| Step | Hyp | Ref
| Expression |
| 1 | | sumss.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
| 2 | 1 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐴 ⊆ 𝐵) |
| 3 | | sumss.2 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 4 | 3 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝐵 = ∅) ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 5 | | sumss.3 |
. . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
| 6 | 5 | adantlr 715 |
. . . 4
⊢ (((𝜑 ∧ 𝐵 = ∅) ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 = 0) |
| 7 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 = ∅) |
| 8 | | 0ss 4400 |
. . . . 5
⊢ ∅
⊆ (ℤ≥‘0) |
| 9 | 7, 8 | eqsstrdi 4028 |
. . . 4
⊢ ((𝜑 ∧ 𝐵 = ∅) → 𝐵 ⊆
(ℤ≥‘0)) |
| 10 | 2, 4, 6, 9 | sumss 15760 |
. . 3
⊢ ((𝜑 ∧ 𝐵 = ∅) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
| 11 | 10 | ex 412 |
. 2
⊢ (𝜑 → (𝐵 = ∅ → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
| 12 | | cnvimass 6100 |
. . . . . . . . 9
⊢ (◡𝑓 “ 𝐴) ⊆ dom 𝑓 |
| 13 | | simprr 773 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) |
| 14 | | f1of 6848 |
. . . . . . . . . 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 6755 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) |
| 17 | 15 | ffnd 6737 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓 Fn (1...(♯‘𝐵))) |
| 18 | | elpreima 7078 |
. . . . . . . . . . . 12
⊢ (𝑓 Fn (1...(♯‘𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
| 19 | 17, 18 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴))) |
| 20 | 15 | ffvelcdmda 7104 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) ∈ 𝐵) |
| 21 | 20 | ex 412 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (1...(♯‘𝐵)) → (𝑓‘𝑛) ∈ 𝐵)) |
| 22 | 21 | adantrd 491 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → ((𝑛 ∈ (1...(♯‘𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
| 23 | 19, 22 | sylbid 240 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑛 ∈ (◡𝑓 “ 𝐴) → (𝑓‘𝑛) ∈ 𝐵)) |
| 24 | 23 | imp 406 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → (𝑓‘𝑛) ∈ 𝐵) |
| 25 | 3 | ex 412 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
| 26 | 25 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
| 27 | | eldif 3961 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ (𝐵 ∖ 𝐴) ↔ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) |
| 28 | | 0cn 11253 |
. . . . . . . . . . . . . . . 16
⊢ 0 ∈
ℂ |
| 29 | 5, 28 | eqeltrdi 2849 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ ℂ) |
| 30 | 27, 29 | sylan2br 595 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑘 ∈ 𝐵 ∧ ¬ 𝑘 ∈ 𝐴)) → 𝐶 ∈ ℂ) |
| 31 | 30 | expr 456 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → (¬ 𝑘 ∈ 𝐴 → 𝐶 ∈ ℂ)) |
| 32 | 26, 31 | pm2.61d 179 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵) → 𝐶 ∈ ℂ) |
| 33 | 32 | fmpttd 7135 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ) |
| 34 | 33 | adantr 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑘 ∈ 𝐵 ↦ 𝐶):𝐵⟶ℂ) |
| 35 | 34 | ffvelcdmda 7104 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ (𝑓‘𝑛) ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
| 36 | 24, 35 | syldan 591 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) ∈ ℂ) |
| 37 | | eldifi 4131 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈
((1...(♯‘𝐵))
∖ (◡𝑓 “ 𝐴)) → 𝑛 ∈ (1...(♯‘𝐵))) |
| 38 | 37, 20 | sylan2 593 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ 𝐵) |
| 39 | | eldifn 4132 |
. . . . . . . . . . . . 13
⊢ (𝑛 ∈
((1...(♯‘𝐵))
∖ (◡𝑓 “ 𝐴)) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
| 40 | 39 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ 𝑛 ∈ (◡𝑓 “ 𝐴)) |
