Step | Hyp | Ref
| Expression |
1 | | sumss.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
2 | 1 | adantr 481 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 ⊆ 𝐵) |
3 | | sumss.2 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
4 | 3 | adantlr 751 |
. . . 4
⊢ (((𝜑 ∧ 𝐵 = ∅) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
5 | | sumss.3 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
6 | 5 | adantlr 751 |
. . . 4
⊢ (((𝜑 ∧ 𝐵 = ∅) ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
7 | | simpr 477 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 = ∅) |
8 | | 0ss 3972 |
. . . . 5
⊢ ∅
⊆ (ℤ≥‘0) |
9 | 7, 8 | syl6eqss 3655 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 ⊆
(ℤ≥‘0)) |
10 | 2, 4, 6, 9 | sumss 14455 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
11 | 10 | ex 450 |
. 2
⊢ (𝜑 → (𝐵 = ∅ → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
12 | | cnvimass 5485 |
. . . . . . . . 9
⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓 |
13 | | simprr 796 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵) |
14 | | f1of 6137 |
. . . . . . . . . . 11
⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))⟶𝐵) |
15 | 13, 14 | syl 17 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))⟶𝐵) |
16 | | fdm 6051 |
. . . . . . . . . 10
⊢ (𝑓:(1...(#‘𝐵))⟶𝐵 → dom 𝑓 = (1...(#‘𝐵))) |
17 | 15, 16 | syl 17 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → dom 𝑓 = (1...(#‘𝐵))) |
18 | 12, 17 | syl5sseq 3653 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(#‘𝐵))) |
19 | | ffn 6045 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(#‘𝐵))⟶𝐵 → 𝑓 Fn (1...(#‘𝐵))) |
20 | 15, 19 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(#‘𝐵))) |
21 | | elpreima 6337 |
. . . . . . . . . . . 12
⊢ (𝑓 Fn (1...(#‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
23 | 15 | ffvelrnda 6359 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵) |
24 | 23 | ex 450 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(#‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵)) |
25 | 24 | adantrd 484 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(#‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
26 | 22, 25 | sylbid 230 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
27 | 26 | imp 445 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵) |
28 | 3 | ex 450 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
29 | 28 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
30 | | eldif 3584 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) |
31 | | 0cn 10032 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℂ |
32 | 5, 31 | syl6eqel 2709 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ) |
33 | 30, 32 | sylan2br 493 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ) |
34 | 33 | expr 643 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
35 | 29, 34 | pm2.61d 170 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
36 | | eqid 2622 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ 𝐵 ↦ 𝐶) = (𝑘 ∈ 𝐵 ↦ 𝐶) |
37 | 35, 36 | fmptd 6385 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ) |
38 | 37 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ) |
39 | 38 | ffvelrnda 6359 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
40 | 27, 39 | syldan 487 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
41 | | eldifi 3732 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(#‘𝐵))) |
42 | 41, 23 | sylan2 491 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵) |
43 | | eldifn 3733 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
44 | 43 | adantl 482 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
45 | 22 | adantr 481 |
. . . . . . . . . . . . 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))) |
46 | 41 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(#‘𝐵))) |
47 | 46 | biantrurd 529 |
. . . . . . . . . . . . 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))) |
48 | 45, 47 | bitr4d 271 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴)) |
49 | 44, 48 | mtbid 314 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴) |
50 | 42, 49 | eldifd 3585 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴)) |
51 | | difss 3737 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 |
52 | | resmpt 5449 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)) |
53 | 51, 52 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) |
54 | 53 | fveq1i 6192 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) |
55 | | fvres 6207 |
. . . . . . . . . . 11
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
56 | 54, 55 | syl5eqr 2670 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
57 | 50, 56 | 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))) |
58 | | c0ex 10034 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
59 | 58 | elsn2 4211 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ {0} ↔ 𝐶 = 0) |
60 | 5, 59 | sylibr 224 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {0}) |
61 | | eqid 2622 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) |
62 | 60, 61 | fmptd 6385 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0}) |
63 | 62 | ad2antrr 762 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0}) |
64 | 63, 50 | ffvelrnd 6360 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0}) |
65 | | elsni 4194 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 0) |
66 | 64, 65 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 0) |
67 | 57, 66 | eqtr3d 2658 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(#‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 0) |
68 | | fzssuz 12382 |
. . . . . . . . 9
⊢
(1...(#‘𝐵))
⊆ (ℤ≥‘1) |
69 | 68 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (1...(#‘𝐵)) ⊆
(ℤ≥‘1)) |
70 | 18, 40, 67, 69 | sumss 14455 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = Σ𝑛 ∈ (1...(#‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
71 | 1 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝐴 ⊆ 𝐵) |
72 | 71 | resmptd 5452 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
73 | 72 | fveq1d 6193 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) |
74 | | fvres 6207 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
75 | 74 | adantl 482 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
76 | 73, 75 | eqtr3d 2658 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
77 | 76 | sumeq2dv 14433 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
78 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
79 | | fzfid 12772 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (1...(#‘𝐵)) ∈ Fin) |
80 | 79, 15 | fisuppfi 8283 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin) |
81 | | f1of1 6136 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))–1-1→𝐵) |
82 | 13, 81 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–1-1→𝐵) |
83 | | f1ores 6151 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(#‘𝐵))–1-1→𝐵 ∧ (◡𝑓 “ 𝐴) ⊆ (1...(#‘𝐵))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
84 | 82, 18, 83 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
85 | | f1ofo 6144 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(#‘𝐵))–onto→𝐵) |
86 | 13, 85 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(#‘𝐵))–onto→𝐵) |
87 | 1 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵) |
88 | | foimacnv 6154 |
. . . . . . . . . . . 12
⊢ ((𝑓:(1...(#‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
89 | 86, 87, 88 | syl2anc 693 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
90 | | f1oeq3 6129 |
. . . . . . . . . . 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)) |
91 | 89, 90 | 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)) |
92 | 84, 91 | mpbid 222 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴) |
93 | | fvres 6207 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
94 | 93 | adantl 482 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
95 | 87 | sselda 3603 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵) |
96 | 38 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
97 | 95, 96 | syldan 487 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
98 | 78, 80, 92, 94, 97 | fsumf1o 14454 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
99 | 77, 98 | eqtrd 2656 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (◡𝑓 “ 𝐴)((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
100 | | eqidd 2623 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(#‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛)) |
101 | 78, 79, 13, 100, 96 | fsumf1o 14454 |
. . . . . . 7
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑛 ∈ (1...(#‘𝐵))((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
102 | 70, 99, 101 | 3eqtr4d 2666 |
. . . . . 6
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
103 | | sumfc 14440 |
. . . . . 6
⊢
Σ𝑚 ∈
𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶 |
104 | | sumfc 14440 |
. . . . . 6
⊢
Σ𝑚 ∈
𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶 |
105 | 102, 103,
104 | 3eqtr3g 2679 |
. . . . 5
⊢ ((𝜑 ∧ ((#‘𝐵) ∈ ℕ ∧ 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
106 | 105 | expr 643 |
. . . 4
⊢ ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
107 | 106 | exlimdv 1861 |
. . 3
⊢ ((𝜑 ∧ (#‘𝐵) ∈ ℕ) → (∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
108 | 107 | expimpd 629 |
. 2
⊢ (𝜑 → (((#‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
109 | | fsumss.4 |
. . 3
⊢ (𝜑 → 𝐵 ∈ Fin) |
110 | | fz1f1o 14441 |
. . 3
⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))) |
111 | 109, 110 | syl 17 |
. 2
⊢ (𝜑 → (𝐵 = ∅ ∨ ((#‘𝐵) ∈ ℕ ∧ ∃𝑓 𝑓:(1...(#‘𝐵))–1-1-onto→𝐵))) |
112 | 11, 108, 111 | mpjaod 396 |
1
⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |