Step | Hyp | Ref
| Expression |
1 | | fprodss.1 |
. . 3
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
2 | | sseq2 3852 |
. . . . 5
⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 ↔ 𝐴 ⊆ ∅)) |
3 | | ss0 4199 |
. . . . 5
⊢ (𝐴 ⊆ ∅ → 𝐴 = ∅) |
4 | 2, 3 | syl6bi 245 |
. . . 4
⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → 𝐴 = ∅)) |
5 | | prodeq1 15012 |
. . . . . 6
⊢ (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ ∅ 𝐶) |
6 | | prodeq1 15012 |
. . . . . . 7
⊢ (𝐵 = ∅ → ∏𝑘 ∈ 𝐵 𝐶 = ∏𝑘 ∈ ∅ 𝐶) |
7 | 6 | eqcomd 2831 |
. . . . . 6
⊢ (𝐵 = ∅ → ∏𝑘 ∈ ∅ 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
8 | 5, 7 | sylan9eq 2881 |
. . . . 5
⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
9 | 8 | expcom 404 |
. . . 4
⊢ (𝐵 = ∅ → (𝐴 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
10 | 4, 9 | syld 47 |
. . 3
⊢ (𝐵 = ∅ → (𝐴 ⊆ 𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
11 | 1, 10 | syl5com 31 |
. 2
⊢ (𝜑 → (𝐵 = ∅ → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
12 | | cnvimass 5726 |
. . . . . . . . 9
⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓 |
13 | | simprr 791 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) |
14 | | f1of 6378 |
. . . . . . . . . 10
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))⟶𝐵) |
15 | 13, 14 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))⟶𝐵) |
16 | 12, 15 | fssdm 6294 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) |
17 | | f1ofn 6379 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓 Fn (1...(♯‘𝐵))) |
18 | 13, 17 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(♯‘𝐵))) |
19 | | elpreima 6586 |
. . . . . . . . . . . 12
⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
20 | 18, 19 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
21 | 15 | ffvelrnda 6608 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵) |
22 | 21 | ex 403 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵)) |
23 | 22 | adantrd 487 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
24 | 20, 23 | sylbid 232 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
25 | 24 | imp 397 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵) |
26 | | fprodss.2 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
27 | 26 | ex 403 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
28 | 27 | adantr 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
29 | | eldif 3808 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) |
30 | | fprodss.3 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 1) |
31 | | ax-1cn 10310 |
. . . . . . . . . . . . . . . 16
⊢ 1 ∈
ℂ |
32 | 30, 31 | syl6eqel 2914 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ) |
33 | 29, 32 | sylan2br 590 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ) |
34 | 33 | expr 450 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
35 | 28, 34 | pm2.61d 172 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
36 | 35 | adantlr 708 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
37 | 36 | fmpttd 6634 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ) |
38 | 37 | ffvelrnda 6608 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
39 | 25, 38 | syldan 587 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
40 | | eqid 2825 |
. . . . . . . . 9
⊢
(ℤ≥‘1) =
(ℤ≥‘1) |
41 | | simprl 789 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈
ℕ) |
42 | | nnuz 12005 |
. . . . . . . . . 10
⊢ ℕ =
(ℤ≥‘1) |
43 | 41, 42 | syl6eleq 2916 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (♯‘𝐵) ∈
(ℤ≥‘1)) |
44 | | ssidd 3849 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) →
(1...(♯‘𝐵))
⊆ (1...(♯‘𝐵))) |
45 | 40, 43, 44 | fprodntriv 15045 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∃𝑚 ∈
(ℤ≥‘1)∃𝑦(𝑦 ≠ 0 ∧ seq𝑚( · , (𝑛 ∈ (ℤ≥‘1)
↦ if(𝑛 ∈
(1...(♯‘𝐵)),
((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)), 1))) ⇝ 𝑦)) |
46 | | eldifi 3959 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
((1...(♯‘𝐵))
∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵))) |
47 | 46, 21 | sylan2 588 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵) |
48 | | eldifn 3960 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
((1...(♯‘𝐵))
∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
49 | 48 | adantl 475 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
50 | 20 | adantr 474 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
51 | 46 | adantl 475 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵))) |
52 | 51 | biantrurd 530 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑓‘𝑛) ∈ 𝐴 ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
53 | 50, 52 | bitr4d 274 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴)) |
54 | 49, 53 | mtbid 316 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴) |
55 | 47, 54 | eldifd 3809 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴)) |
56 | | difss 3964 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 |
57 | | resmpt 5686 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)) |
58 | 56, 57 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) |
