Step | Hyp | Ref
| Expression |
1 | | prod0 15651 |
. . . 4
⊢
∏𝑘 ∈
∅ 𝐵 =
1 |
2 | | fprodf1o.3 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹:𝐶–1-1-onto→𝐴) |
3 | 2 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:𝐶–1-1-onto→𝐴) |
4 | | f1oeq2 6703 |
. . . . . . . . 9
⊢ (𝐶 = ∅ → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴)) |
5 | 4 | adantl 482 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝐶 = ∅) → (𝐹:𝐶–1-1-onto→𝐴 ↔ 𝐹:∅–1-1-onto→𝐴)) |
6 | 3, 5 | mpbid 231 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–1-1-onto→𝐴) |
7 | | f1ofo 6721 |
. . . . . . 7
⊢ (𝐹:∅–1-1-onto→𝐴 → 𝐹:∅–onto→𝐴) |
8 | 6, 7 | syl 17 |
. . . . . 6
⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐹:∅–onto→𝐴) |
9 | | fo00 6749 |
. . . . . . 7
⊢ (𝐹:∅–onto→𝐴 ↔ (𝐹 = ∅ ∧ 𝐴 = ∅)) |
10 | 9 | simprbi 497 |
. . . . . 6
⊢ (𝐹:∅–onto→𝐴 → 𝐴 = ∅) |
11 | 8, 10 | syl 17 |
. . . . 5
⊢ ((𝜑 ∧ 𝐶 = ∅) → 𝐴 = ∅) |
12 | 11 | prodeq1d 15629 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ ∅ 𝐵) |
13 | | prodeq1 15617 |
. . . . . 6
⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = ∏𝑛 ∈ ∅ 𝐷) |
14 | | prod0 15651 |
. . . . . 6
⊢
∏𝑛 ∈
∅ 𝐷 =
1 |
15 | 13, 14 | eqtrdi 2796 |
. . . . 5
⊢ (𝐶 = ∅ → ∏𝑛 ∈ 𝐶 𝐷 = 1) |
16 | 15 | adantl 482 |
. . . 4
⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑛 ∈ 𝐶 𝐷 = 1) |
17 | 1, 12, 16 | 3eqtr4a 2806 |
. . 3
⊢ ((𝜑 ∧ 𝐶 = ∅) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷) |
18 | 17 | ex 413 |
. 2
⊢ (𝜑 → (𝐶 = ∅ → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)) |
19 | | 2fveq3 6776 |
. . . . . . . 8
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛)))) |
20 | | simprl 768 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (♯‘𝐶) ∈
ℕ) |
21 | | simprr 770 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) |
22 | | f1of 6714 |
. . . . . . . . . . . 12
⊢ (𝐹:𝐶–1-1-onto→𝐴 → 𝐹:𝐶⟶𝐴) |
23 | 2, 22 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐶⟶𝐴) |
24 | 23 | ffvelrnda 6958 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → (𝐹‘𝑚) ∈ 𝐴) |
25 | | fprodf1o.5 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
26 | 25 | fmpttd 6986 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
27 | 26 | ffvelrnda 6958 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝐹‘𝑚) ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ) |
28 | 24, 27 | syldan 591 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ) |
29 | 28 | adantlr 712 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) ∈ ℂ) |
30 | | simpr 485 |
. . . . . . . . . . . 12
⊢
(((♯‘𝐶)
∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) |
31 | | f1oco 6736 |
. . . . . . . . . . . 12
⊢ ((𝐹:𝐶–1-1-onto→𝐴 ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴) |
32 | 2, 30, 31 | syl2an 596 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴) |
33 | | f1of 6714 |
. . . . . . . . . . 11
⊢ ((𝐹 ∘ 𝑓):(1...(♯‘𝐶))–1-1-onto→𝐴 → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴) |
34 | 32, 33 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴) |
35 | | fvco3 6864 |
. . . . . . . . . 10
⊢ (((𝐹 ∘ 𝑓):(1...(♯‘𝐶))⟶𝐴 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛))) |
36 | 34, 35 | sylan 580 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛))) |
37 | | f1of 6714 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → 𝑓:(1...(♯‘𝐶))⟶𝐶) |
38 | 37 | adantl 482 |
. . . . . . . . . . . 12
⊢
(((♯‘𝐶)
∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → 𝑓:(1...(♯‘𝐶))⟶𝐶) |
39 | 38 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → 𝑓:(1...(♯‘𝐶))⟶𝐶) |
40 | | fvco3 6864 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘𝐶))⟶𝐶 ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛))) |
41 | 39, 40 | sylan 580 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝐹 ∘ 𝑓)‘𝑛) = (𝐹‘(𝑓‘𝑛))) |
42 | 41 | fveq2d 6775 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛)))) |
43 | 36, 42 | eqtrd 2780 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑛 ∈ (1...(♯‘𝐶))) → (((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓))‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘(𝑓‘𝑛)))) |
44 | 19, 20, 21, 29, 43 | fprod 15649 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(♯‘𝐶))) |
