ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sumeq2 GIF version

Theorem sumeq2 10585
Description: Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)
Assertion
Ref Expression
sumeq2 (∀𝑘𝐴 𝐵 = 𝐶 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Distinct variable group:   𝐴,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem sumeq2
Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpr 108 . . . . . . . . . . . 12 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → 𝑛𝐴)
2 simp-4l 508 . . . . . . . . . . . 12 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → ∀𝑘𝐴 𝐵 = 𝐶)
3 nfcsb1v 2951 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝐵
4 nfcsb1v 2951 . . . . . . . . . . . . . 14 𝑘𝑛 / 𝑘𝐶
53, 4nfeq 2232 . . . . . . . . . . . . 13 𝑘𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶
6 csbeq1a 2929 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝐵 = 𝑛 / 𝑘𝐵)
7 csbeq1a 2929 . . . . . . . . . . . . . 14 (𝑘 = 𝑛𝐶 = 𝑛 / 𝑘𝐶)
86, 7eqeq12d 2099 . . . . . . . . . . . . 13 (𝑘 = 𝑛 → (𝐵 = 𝐶𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶))
95, 8rspc 2708 . . . . . . . . . . . 12 (𝑛𝐴 → (∀𝑘𝐴 𝐵 = 𝐶𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶))
101, 2, 9sylc 61 . . . . . . . . . . 11 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛𝐴) → 𝑛 / 𝑘𝐵 = 𝑛 / 𝑘𝐶)
11 simpllr 501 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑚 ∈ ℤ)
12 simplrl 502 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝐴 ⊆ (ℤ𝑚))
13 simplrr 503 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
14 simpr 108 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
1511, 12, 13, 14sumdc 10583 . . . . . . . . . . 11 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → DECID 𝑛𝐴)
1610, 15ifeq1dadc 3404 . . . . . . . . . 10 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ ℤ) → if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0) = if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0))
1716mpteq2dva 3897 . . . . . . . . 9 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛𝐴, 𝑛 / 𝑘𝐶, 0)))
18 iseqeq3 9759 . . . . . . . . 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))
1917, 18syl 14 . . . . . . . 8 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))
2019breq1d 3824 . . . . . . 7 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))
2120pm5.32da 440 . . . . . 6 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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)))
22 df-3an 924 . . . . . 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))
23 df-3an 924 . . . . . 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))
2421, 22, 233bitr4g 221 . . . . 5 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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)))
2524rexbidva 2373 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (∃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)))
26 f1of 5204 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
2726ad3antlr 477 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑓:(1...𝑚)⟶𝐴)
28 simplr 497 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ ℕ)
29 simpr 108 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛𝑚)
30 simp-4r 509 . . . . . . . . . . . . . . . . 17 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℕ)
3130nnzd 8777 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℤ)
32 fznn 9410 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℤ → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
3331, 32syl 14 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
3428, 29, 33mpbir2and 888 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ (1...𝑚))
3527, 34ffvelrnd 5383 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) ∈ 𝐴)
36 simp-4l 508 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → ∀𝑘𝐴 𝐵 = 𝐶)
37 nfcsb1v 2951 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐵
38 nfcsb1v 2951 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐶
3937, 38nfeq 2232 . . . . . . . . . . . . . 14 𝑘(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶
40 csbeq1a 2929 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐵 = (𝑓𝑛) / 𝑘𝐵)
41 csbeq1a 2929 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐶 = (𝑓𝑛) / 𝑘𝐶)
4240, 41eqeq12d 2099 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑛) → (𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
4339, 42rspc 2708 . . . . . . . . . . . . 13 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
4435, 36, 43sylc 61 . . . . . . . . . . . 12 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶)
45 simpr 108 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
4645nnzd 8777 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
47 simpllr 501 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
4847nnzd 8777 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
49 zdcle 8733 . . . . . . . . . . . . 13 ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛𝑚)
5046, 48, 49syl2anc 403 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛𝑚)
5144, 50ifeq1dadc 3404 . . . . . . . . . . 11 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0))
5251mpteq2dva 3897 . . . . . . . . . 10 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 0)))
53 iseqeq3 9759 . . . . . . . . . 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))
5452, 53syl 14 . . . . . . . . 9 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))
5554fveq1d 5258 . . . . . . . 8 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))
5655eqeq2d 2096 . . . . . . 7 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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)))
5756pm5.32da 440 . . . . . 6 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))))
5857exbidv 1750 . . . . 5 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ 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))))
5958rexbidva 2373 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (∃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))))
6025, 59orbi12d 740 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((∃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)))))
6160iotabidv 4958 . 2 (∀𝑘𝐴 𝐵 = 𝐶 → (℩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)))))
62 df-isum 10579 . 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))))
63 df-isum 10579 . 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))))
6461, 62, 633eqtr4g 2142 1 (∀𝑘𝐴 𝐵 = 𝐶 → Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wo 662  DECID wdc 778  w3a 922   = wceq 1287  wex 1424  wcel 1436  wral 2355  wrex 2356  csb 2921  wss 2986  ifcif 3376   class class class wbr 3814  cmpt 3868  cio 4935  wf 4968  1-1-ontowf1o 4971  cfv 4972  (class class class)co 5594  cc 7269  0cc0 7271  1c1 7272   + caddc 7274  cle 7444  cn 8334  cz 8660  cuz 8928  ...cfz 9333  seqcseq 9754  cli 10505  Σcsu 10578
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3925  ax-pow 3977  ax-pr 4003  ax-un 4227  ax-setind 4319  ax-cnex 7357  ax-resscn 7358  ax-1cn 7359  ax-1re 7360  ax-icn 7361  ax-addcl 7362  ax-addrcl 7363  ax-mulcl 7364  ax-addcom 7366  ax-addass 7368  ax-distr 7370  ax-i2m1 7371  ax-0lt1 7372  ax-0id 7374  ax-rnegex 7375  ax-cnre 7377  ax-pre-ltirr 7378  ax-pre-ltwlin 7379  ax-pre-lttrn 7380  ax-pre-ltadd 7382
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2616  df-sbc 2829  df-csb 2922  df-dif 2988  df-un 2990  df-in 2992  df-ss 2999  df-if 3377  df-pw 3411  df-sn 3431  df-pr 3432  df-op 3434  df-uni 3631  df-int 3666  df-br 3815  df-opab 3869  df-mpt 3870  df-id 4087  df-xp 4410  df-rel 4411  df-cnv 4412  df-co 4413  df-dm 4414  df-rn 4415  df-res 4416  df-ima 4417  df-iota 4937  df-fun 4974  df-fn 4975  df-f 4976  df-f1 4977  df-f1o 4979  df-fv 4980  df-riota 5550  df-ov 5597  df-oprab 5598  df-mpt2 5599  df-recs 6005  df-frec 6091  df-pnf 7445  df-mnf 7446  df-xr 7447  df-ltxr 7448  df-le 7449  df-sub 7576  df-neg 7577  df-inn 8335  df-n0 8584  df-z 8661  df-uz 8929  df-fz 9334  df-iseq 9755  df-isum 10579
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator