396 | adantllr 751 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌)) |
398 | | iftrue 4042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑌 ≤ ((𝐶‘𝑗)‘𝑘) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) = ((𝐶‘𝑗)‘𝑘)) |
399 | 398 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) = ((𝐶‘𝑗)‘𝑘)) |
400 | 199 | adantl3r 782 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
401 | 400 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
402 | 399, 401 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) ≤ (𝑓‘𝑘)) |
403 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑌 ≤ ((𝐶‘𝑗)‘𝑘) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) = 𝑌) |
404 | 403 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) = 𝑌) |
405 | | simpl1 1057 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → (𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) |
406 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑘 = 𝐾 ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → ¬ 𝑌 ≤ (𝑓‘𝑘)) |
407 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑘 = 𝐾 → (𝑓‘𝑘) = (𝑓‘𝐾)) |
408 | 407 | breq2d 4595 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝐾 → (𝑌 ≤ (𝑓‘𝑘) ↔ 𝑌 ≤ (𝑓‘𝐾))) |
409 | 408 | notbid 307 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (𝑘 = 𝐾 → (¬ 𝑌 ≤ (𝑓‘𝑘) ↔ ¬ 𝑌 ≤ (𝑓‘𝐾))) |
410 | 409 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ ((𝑘 = 𝐾 ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → (¬ 𝑌 ≤ (𝑓‘𝑘) ↔ ¬ 𝑌 ≤ (𝑓‘𝐾))) |
411 | 406, 410 | mpbid 221 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑘 = 𝐾 ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → ¬ 𝑌 ≤ (𝑓‘𝐾)) |
412 | 411 | 3ad2antl3 1218 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → ¬ 𝑌 ≤ (𝑓‘𝐾)) |
413 | 407 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝐾 → (𝑓‘𝐾) = (𝑓‘𝑘)) |
414 | 413 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝐾) = (𝑓‘𝑘)) |
415 | 373 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
416 | | id 22 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → (𝜑 ∧ 𝑗 ∈ ℕ)) |
417 | 416 | ad4ant13 1284 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝜑 ∧ 𝑗 ∈ ℕ)) |
418 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
419 | 254 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑘 ∈ 𝑋) |
420 | 417, 418,
419, 387 | syl21anc 1317 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑓‘𝑘) ∈ ℝ) |
421 | 420 | ex 449 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (((𝜑 ∧ 𝑘 ∈ 𝑋) ∧ 𝑗 ∈ ℕ) → (𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝑓‘𝑘) ∈ ℝ)) |
422 | 421 | rexlimdva 3013 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑋) → (∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝑓‘𝑘) ∈ ℝ)) |
423 | 422 | adantlr 747 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋) → (∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → (𝑓‘𝑘) ∈ ℝ)) |
424 | 415, 423 | mpd 15 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ ℝ) |
425 | 424 | 3adant3 1074 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ ℝ) |
426 | 414, 425 | eqeltrd 2688 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → (𝑓‘𝐾) ∈ ℝ) |
427 | 426 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → (𝑓‘𝐾) ∈ ℝ) |
428 | 405, 370,
37 | 3syl 18 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → 𝑌 ∈ ℝ) |
429 | 427, 428 | ltnled 10063 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → ((𝑓‘𝐾) < 𝑌 ↔ ¬ 𝑌 ≤ (𝑓‘𝐾))) |
430 | 412, 429 | mpbird 246 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → (𝑓‘𝐾) < 𝑌) |
431 | 374, 373 | r19.29a 3060 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 Fn 𝑋) |
432 | 431 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) → 𝑓 Fn 𝑋) |
433 | 280 | a1i 11 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → -∞ ∈
ℝ*) |
434 | 282 | ad4antr 764 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → 𝑌 ∈
ℝ*) |
435 | 424 | ad4ant13 1284 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ ℝ) |
436 | 435 | mnfltd 11834 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → -∞ < (𝑓‘𝑘)) |
437 | 407 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑓‘𝐾) < 𝑌 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) = (𝑓‘𝐾)) |
438 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (((𝑓‘𝐾) < 𝑌 ∧ 𝑘 = 𝐾) → (𝑓‘𝐾) < 𝑌) |
439 | 437, 438 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑓‘𝐾) < 𝑌 ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) < 𝑌) |
440 | 439 | ad4ant24 1290 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) < 𝑌) |
441 | 433, 434,
435, 436, 440 | eliood 38567 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ (-∞(,)𝑌)) |
442 | 157 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑘 = 𝐾 → (-∞(,)𝑌) = if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
443 | 442 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (-∞(,)𝑌) = if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
444 | 441, 443 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
445 | 424 | ad4ant13 1284 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ ℝ) |
446 | 161 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (¬
𝑘 = 𝐾 → ℝ = if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
447 | 446 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → ℝ = if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
448 | 445, 447 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
449 | 444, 448 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
450 | 449 | ralrimiva 2949 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
451 | 432, 450 | jca 553 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ (𝑓‘𝐾) < 𝑌) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ))) |
452 | 405, 430,
451 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ))) |
453 | 452, 174 | sylibr 223 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → 𝑓 ∈ X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ)) |
454 | 168 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝜑 → X𝑘 ∈
𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = (𝐾(𝐻‘𝑋)𝑌)) |
455 | 454 | ad2antrr 758 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = (𝐾(𝐻‘𝑋)𝑌)) |
456 | 455 | 3ad2antl1 1216 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → X𝑘 ∈ 𝑋 if(𝑘 = 𝐾, (-∞(,)𝑌), ℝ) = (𝐾(𝐻‘𝑋)𝑌)) |
457 | 453, 456 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
458 | | eldifn 3695 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) → ¬ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
459 | 458 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → ¬ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
460 | 459 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → ¬ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
461 | 460 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ (𝑓‘𝑘)) → ¬ 𝑓 ∈ (𝐾(𝐻‘𝑋)𝑌)) |
462 | 457, 461 | condan 831 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑘 ∈ 𝑋 ∧ 𝑘 = 𝐾) → 𝑌 ≤ (𝑓‘𝑘)) |
463 | 462 | ad5ant145 1307 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → 𝑌 ≤ (𝑓‘𝑘)) |
464 | 463 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → 𝑌 ≤ (𝑓‘𝑘)) |
465 | 404, 464 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) ∧ ¬ 𝑌 ≤ ((𝐶‘𝑗)‘𝑘)) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) ≤ (𝑓‘𝑘)) |
466 | 402, 465 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌) ≤ (𝑓‘𝑘)) |
467 | 397, 466 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ≤ (𝑓‘𝑘)) |
468 | 392 | ad5ant124 1303 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘))) |
469 | | iffalse 4045 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (¬
𝑘 = 𝐾 → if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘)) = ((𝐶‘𝑗)‘𝑘)) |
470 | 469 | adantl 481 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → if(𝑘 = 𝐾, if(𝑌 ≤ ((𝐶‘𝑗)‘𝑘), ((𝐶‘𝑗)‘𝑘), 𝑌), ((𝐶‘𝑗)‘𝑘)) = ((𝐶‘𝑗)‘𝑘)) |
471 | 468, 470 | eqtrd 2644 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) = ((𝐶‘𝑗)‘𝑘)) |
472 | 199 | adantr 480 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → ((𝐶‘𝑗)‘𝑘) ≤ (𝑓‘𝑘)) |
473 | 471, 472 | eqbrtrd 4605 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈
𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ≤ (𝑓‘𝑘)) |
474 | 473 | adantl4r 783 |
. . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) ∧ ¬ 𝑘 = 𝐾) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ≤ (𝑓‘𝑘)) |
475 | 467, 474 | pm2.61dan 828 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘) ≤ (𝑓‘𝑘)) |
476 | 202 | adantl3r 782 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) < ((𝐷‘𝑗)‘𝑘)) |
477 | 380, 381,
389, 475, 476 | elicod 12095 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) ∧ 𝑘 ∈ 𝑋) → (𝑓‘𝑘) ∈ ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
478 | 477 | ex 449 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑘 ∈ 𝑋 → (𝑓‘𝑘) ∈ ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
479 | 376, 478 | ralrimi 2940 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
480 | 374, 479 | jca 553 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
481 | 173 | elixp 7801 |
. . . . . . . . . . . . . 14
⊢ (𝑓 ∈ X𝑘 ∈
𝑋 ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ (𝑓 Fn 𝑋 ∧ ∀𝑘 ∈ 𝑋 (𝑓‘𝑘) ∈ ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
482 | 480, 481 | sylibr 223 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
483 | | eqidd 2611 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ ℕ → (𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙))) = (𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))) |
484 | | fveq2 6103 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑙 = 𝑗 → (𝐶‘𝑙) = (𝐶‘𝑗)) |
485 | 484 | fveq2d 6107 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑙 = 𝑗 → ((𝑆‘𝑌)‘(𝐶‘𝑙)) = ((𝑆‘𝑌)‘(𝐶‘𝑗))) |
486 | 485 | adantl 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 ∈ ℕ ∧ 𝑙 = 𝑗) → ((𝑆‘𝑌)‘(𝐶‘𝑙)) = ((𝑆‘𝑌)‘(𝐶‘𝑗))) |
487 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑆‘𝑌)‘(𝐶‘𝑗)) ∈ V |
488 | 487 | a1i 11 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑗 ∈ ℕ → ((𝑆‘𝑌)‘(𝐶‘𝑗)) ∈ V) |
489 | 483, 486,
345, 488 | fvmptd 6197 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑗 ∈ ℕ → ((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗) = ((𝑆‘𝑌)‘(𝐶‘𝑗))) |
490 | 489 | fveq1d 6105 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑗 ∈ ℕ → (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘) = (((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)) |
491 | 490 | oveq1d 6564 |
. . . . . . . . . . . . . . . 16
⊢ (𝑗 ∈ ℕ → ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
492 | 491 | ixpeq2dv 7810 |
. . . . . . . . . . . . . . 15
⊢ (𝑗 ∈ ℕ → X𝑘 ∈
𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
493 | 492 | ad2antlr 759 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) = X𝑘 ∈ 𝑋 ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
494 | 493 | eleq2d 2673 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → (𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑆‘𝑌)‘(𝐶‘𝑗))‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
495 | 482, 494 | mpbird 246 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) ∧ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) → 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
496 | 495 | ex 449 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ∧ 𝑗 ∈ ℕ) → (𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
497 | 496 | reximdva 3000 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → (∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 (((𝐶‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)))) |
498 | 373, 497 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
499 | | eliun 4460 |
. . . . . . . . 9
⊢ (𝑓 ∈ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∃𝑗 ∈ ℕ 𝑓 ∈ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
500 | 498, 499 | sylibr 223 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) → 𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
501 | 500 | ralrimiva 2949 |
. . . . . . 7
⊢ (𝜑 → ∀𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
502 | | dfss3 3558 |
. . . . . . 7
⊢ ((𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘)) ↔ ∀𝑓 ∈ (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))𝑓 ∈ ∪
𝑗 ∈ ℕ X𝑘 ∈
𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
503 | 501, 502 | sylibr 223 |
. . . . . 6
⊢ (𝜑 → (𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)) ⊆ ∪ 𝑗 ∈ ℕ X𝑘 ∈ 𝑋 ((((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)‘𝑘)[,)((𝐷‘𝑗)‘𝑘))) |
504 | 13, 369, 19, 503, 12 | ovnlecvr2 39500 |
. . . . 5
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
505 | 489 | oveq1d 6564 |
. . . . . . . 8
⊢ (𝑗 ∈ ℕ → (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))) |
506 | 505 | mpteq2ia 4668 |
. . . . . . 7
⊢ (𝑗 ∈ ℕ ↦ (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) = (𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))) |
507 | 506 | fveq2i 6106 |
. . . . . 6
⊢
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))) |
508 | 507 | a1i 11 |
. . . . 5
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑙 ∈ ℕ ↦ ((𝑆‘𝑌)‘(𝐶‘𝑙)))‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) |
509 | 504, 508 | breqtrd 4609 |
. . . 4
⊢ (𝜑 → ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) |
510 | 1, 2, 96, 109, 359, 509 | leadd12dd 38473 |
. . 3
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) +
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))))) |
511 | 14, 106, 38, 18, 22, 12, 36, 97 | hspmbllem1 39516 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ) → ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)) = (((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) +𝑒 (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))) |
512 | 511 | mpteq2dva 4672 |
. . . . 5
⊢ (𝜑 → (𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))) = (𝑗 ∈ ℕ ↦ (((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) +𝑒 (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) |
513 | 512 | fveq2d 6107 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗)))) =
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) +𝑒 (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))))) |
514 | 8, 10, 41, 105 | sge0xadd 39328 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))) +𝑒 (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) =
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) +𝑒
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))))) |
515 | 96, 109 | rexaddd 11939 |
. . . 4
⊢ (𝜑 →
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) +𝑒
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) =
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) +
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗)))))) |
516 | 513, 514,
515 | 3eqtrrd 2649 |
. . 3
⊢ (𝜑 →
((Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)((𝑇‘𝑌)‘(𝐷‘𝑗))))) +
(Σ^‘(𝑗 ∈ ℕ ↦ (((𝑆‘𝑌)‘(𝐶‘𝑗))(𝐿‘𝑋)(𝐷‘𝑗))))) =
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
517 | 510, 516 | breqtrd 4609 |
. 2
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤
(Σ^‘(𝑗 ∈ ℕ ↦ ((𝐶‘𝑗)(𝐿‘𝑋)(𝐷‘𝑗))))) |
518 | 3, 35, 7, 517, 34 | letrd 10073 |
1
⊢ (𝜑 → (((voln*‘𝑋)‘(𝐴 ∩ (𝐾(𝐻‘𝑋)𝑌))) + ((voln*‘𝑋)‘(𝐴 ∖ (𝐾(𝐻‘𝑋)𝑌)))) ≤ (((voln*‘𝑋)‘𝐴) + 𝐸)) |