45 | | fprodf1o.4 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) = 𝐺) |
46 | 23 | ffvelrnda 6958 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → (𝐹‘𝑛) ∈ 𝐴) |
47 | 45, 46 | eqeltrrd 2842 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → 𝐺 ∈ 𝐴) |
48 | | fprodf1o.1 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = 𝐺 → 𝐵 = 𝐷) |
49 | | eqid 2740 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ 𝐴 ↦ 𝐵) = (𝑘 ∈ 𝐴 ↦ 𝐵) |
50 | 48, 49 | fvmpti 6871 |
. . . . . . . . . . . . 13
⊢ (𝐺 ∈ 𝐴 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷)) |
51 | 47, 50 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺) = ( I ‘𝐷)) |
52 | 45 | fveq2d 6775 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝐺)) |
53 | | eqid 2740 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ 𝐶 ↦ 𝐷) = (𝑛 ∈ 𝐶 ↦ 𝐷) |
54 | 53 | fvmpt2i 6882 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ 𝐶 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷)) |
55 | 54 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ( I ‘𝐷)) |
56 | 51, 52, 55 | 3eqtr4rd 2791 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛))) |
57 | 56 | ralrimiva 3110 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛))) |
58 | | nffvmpt1 6782 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) |
59 | 58 | nfeq1 2924 |
. . . . . . . . . . 11
⊢
Ⅎ𝑛((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)) |
60 | | fveq2 6771 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)) |
61 | | 2fveq3 6776 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑚 → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))) |
62 | 60, 61 | eqeq12d 2756 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑚 → (((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) ↔ ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))) |
63 | 59, 62 | rspc 3548 |
. . . . . . . . . 10
⊢ (𝑚 ∈ 𝐶 → (∀𝑛 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑛) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑛)) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚)))) |
64 | 57, 63 | mpan9 507 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))) |
65 | 64 | adantlr 712 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐶) → ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))) |
66 | 65 | prodeq2dv 15631 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘(𝐹‘𝑚))) |
67 | | fveq2 6771 |
. . . . . . . 8
⊢ (𝑚 = ((𝐹 ∘ 𝑓)‘𝑛) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐵)‘((𝐹 ∘ 𝑓)‘𝑛))) |
68 | 26 | adantr 481 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → (𝑘 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℂ) |
69 | 68 | ffvelrnda 6958 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) ∈ ℂ) |
70 | 67, 20, 32, 69, 36 | fprod 15649 |
. . . . . . 7
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = (seq1( · , ((𝑘 ∈ 𝐴 ↦ 𝐵) ∘ (𝐹 ∘ 𝑓)))‘(♯‘𝐶))) |
71 | 44, 66, 70 | 3eqtr4rd 2791 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑚 ∈ 𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚)) |
72 | | prodfc 15653 |
. . . . . 6
⊢
∏𝑚 ∈
𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐵)‘𝑚) = ∏𝑘 ∈ 𝐴 𝐵 |
73 | | prodfc 15653 |
. . . . . 6
⊢
∏𝑚 ∈
𝐶 ((𝑛 ∈ 𝐶 ↦ 𝐷)‘𝑚) = ∏𝑛 ∈ 𝐶 𝐷 |
74 | 71, 72, 73 | 3eqtr3g 2803 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝐶) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶)) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷) |
75 | 74 | expr 457 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) → (𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)) |
76 | 75 | exlimdv 1940 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝐶) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)) |
77 | 76 | expimpd 454 |
. 2
⊢ (𝜑 → (((♯‘𝐶) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷)) |
78 | | fprodf1o.2 |
. . 3
⊢ (𝜑 → 𝐶 ∈ Fin) |
79 | | fz1f1o 15420 |
. . 3
⊢ (𝐶 ∈ Fin → (𝐶 = ∅ ∨
((♯‘𝐶) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶))) |
80 | 78, 79 | syl 17 |
. 2
⊢ (𝜑 → (𝐶 = ∅ ∨ ((♯‘𝐶) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐶))–1-1-onto→𝐶))) |
81 | 18, 77, 80 | mpjaod 857 |
1
⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑛 ∈ 𝐶 𝐷) |