59 | 58 | fveq1i 6434 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) |
60 | | fvres 6452 |
. . . . . . . . . . 11
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
61 | 59, 60 | syl5eqr 2875 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
62 | 55, 61 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
63 | | 1ex 10352 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
V |
64 | 63 | elsn2 4432 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ {1} ↔ 𝐶 = 1) |
65 | 30, 64 | sylibr 226 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {1}) |
66 | 65 | fmpttd 6634 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1}) |
67 | 66 | ad2antrr 719 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{1}) |
68 | 67, 55 | ffvelrnd 6609 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1}) |
69 | | elsni 4414 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {1} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1) |
70 | 68, 69 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 1) |
71 | 62, 70 | eqtr3d 2863 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 1) |
72 | | fzssuz 12675 |
. . . . . . . . 9
⊢
(1...(♯‘𝐵)) ⊆
(ℤ≥‘1) |
73 | 72 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) →
(1...(♯‘𝐵))
⊆ (ℤ≥‘1)) |
74 | 16, 39, 45, 71, 73 | prodss 15050 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
75 | 1 | adantr 474 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵) |
76 | 75 | resmptd 5689 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
77 | 76 | fveq1d 6435 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) |
78 | | fvres 6452 |
. . . . . . . . . 10
⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
79 | 77, 78 | sylan9req 2882 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
80 | 79 | prodeq2dv 15026 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
81 | | fveq2 6433 |
. . . . . . . . 9
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
82 | | fzfid 13067 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) →
(1...(♯‘𝐵))
∈ Fin) |
83 | 82, 15 | fisuppfi 8552 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin) |
84 | | f1of1 6377 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–1-1→𝐵) |
85 | 13, 84 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1→𝐵) |
86 | | f1ores 6392 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘𝐵))–1-1→𝐵 ∧ (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
87 | 85, 16, 86 | syl2anc 581 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
88 | | f1ofo 6385 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–onto→𝐵) |
89 | 13, 88 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–onto→𝐵) |
90 | | foimacnv 6395 |
. . . . . . . . . . . 12
⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
91 | 89, 75, 90 | syl2anc 581 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
92 | | f1oeq3 6369 |
. . . . . . . . . . 11
⊢ ((𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴 → ((𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)) |
93 | 91, 92 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴)) ↔ (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴)) |
94 | 87, 93 | mpbid 224 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴) |
95 | | fvres 6452 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
96 | 95 | adantl 475 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
97 | 75 | sselda 3827 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵) |
98 | 37 | ffvelrnda 6608 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
99 | 97, 98 | syldan 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
100 | 81, 83, 94, 96, 99 | fprodf1o 15049 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
101 | 80, 100 | eqtrd 2861 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
102 | | eqidd 2826 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛)) |
103 | 81, 82, 13, 102, 98 | fprodf1o 15049 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑛 ∈ (1...(♯‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
104 | 74, 101, 103 | 3eqtr4d 2871 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
105 | | prodfc 15048 |
. . . . . 6
⊢
∏𝑚 ∈
𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐶 |
106 | | prodfc 15048 |
. . . . . 6
⊢
∏𝑚 ∈
𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ∏𝑘 ∈ 𝐵 𝐶 |
107 | 104, 105,
106 | 3eqtr3g 2884 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |
108 | 107 | expr 450 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
109 | 108 | exlimdv 2034 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
110 | 109 | expimpd 447 |
. 2
⊢ (𝜑 → (((♯‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶)) |
111 | | fprodss.4 |
. . 3
⊢ (𝜑 → 𝐵 ∈ Fin) |
112 | | fz1f1o 14818 |
. . 3
⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨
((♯‘𝐵) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵))) |
113 | 111, 112 | syl 17 |
. 2
⊢ (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵))) |
114 | 11, 110, 113 | mpjaod 893 |
1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐶 = ∏𝑘 ∈ 𝐵 𝐶) |