Step | Hyp | Ref
| Expression |
1 | | nfv 1830 |
. . . 4
⊢
Ⅎ𝑘𝜑 |
2 | | meadjiunlem.g |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑋⟶𝑆) |
3 | | meadjiunlem.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
4 | 2, 3 | jca 553 |
. . . . 5
⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉)) |
5 | | fex 6394 |
. . . . 5
⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V) |
6 | | rnexg 6990 |
. . . . 5
⊢ (𝐺 ∈ V → ran 𝐺 ∈ V) |
7 | 4, 5, 6 | 3syl 18 |
. . . 4
⊢ (𝜑 → ran 𝐺 ∈ V) |
8 | | difssd 3700 |
. . . 4
⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺) |
9 | | meadjiunlem.f |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ Meas) |
10 | | meadjiunlem.3 |
. . . . . . 7
⊢ 𝑆 = dom 𝑀 |
11 | 9, 10 | meaf 39346 |
. . . . . 6
⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
12 | 11 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞)) |
13 | | frn 5966 |
. . . . . . . 8
⊢ (𝐺:𝑋⟶𝑆 → ran 𝐺 ⊆ 𝑆) |
14 | 2, 13 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ran 𝐺 ⊆ 𝑆) |
15 | 14 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆) |
16 | 8 | sselda 3568 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺) |
17 | 15, 16 | sseldd 3569 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆) |
18 | 12, 17 | ffvelrnd 6268 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞)) |
19 | | simpl 472 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑) |
20 | | id 22 |
. . . . . . . 8
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) |
21 | | dfin4 3826 |
. . . . . . . . 9
⊢ (ran
𝐺 ∩ {∅}) = (ran
𝐺 ∖ (ran 𝐺 ∖
{∅})) |
22 | 21 | eqcomi 2619 |
. . . . . . . 8
⊢ (ran
𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran
𝐺 ∩
{∅}) |
23 | 20, 22 | syl6eleq 2698 |
. . . . . . 7
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅})) |
24 | | elinel2 3762 |
. . . . . . . 8
⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅}) |
25 | | elsni 4142 |
. . . . . . . 8
⊢ (𝑘 ∈ {∅} → 𝑘 = ∅) |
26 | 24, 25 | syl 17 |
. . . . . . 7
⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 = ∅) |
27 | 23, 26 | syl 17 |
. . . . . 6
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 = ∅) |
28 | 27 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅) |
29 | | simpr 476 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅) |
30 | 29 | fveq2d 6107 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅)) |
31 | 9 | mea0 39347 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘∅) = 0) |
32 | 31 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0) |
33 | 30, 32 | eqtrd 2644 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = 0) |
34 | 19, 28, 33 | syl2anc 691 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → (𝑀‘𝑘) = 0) |
35 | 1, 7, 8, 18, 34 | sge0ss 39305 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))) |
36 | 35 | eqcomd 2616 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
37 | 11, 14 | feqresmpt 6160 |
. . 3
⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) |
38 | 37 | fveq2d 6107 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑀 ↾ ran 𝐺)) =
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))) |
39 | 2 | ffvelrnda 6267 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆) |
40 | 2 | feqmptd 6159 |
. . . . 5
⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗))) |
41 | 11 | feqmptd 6159 |
. . . . 5
⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘))) |
42 | | fveq2 6103 |
. . . . 5
⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗))) |
43 | 39, 40, 41, 42 | fmptco 6303 |
. . . 4
⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) |
44 | 43 | fveq2d 6107 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑀 ∘ 𝐺)) =
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))) |
45 | | nfv 1830 |
. . . . 5
⊢
Ⅎ𝑗𝜑 |
46 | | meadjiunlem.y |
. . . . . 6
⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} |
47 | | ssrab2 3650 |
. . . . . . 7
⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋 |
48 | 47 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋) |
49 | 46, 48 | syl5eqss 3612 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
50 | 11 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞)) |
51 | 2 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆) |
52 | 49 | sselda 3568 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋) |
53 | 51, 52 | ffvelrnd 6268 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆) |
54 | 50, 53 | ffvelrnd 6268 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞)) |
55 | | eldifi 3694 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋) |
56 | 55 | ad2antlr 759 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋) |
57 | | fveq2 6103 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅)) |
58 | 57 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅)) |
59 | 9 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas) |
60 | 59 | mea0 39347 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0) |
61 | 58, 60 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0) |
62 | 61 | ad4ant14 1285 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0) |
63 | | neneq 2788 |
. . . . . . . . . . . . 13
⊢ ((𝑀‘(𝐺‘𝑗)) ≠ 0 → ¬ (𝑀‘(𝐺‘𝑗)) = 0) |
64 | 63 | ad2antlr 759 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → ¬ (𝑀‘(𝐺‘𝑗)) = 0) |
65 | 62, 64 | pm2.65da 598 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ (𝐺‘𝑗) = ∅) |
66 | 65 | neqned 2789 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅) |
67 | 56, 66 | jca 553 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅)) |
68 | | fveq2 6103 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗)) |
69 | 68 | neeq1d 2841 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅)) |
70 | 69 | elrab 3331 |
. . . . . . . . 9
⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅)) |
71 | 67, 70 | sylibr 223 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}) |
72 | 71, 46 | syl6eleqr 2699 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌) |
73 | | eldifn 3695 |
. . . . . . . 8
⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑗 ∈ 𝑌) |
74 | 73 | ad2antlr 759 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ 𝑗 ∈ 𝑌) |
75 | 72, 74 | pm2.65da 598 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → ¬ (𝑀‘(𝐺‘𝑗)) ≠ 0) |
76 | | nne 2786 |
. . . . . 6
⊢ (¬
(𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0) |
77 | 75, 76 | sylib 207 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0) |
78 | 45, 3, 49, 54, 77 | sge0ss 39305 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))) |
79 | 78 | eqcomd 2616 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗))))) |
80 | 3, 49 | ssexd 4733 |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ V) |
81 | | nfv 1830 |
. . . . . . . . 9
⊢
Ⅎ𝑖𝜑 |
82 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) |
83 | 2 | ffnd 5959 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 Fn 𝑋) |
84 | | dffn3 5967 |
. . . . . . . . . . . . 13
⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺) |
85 | 83, 84 | sylib 207 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺) |
86 | 85 | adantr 480 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺) |
87 | 49 | sselda 3568 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋) |
88 | 86, 87 | ffvelrnd 6268 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺) |
89 | 46 | eleq2i 2680 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}) |
90 | | rabid 3095 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
91 | 89, 90 | bitri 263 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
92 | 91 | biimpi 205 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
93 | 92 | simprd 478 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅) |
94 | 93 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅) |
95 | | nelsn 4159 |
. . . . . . . . . . 11
⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅}) |
96 | 94, 95 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅}) |
97 | 88, 96 | eldifd 3551 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅})) |
98 | | meadjiunlem.dj |
. . . . . . . . . 10
⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖)) |
99 | | disjss1 4559 |
. . . . . . . . . 10
⊢ (𝑌 ⊆ 𝑋 → (Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖) → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖))) |
100 | 49, 98, 99 | sylc 63 |
. . . . . . . . 9
⊢ (𝜑 → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖)) |
101 | 81, 82, 97, 94, 100 | disjf1 38364 |
. . . . . . . 8
⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅})) |
102 | 2, 49 | feqresmpt 6160 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
103 | | f1eq1 6009 |
. . . . . . . . 9
⊢ ((𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))) |
104 | 102, 103 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))) |
105 | 101, 104 | mpbird 246 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅})) |
106 | 102 | rneqd 5274 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
107 | 97 | ralrimiva 2949 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅})) |
108 | 82 | rnmptss 6299 |
. . . . . . . . . 10
⊢
(∀𝑖 ∈
𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅})) |
109 | 107, 108 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅})) |
110 | 106, 109 | eqsstrd 3602 |
. . . . . . . 8
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅})) |
111 | | simpl 472 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑) |
112 | | eldifi 3694 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺) |
113 | 112 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺) |
114 | | eldifsni 4261 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅) |
115 | 114 | adantl 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅) |
116 | | simpr 476 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺) |
117 | | fvelrnb 6153 |
. . . . . . . . . . . . . . . 16
⊢ (𝐺 Fn 𝑋 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
118 | 83, 117 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
119 | 118 | adantr 480 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
120 | 116, 119 | mpbid 221 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥) |
121 | 120 | 3adant3 1074 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥) |
122 | | id 22 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐺‘𝑖) = 𝑥 → (𝐺‘𝑖) = 𝑥) |
123 | 122 | eqcomd 2616 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖)) |
124 | 123 | 3ad2ant3 1077 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖)) |
125 | | simp1l 1078 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑) |
126 | | simp2 1055 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋) |
127 | | simpr 476 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥) |
128 | | simpl 472 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅) |
129 | 127, 128 | eqnetrd 2849 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
130 | 129 | adantll 746 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
131 | 130 | 3adant2 1073 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
132 | 91 | biimpri 217 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌) |
133 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝐺‘𝑖) ∈ V |
134 | 133 | a1i 11 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V) |
135 | 82 | elrnmpt1 5295 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑖 ∈ 𝑌 ∧ (𝐺‘𝑖) ∈ V) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
136 | 132, 134,
135 | syl2anc 691 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
137 | 136 | 3adant1 1072 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
138 | 106 | eqcomd 2616 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌)) |
139 | 138 | 3ad2ant1 1075 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌)) |
140 | 137, 139 | eleqtrd 2690 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌)) |
141 | 125, 126,
131, 140 | syl3anc 1318 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌)) |
142 | 124, 141 | eqeltrd 2688 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
143 | 142 | 3exp 1256 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (𝑖 ∈ 𝑋 → ((𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))) |
144 | 143 | rexlimdv 3012 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))) |
145 | 144 | 3adant2 1073 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))) |
146 | 121, 145 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
147 | 111, 113,
115, 146 | syl3anc 1318 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
148 | 147 | ralrimiva 2949 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐺 ∖ {∅})𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
149 | | dfss3 3558 |
. . . . . . . . 9
⊢ ((ran
𝐺 ∖ {∅})
⊆ ran (𝐺 ↾
𝑌) ↔ ∀𝑥 ∈ (ran 𝐺 ∖ {∅})𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
150 | 148, 149 | sylibr 223 |
. . . . . . . 8
⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran (𝐺 ↾ 𝑌)) |
151 | 110, 150 | eqssd 3585 |
. . . . . . 7
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})) |
152 | 105, 151 | jca 553 |
. . . . . 6
⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))) |
153 | | dff1o5 6059 |
. . . . . 6
⊢ ((𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran
𝐺 ∖ {∅}) ↔
((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))) |
154 | 152, 153 | sylibr 223 |
. . . . 5
⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran
𝐺 ∖
{∅})) |
155 | | fvres 6117 |
. . . . . 6
⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗)) |
156 | 155 | adantl 481 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗)) |
157 | 1, 45, 42, 80, 154, 156, 18 | sge0f1o 39275 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗))))) |
158 | 157 | eqcomd 2616 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
159 | 44, 79, 158 | 3eqtrd 2648 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑀 ∘ 𝐺)) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
160 | 36, 38, 159 | 3eqtr4d 2654 |
1
⊢ (𝜑 →
(Σ^‘(𝑀 ↾ ran 𝐺)) =
(Σ^‘(𝑀 ∘ 𝐺))) |