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Theorem sumeq2d 10397
Description: Equality theorem for sum. (Contributed by Jim Kingdon, 11-Feb-2022.)
Hypothesis
Ref Expression
sumeq2d.bc ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
Assertion
Ref Expression
sumeq2d (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Distinct variable groups:   𝐴,𝑘   𝜑,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem sumeq2d
Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → 𝑛𝐴)
2 sumeq2d.bc . . . . . . . . . . . . . 14 ((𝜑𝑘𝐴) → 𝐵 = 𝐶)
32ralrimiva 2439 . . . . . . . . . . . . 13 (𝜑 → ∀𝑘𝐴 𝐵 = 𝐶)
43ad4antr 478 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → ∀𝑘𝐴 𝐵 = 𝐶)
5 nfcsb1v 2947 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝐵
6 nfcsb1v 2947 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝐶
75, 6nfeq 2230 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶
8 csbeq1a 2925 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
9 csbeq1a 2925 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝐶 = 𝑛 / 𝑘𝐶)
108, 9eqeq12d 2097 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝐵 = 𝐶𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶))
117, 10rspc 2704 . . . . . . . . . . . 12 (𝑛𝐴 → (∀𝑘𝐴 𝐵 = 𝐶𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶))
121, 4, 11sylc 61 . . . . . . . . . . 11 (((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → 𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶)
13 simpllr 501 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑚 ∈ ℤ)
14 simplrl 502 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝐴 ⊆ (ℤ𝑚))
15 simplrr 503 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
16 simpr 108 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
1713, 14, 15, 16sumdc 10396 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → DECID 𝑛𝐴)
1812, 17ifeq1dadc 3396 . . . . . . . . . 10 ((((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
1918mpteq2dva 3888 . . . . . . . . 9 (((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
20 iseqeq3 9578 . . . . . . . . 9 ((𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ))
2119, 20syl 14 . . . . . . . 8 (((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ))
2221breq1d 3815 . . . . . . 7 (((𝜑𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥))
2322pm5.32da 440 . . . . . 6 ((𝜑𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥)))
24 df-3an 922 . . . . . 6 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥))
25 df-3an 922 . . . . . 6 ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥))
2623, 24, 253bitr4g 221 . . . . 5 ((𝜑𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥)))
2726rexbidva 2370 . . . 4 (𝜑 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥)))
28 f1of 5177 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
2928ad3antlr 477 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑓:(1...𝑚)⟶𝐴)
30 simplr 497 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ ℕ)
31 simpr 108 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛𝑚)
32 simp-4r 509 . . . . . . . . . . . . . . . . 17 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℕ)
3332nnzd 8601 . . . . . . . . . . . . . . . 16 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℤ)
34 fznn 9234 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℤ → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
3533, 34syl 14 . . . . . . . . . . . . . . 15 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
3630, 31, 35mpbir2and 886 . . . . . . . . . . . . . 14 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ (1...𝑚))
3729, 36ffvelrnd 5355 . . . . . . . . . . . . 13 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) ∈ 𝐴)
383ad4antr 478 . . . . . . . . . . . . 13 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → ∀𝑘𝐴 𝐵 = 𝐶)
39 nfcsb1v 2947 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐵
40 nfcsb1v 2947 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐶
4139, 40nfeq 2230 . . . . . . . . . . . . . 14 𝑘(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶
42 csbeq1a 2925 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐵 = (𝑓𝑛) / 𝑘𝐵)
43 csbeq1a 2925 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐶 = (𝑓𝑛) / 𝑘𝐶)
4442, 43eqeq12d 2097 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑛) → (𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
4541, 44rspc 2704 . . . . . . . . . . . . 13 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
4637, 38, 45sylc 61 . . . . . . . . . . . 12 (((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶)
47 simpr 108 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4847nnzd 8601 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
49 simpllr 501 . . . . . . . . . . . . . 14 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
5049nnzd 8601 . . . . . . . . . . . . 13 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
51 zdcle 8557 . . . . . . . . . . . . 13 ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛𝑚)
5248, 50, 51syl2anc 403 . . . . . . . . . . . 12 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛𝑚)
5346, 52ifeq1dadc 3396 . . . . . . . . . . 11 ((((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0))
5453mpteq2dva 3888 . . . . . . . . . 10 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))
55 iseqeq3 9578 . . . . . . . . . 10 ((𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ))
5654, 55syl 14 . . . . . . . . 9 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ))
5756fveq1d 5231 . . . . . . . 8 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))
5857eqeq2d 2094 . . . . . . 7 (((𝜑𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))
5958pm5.32da 440 . . . . . 6 ((𝜑𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
6059exbidv 1748 . . . . 5 ((𝜑𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
6160rexbidva 2370 . . . 4 (𝜑 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
6227, 61orbi12d 740 . . 3 (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))))
6362iotabidv 4938 . 2 (𝜑 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚)))))
64 df-isum 10392 . 2 Σ𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)), ℂ)‘𝑚))))
65 df-isum 10392 . 2 Σ𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)), ℂ) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)), ℂ)‘𝑚))))
6663, 64, 653eqtr4g 2140 1 (𝜑 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  DECID wdc 776  w3a 920   = wceq 1285  wex 1422  wcel 1434  wral 2353  wrex 2354  csb 2917  wss 2982  ifcif 3368   class class class wbr 3805  cmpt 3859  cio 4915  wf 4948  1-1-ontowf1o 4951  cfv 4952  (class class class)co 5563  cc 7093  0cc0 7095  1c1 7096   + caddc 7098  cle 7268  cn 8158  cz 8484  cuz 8752  ...cfz 9157  seqcseq 9573  cli 10318  Σcsu 10391
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-addcom 7190  ax-addass 7192  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-0id 7198  ax-rnegex 7199  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-ltadd 7206
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-if 3369  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-int 3657  df-br 3806  df-opab 3860  df-mpt 3861  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-f1o 4959  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-recs 5974  df-frec 6060  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-inn 8159  df-n0 8408  df-z 8485  df-uz 8753  df-fz 9158  df-iseq 9574  df-isum 10392
This theorem is referenced by: (None)
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