| 41 | 37 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → 𝑛 ∈ (1...(♯‘𝐵))) |
| 42 | 19 | adantr 480 |
. . . . . . . . . . . . 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))) |
| 43 | 41, 42 | mpbirand 707 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑛 ∈ (◡𝑓 “ 𝐴) ↔ (𝑓‘𝑛) ∈ 𝐴)) |
| 44 | 40, 43 | mtbid 324 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ¬ (𝑓‘𝑛) ∈ 𝐴) |
| 45 | 38, 44 | eldifd 3962 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴)) |
| 46 | | difss 4136 |
. . . . . . . . . . . . 13
⊢ (𝐵 ∖ 𝐴) ⊆ 𝐵 |
| 47 | | resmpt 6055 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∖ 𝐴) ⊆ 𝐵 → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)) |
| 48 | 46, 47 | ax-mp 5 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴)) = (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶) |
| 49 | 48 | fveq1i 6907 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) |
| 50 | | fvres 6925 |
. . . . . . . . . . 11
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ (𝐵 ∖ 𝐴))‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
| 51 | 49, 50 | eqtr3id 2791 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑛) ∈ (𝐵 ∖ 𝐴) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
| 52 | 45, 51 | 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))) |
| 53 | | c0ex 11255 |
. . . . . . . . . . . . . . 15
⊢ 0 ∈
V |
| 54 | 53 | elsn2 4665 |
. . . . . . . . . . . . . 14
⊢ (𝐶 ∈ {0} ↔ 𝐶 = 0) |
| 55 | 5, 54 | sylibr 234 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝐵 ∖ 𝐴)) → 𝐶 ∈ {0}) |
| 56 | 55 | fmpttd 7135 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0}) |
| 57 | 56 | ad2antrr 726 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → (𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶):(𝐵 ∖ 𝐴)⟶{0}) |
| 58 | 57, 45 | ffvelcdmd 7105 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0}) |
| 59 | | elsni 4643 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) ∈ {0} → ((𝑘 ∈ (𝐵 ∖ 𝐴) ↦ 𝐶)‘(𝑓‘𝑛)) = 0) |
| 60 | 58, 59 | 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) |
| 61 | 52, 60 | eqtr3d 2779 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ ((1...(♯‘𝐵)) ∖ (◡𝑓 “ 𝐴))) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛)) = 0) |
| 62 | | fzssuz 13605 |
. . . . . . . . 9
⊢
(1...(♯‘𝐵)) ⊆
(ℤ≥‘1) |
| 63 | 62 | a1i 11 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) →
(1...(♯‘𝐵))
⊆ (ℤ≥‘1)) |
| 64 | 16, 36, 61, 63 | sumss 15760 |
. . . . . . 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))) |
| 65 | 1 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝐴 ⊆ 𝐵) |
| 66 | 65 | resmptd 6058 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴) = (𝑘 ∈ 𝐴 ↦ 𝐶)) |
| 67 | 66 | fveq1d 6908 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚)) |
| 68 | | fvres 6925 |
. . . . . . . . . . 11
⊢ (𝑚 ∈ 𝐴 → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
| 69 | 68 | adantl 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → (((𝑘 ∈ 𝐵 ↦ 𝐶) ↾ 𝐴)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
| 70 | 67, 69 | eqtr3d 2779 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
| 71 | 70 | sumeq2dv 15738 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
| 72 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑚 = (𝑓‘𝑛) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = ((𝑘 ∈ 𝐵 ↦ 𝐶)‘(𝑓‘𝑛))) |
| 73 | | fzfid 14014 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) →
(1...(♯‘𝐵))
∈ Fin) |
| 74 | 73, 15 | fisuppfi 9411 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (◡𝑓 “ 𝐴) ∈ Fin) |
| 75 | | f1of1 6847 |
. . . . . . . . . . . 12
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–1-1→𝐵) |
| 76 | 13, 75 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–1-1→𝐵) |
| 77 | | f1ores 6862 |
. . . . . . . . . . 11
⊢ ((𝑓:(1...(♯‘𝐵))–1-1→𝐵 ∧ (◡𝑓 “ 𝐴) ⊆ (1...(♯‘𝐵))) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
| 78 | 76, 16, 77 | syl2anc 584 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→(𝑓 “ (◡𝑓 “ 𝐴))) |
| 79 | | f1ofo 6855 |
. . . . . . . . . . . . 13
⊢ (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → 𝑓:(1...(♯‘𝐵))–onto→𝐵) |
| 80 | 13, 79 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝑓:(1...(♯‘𝐵))–onto→𝐵) |
| 81 | 1 | adantr 480 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → 𝐴 ⊆ 𝐵) |
| 82 | | foimacnv 6865 |
. . . . . . . . . . . 12
⊢ ((𝑓:(1...(♯‘𝐵))–onto→𝐵 ∧ 𝐴 ⊆ 𝐵) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
| 83 | 80, 81, 82 | syl2anc 584 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 “ (◡𝑓 “ 𝐴)) = 𝐴) |
| 84 | 83 | f1oeq3d 6845 |
. . . . . . . . . 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)) |
| 85 | 78, 84 | mpbid 232 |
. . . . . . . . 9
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → (𝑓 ↾ (◡𝑓 “ 𝐴)):(◡𝑓 “ 𝐴)–1-1-onto→𝐴) |
| 86 | | fvres 6925 |
. . . . . . . . . 10
⊢ (𝑛 ∈ (◡𝑓 “ 𝐴) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
| 87 | 86 | adantl 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (◡𝑓 “ 𝐴)) → ((𝑓 ↾ (◡𝑓 “ 𝐴))‘𝑛) = (𝑓‘𝑛)) |
| 88 | 81 | sselda 3983 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → 𝑚 ∈ 𝐵) |
| 89 | 34 | ffvelcdmda 7104 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐵) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
| 90 | 88, 89 | syldan 591 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑚 ∈ 𝐴) → ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) ∈ ℂ) |
| 91 | 72, 74, 85, 87, 90 | fsumf1o 15759 |
. . . . . . . 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))) |
| 92 | 71, 91 | eqtrd 2777 |
. . . . . . 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))) |
| 93 | | eqidd 2738 |
. . . . . . . 8
⊢ (((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) ∧ 𝑛 ∈ (1...(♯‘𝐵))) → (𝑓‘𝑛) = (𝑓‘𝑛)) |
| 94 | 72, 73, 13, 93, 89 | fsumf1o 15759 |
. . . . . . 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))) |
| 95 | 64, 92, 94 | 3eqtr4d 2787 |
. . . . . 6
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑚 ∈ 𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑚 ∈ 𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚)) |
| 96 | | sumfc 15745 |
. . . . . 6
⊢
Σ𝑚 ∈
𝐴 ((𝑘 ∈ 𝐴 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐴 𝐶 |
| 97 | | sumfc 15745 |
. . . . . 6
⊢
Σ𝑚 ∈
𝐵 ((𝑘 ∈ 𝐵 ↦ 𝐶)‘𝑚) = Σ𝑘 ∈ 𝐵 𝐶 |
| 98 | 95, 96, 97 | 3eqtr3g 2800 |
. . . . 5
⊢ ((𝜑 ∧ ((♯‘𝐵) ∈ ℕ ∧ 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵)) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |
| 99 | 98 | expr 456 |
. . . 4
⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) → (𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
| 100 | 99 | exlimdv 1933 |
. . 3
⊢ ((𝜑 ∧ (♯‘𝐵) ∈ ℕ) →
(∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
| 101 | 100 | expimpd 453 |
. 2
⊢ (𝜑 → (((♯‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵) → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶)) |
| 102 | | fsumss.4 |
. . 3
⊢ (𝜑 → 𝐵 ∈ Fin) |
| 103 | | fz1f1o 15746 |
. . 3
⊢ (𝐵 ∈ Fin → (𝐵 = ∅ ∨
((♯‘𝐵) ∈
ℕ ∧ ∃𝑓
𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵))) |
| 104 | 102, 103 | syl 17 |
. 2
⊢ (𝜑 → (𝐵 = ∅ ∨ ((♯‘𝐵) ∈ ℕ ∧
∃𝑓 𝑓:(1...(♯‘𝐵))–1-1-onto→𝐵))) |
| 105 | 11, 101, 104 | mpjaod 861 |
1
⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 = Σ𝑘 ∈ 𝐵 𝐶) |