Step | Hyp | Ref
| Expression |
1 | | nfv 1922 |
. . . 4
⊢
Ⅎ𝑘𝜑 |
2 | | meadjiunlem.g |
. . . . . 6
⊢ (𝜑 → 𝐺:𝑋⟶𝑆) |
3 | | meadjiunlem.x |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ 𝑉) |
4 | 2, 3 | jca 515 |
. . . . 5
⊢ (𝜑 → (𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉)) |
5 | | fex 7063 |
. . . . 5
⊢ ((𝐺:𝑋⟶𝑆 ∧ 𝑋 ∈ 𝑉) → 𝐺 ∈ V) |
6 | | rnexg 7703 |
. . . . 5
⊢ (𝐺 ∈ V → ran 𝐺 ∈ V) |
7 | 4, 5, 6 | 3syl 18 |
. . . 4
⊢ (𝜑 → ran 𝐺 ∈ V) |
8 | | difssd 4063 |
. . . 4
⊢ (𝜑 → (ran 𝐺 ∖ {∅}) ⊆ ran 𝐺) |
9 | | meadjiunlem.f |
. . . . . . 7
⊢ (𝜑 → 𝑀 ∈ Meas) |
10 | | meadjiunlem.3 |
. . . . . . 7
⊢ 𝑆 = dom 𝑀 |
11 | 9, 10 | meaf 43711 |
. . . . . 6
⊢ (𝜑 → 𝑀:𝑆⟶(0[,]+∞)) |
12 | 11 | adantr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑀:𝑆⟶(0[,]+∞)) |
13 | 2 | frnd 6574 |
. . . . . . 7
⊢ (𝜑 → ran 𝐺 ⊆ 𝑆) |
14 | 13 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → ran 𝐺 ⊆ 𝑆) |
15 | 8 | sselda 3917 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ ran 𝐺) |
16 | 14, 15 | sseldd 3918 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ 𝑆) |
17 | 12, 16 | ffvelrnd 6926 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ {∅})) → (𝑀‘𝑘) ∈ (0[,]+∞)) |
18 | | simpl 486 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝜑) |
19 | | id 22 |
. . . . . . . 8
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) |
20 | | dfin4 4198 |
. . . . . . . . 9
⊢ (ran
𝐺 ∩ {∅}) = (ran
𝐺 ∖ (ran 𝐺 ∖
{∅})) |
21 | 20 | eqcomi 2748 |
. . . . . . . 8
⊢ (ran
𝐺 ∖ (ran 𝐺 ∖ {∅})) = (ran
𝐺 ∩
{∅}) |
22 | 19, 21 | eleqtrdi 2850 |
. . . . . . 7
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 ∈ (ran 𝐺 ∩ {∅})) |
23 | | elinel2 4126 |
. . . . . . . 8
⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 ∈ {∅}) |
24 | | elsni 4574 |
. . . . . . . 8
⊢ (𝑘 ∈ {∅} → 𝑘 = ∅) |
25 | 23, 24 | syl 17 |
. . . . . . 7
⊢ (𝑘 ∈ (ran 𝐺 ∩ {∅}) → 𝑘 = ∅) |
26 | 22, 25 | syl 17 |
. . . . . 6
⊢ (𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅})) → 𝑘 = ∅) |
27 | 26 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → 𝑘 = ∅) |
28 | | simpr 488 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 = ∅) → 𝑘 = ∅) |
29 | 28 | fveq2d 6742 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = (𝑀‘∅)) |
30 | 9 | mea0 43712 |
. . . . . . 7
⊢ (𝜑 → (𝑀‘∅) = 0) |
31 | 30 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘∅) = 0) |
32 | 29, 31 | eqtrd 2779 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 = ∅) → (𝑀‘𝑘) = 0) |
33 | 18, 27, 32 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ (ran 𝐺 ∖ (ran 𝐺 ∖ {∅}))) → (𝑀‘𝑘) = 0) |
34 | 1, 7, 8, 17, 33 | sge0ss 43670 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))) |
35 | 34 | eqcomd 2745 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
36 | 11, 13 | feqresmpt 6802 |
. . 3
⊢ (𝜑 → (𝑀 ↾ ran 𝐺) = (𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘))) |
37 | 36 | fveq2d 6742 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑀 ↾ ran 𝐺)) =
(Σ^‘(𝑘 ∈ ran 𝐺 ↦ (𝑀‘𝑘)))) |
38 | 2 | ffvelrnda 6925 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑋) → (𝐺‘𝑗) ∈ 𝑆) |
39 | 2 | feqmptd 6801 |
. . . . 5
⊢ (𝜑 → 𝐺 = (𝑗 ∈ 𝑋 ↦ (𝐺‘𝑗))) |
40 | 11 | feqmptd 6801 |
. . . . 5
⊢ (𝜑 → 𝑀 = (𝑘 ∈ 𝑆 ↦ (𝑀‘𝑘))) |
41 | | fveq2 6738 |
. . . . 5
⊢ (𝑘 = (𝐺‘𝑗) → (𝑀‘𝑘) = (𝑀‘(𝐺‘𝑗))) |
42 | 38, 39, 40, 41 | fmptco 6965 |
. . . 4
⊢ (𝜑 → (𝑀 ∘ 𝐺) = (𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) |
43 | 42 | fveq2d 6742 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑀 ∘ 𝐺)) =
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))) |
44 | | nfv 1922 |
. . . . 5
⊢
Ⅎ𝑗𝜑 |
45 | | meadjiunlem.y |
. . . . . 6
⊢ 𝑌 = {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} |
46 | | ssrab2 4009 |
. . . . . . 7
⊢ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋 |
47 | 46 | a1i 11 |
. . . . . 6
⊢ (𝜑 → {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ⊆ 𝑋) |
48 | 45, 47 | eqsstrid 3965 |
. . . . 5
⊢ (𝜑 → 𝑌 ⊆ 𝑋) |
49 | 11 | adantr 484 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑀:𝑆⟶(0[,]+∞)) |
50 | 2 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝐺:𝑋⟶𝑆) |
51 | 48 | sselda 3917 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → 𝑗 ∈ 𝑋) |
52 | 50, 51 | ffvelrnd 6926 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝐺‘𝑗) ∈ 𝑆) |
53 | 49, 52 | ffvelrnd 6926 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → (𝑀‘(𝐺‘𝑗)) ∈ (0[,]+∞)) |
54 | | eldifi 4057 |
. . . . . . . . . . 11
⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → 𝑗 ∈ 𝑋) |
55 | 54 | ad2antlr 727 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑋) |
56 | | fveq2 6738 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑗) = ∅ → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅)) |
57 | 56 | adantl 485 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = (𝑀‘∅)) |
58 | 9 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → 𝑀 ∈ Meas) |
59 | 58 | mea0 43712 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘∅) = 0) |
60 | 57, 59 | eqtrd 2779 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0) |
61 | 60 | ad4ant14 752 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → (𝑀‘(𝐺‘𝑗)) = 0) |
62 | | neneq 2949 |
. . . . . . . . . . . . 13
⊢ ((𝑀‘(𝐺‘𝑗)) ≠ 0 → ¬ (𝑀‘(𝐺‘𝑗)) = 0) |
63 | 62 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) ∧ (𝐺‘𝑗) = ∅) → ¬ (𝑀‘(𝐺‘𝑗)) = 0) |
64 | 61, 63 | pm2.65da 817 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ (𝐺‘𝑗) = ∅) |
65 | 64 | neqned 2950 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝐺‘𝑗) ≠ ∅) |
66 | 55, 65 | jca 515 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅)) |
67 | | fveq2 6738 |
. . . . . . . . . . 11
⊢ (𝑖 = 𝑗 → (𝐺‘𝑖) = (𝐺‘𝑗)) |
68 | 67 | neeq1d 3003 |
. . . . . . . . . 10
⊢ (𝑖 = 𝑗 → ((𝐺‘𝑖) ≠ ∅ ↔ (𝐺‘𝑗) ≠ ∅)) |
69 | 68 | elrab 3617 |
. . . . . . . . 9
⊢ (𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑗 ∈ 𝑋 ∧ (𝐺‘𝑗) ≠ ∅)) |
70 | 66, 69 | sylibr 237 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}) |
71 | 70, 45 | eleqtrrdi 2851 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → 𝑗 ∈ 𝑌) |
72 | | eldifn 4058 |
. . . . . . . 8
⊢ (𝑗 ∈ (𝑋 ∖ 𝑌) → ¬ 𝑗 ∈ 𝑌) |
73 | 72 | ad2antlr 727 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) ∧ (𝑀‘(𝐺‘𝑗)) ≠ 0) → ¬ 𝑗 ∈ 𝑌) |
74 | 71, 73 | pm2.65da 817 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → ¬ (𝑀‘(𝐺‘𝑗)) ≠ 0) |
75 | | nne 2947 |
. . . . . 6
⊢ (¬
(𝑀‘(𝐺‘𝑗)) ≠ 0 ↔ (𝑀‘(𝐺‘𝑗)) = 0) |
76 | 74, 75 | sylib 221 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ (𝑋 ∖ 𝑌)) → (𝑀‘(𝐺‘𝑗)) = 0) |
77 | 44, 3, 48, 53, 76 | sge0ss 43670 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗))))) |
78 | 77 | eqcomd 2745 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑋 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗))))) |
79 | 3, 48 | ssexd 5233 |
. . . . 5
⊢ (𝜑 → 𝑌 ∈ V) |
80 | | nfv 1922 |
. . . . . . . . 9
⊢
Ⅎ𝑖𝜑 |
81 | | eqid 2739 |
. . . . . . . . 9
⊢ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) |
82 | 2 | ffnd 6567 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐺 Fn 𝑋) |
83 | | dffn3 6579 |
. . . . . . . . . . . . 13
⊢ (𝐺 Fn 𝑋 ↔ 𝐺:𝑋⟶ran 𝐺) |
84 | 82, 83 | sylib 221 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐺:𝑋⟶ran 𝐺) |
85 | 84 | adantr 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝐺:𝑋⟶ran 𝐺) |
86 | 48 | sselda 3917 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → 𝑖 ∈ 𝑋) |
87 | 85, 86 | ffvelrnd 6926 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ ran 𝐺) |
88 | 45 | eleq2i 2831 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ 𝑌 ↔ 𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅}) |
89 | | rabid 3305 |
. . . . . . . . . . . . . . 15
⊢ (𝑖 ∈ {𝑖 ∈ 𝑋 ∣ (𝐺‘𝑖) ≠ ∅} ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
90 | 88, 89 | bitri 278 |
. . . . . . . . . . . . . 14
⊢ (𝑖 ∈ 𝑌 ↔ (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
91 | 90 | biimpi 219 |
. . . . . . . . . . . . 13
⊢ (𝑖 ∈ 𝑌 → (𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅)) |
92 | 91 | simprd 499 |
. . . . . . . . . . . 12
⊢ (𝑖 ∈ 𝑌 → (𝐺‘𝑖) ≠ ∅) |
93 | 92 | adantl 485 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ≠ ∅) |
94 | | nelsn 4597 |
. . . . . . . . . . 11
⊢ ((𝐺‘𝑖) ≠ ∅ → ¬ (𝐺‘𝑖) ∈ {∅}) |
95 | 93, 94 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → ¬ (𝐺‘𝑖) ∈ {∅}) |
96 | 87, 95 | eldifd 3894 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑌) → (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅})) |
97 | | meadjiunlem.dj |
. . . . . . . . . 10
⊢ (𝜑 → Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖)) |
98 | | disjss1 5040 |
. . . . . . . . . 10
⊢ (𝑌 ⊆ 𝑋 → (Disj 𝑖 ∈ 𝑋 (𝐺‘𝑖) → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖))) |
99 | 48, 97, 98 | sylc 65 |
. . . . . . . . 9
⊢ (𝜑 → Disj 𝑖 ∈ 𝑌 (𝐺‘𝑖)) |
100 | 80, 81, 96, 93, 99 | disjf1 42440 |
. . . . . . . 8
⊢ (𝜑 → (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅})) |
101 | 2, 48 | feqresmpt 6802 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
102 | | f1eq1 6631 |
. . . . . . . . 9
⊢ ((𝐺 ↾ 𝑌) = (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))) |
103 | 101, 102 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ↔ (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)):𝑌–1-1→(ran 𝐺 ∖ {∅}))) |
104 | 100, 103 | mpbird 260 |
. . . . . . 7
⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅})) |
105 | 101 | rneqd 5824 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
106 | 96 | ralrimiva 3108 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑖 ∈ 𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅})) |
107 | 81 | rnmptss 6960 |
. . . . . . . . . 10
⊢
(∀𝑖 ∈
𝑌 (𝐺‘𝑖) ∈ (ran 𝐺 ∖ {∅}) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅})) |
108 | 106, 107 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) ⊆ (ran 𝐺 ∖ {∅})) |
109 | 105, 108 | eqsstrd 3955 |
. . . . . . . 8
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) ⊆ (ran 𝐺 ∖ {∅})) |
110 | | simpl 486 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝜑) |
111 | | eldifi 4057 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ∈ ran 𝐺) |
112 | 111 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran 𝐺) |
113 | | eldifsni 4719 |
. . . . . . . . . 10
⊢ (𝑥 ∈ (ran 𝐺 ∖ {∅}) → 𝑥 ≠ ∅) |
114 | 113 | adantl 485 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ≠ ∅) |
115 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → 𝑥 ∈ ran 𝐺) |
116 | | fvelrnb 6794 |
. . . . . . . . . . . . . 14
⊢ (𝐺 Fn 𝑋 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
117 | 82, 116 | syl 17 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
118 | 117 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → (𝑥 ∈ ran 𝐺 ↔ ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥)) |
119 | 115, 118 | mpbid 235 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥) |
120 | 119 | 3adant3 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → ∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥) |
121 | | id 22 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐺‘𝑖) = 𝑥 → (𝐺‘𝑖) = 𝑥) |
122 | 121 | eqcomd 2745 |
. . . . . . . . . . . . . . 15
⊢ ((𝐺‘𝑖) = 𝑥 → 𝑥 = (𝐺‘𝑖)) |
123 | 122 | 3ad2ant3 1137 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 = (𝐺‘𝑖)) |
124 | | simp1l 1199 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝜑) |
125 | | simp2 1139 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑖 ∈ 𝑋) |
126 | | simpr 488 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) = 𝑥) |
127 | | simpl 486 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ≠ ∅) |
128 | 126, 127 | eqnetrd 3011 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑥 ≠ ∅ ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
129 | 128 | adantll 714 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
130 | 129 | 3adant2 1133 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ≠ ∅) |
131 | 90 | biimpri 231 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → 𝑖 ∈ 𝑌) |
132 | | fvexd 6753 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ V) |
133 | 81 | elrnmpt1 5844 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑖 ∈ 𝑌 ∧ (𝐺‘𝑖) ∈ V) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
134 | 131, 132,
133 | syl2anc 587 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
135 | 134 | 3adant1 1132 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖))) |
136 | 105 | eqcomd 2745 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌)) |
137 | 136 | 3ad2ant1 1135 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → ran (𝑖 ∈ 𝑌 ↦ (𝐺‘𝑖)) = ran (𝐺 ↾ 𝑌)) |
138 | 135, 137 | eleqtrd 2842 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) ≠ ∅) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌)) |
139 | 124, 125,
130, 138 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → (𝐺‘𝑖) ∈ ran (𝐺 ↾ 𝑌)) |
140 | 123, 139 | eqeltrd 2840 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ≠ ∅) ∧ 𝑖 ∈ 𝑋 ∧ (𝐺‘𝑖) = 𝑥) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
141 | 140 | 3exp 1121 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (𝑖 ∈ 𝑋 → ((𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌)))) |
142 | 141 | rexlimdv 3212 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))) |
143 | 142 | 3adant2 1133 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → (∃𝑖 ∈ 𝑋 (𝐺‘𝑖) = 𝑥 → 𝑥 ∈ ran (𝐺 ↾ 𝑌))) |
144 | 120, 143 | mpd 15 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ ran 𝐺 ∧ 𝑥 ≠ ∅) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
145 | 110, 112,
114, 144 | syl3anc 1373 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑥 ∈ (ran 𝐺 ∖ {∅})) → 𝑥 ∈ ran (𝐺 ↾ 𝑌)) |
146 | 109, 145 | eqelssd 3938 |
. . . . . . 7
⊢ (𝜑 → ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅})) |
147 | 104, 146 | jca 515 |
. . . . . 6
⊢ (𝜑 → ((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))) |
148 | | dff1o5 6691 |
. . . . . 6
⊢ ((𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran
𝐺 ∖ {∅}) ↔
((𝐺 ↾ 𝑌):𝑌–1-1→(ran 𝐺 ∖ {∅}) ∧ ran (𝐺 ↾ 𝑌) = (ran 𝐺 ∖ {∅}))) |
149 | 147, 148 | sylibr 237 |
. . . . 5
⊢ (𝜑 → (𝐺 ↾ 𝑌):𝑌–1-1-onto→(ran
𝐺 ∖
{∅})) |
150 | | fvres 6757 |
. . . . . 6
⊢ (𝑗 ∈ 𝑌 → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗)) |
151 | 150 | adantl 485 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ 𝑌) → ((𝐺 ↾ 𝑌)‘𝑗) = (𝐺‘𝑗)) |
152 | 1, 44, 41, 79, 149, 151, 17 | sge0f1o 43640 |
. . . 4
⊢ (𝜑 →
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘))) =
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗))))) |
153 | 152 | eqcomd 2745 |
. . 3
⊢ (𝜑 →
(Σ^‘(𝑗 ∈ 𝑌 ↦ (𝑀‘(𝐺‘𝑗)))) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
154 | 43, 78, 153 | 3eqtrd 2783 |
. 2
⊢ (𝜑 →
(Σ^‘(𝑀 ∘ 𝐺)) =
(Σ^‘(𝑘 ∈ (ran 𝐺 ∖ {∅}) ↦ (𝑀‘𝑘)))) |
155 | 35, 37, 154 | 3eqtr4d 2789 |
1
⊢ (𝜑 →
(Σ^‘(𝑀 ↾ ran 𝐺)) =
(Σ^‘(𝑀 ∘ 𝐺))) |