Theorem sumeq2 11380

Description: Equality theorem for sum. (Contributed by NM, 11-Dec-2005.) (Revised by Mario Carneiro, 13-Jul-2013.)

Assertion

Ref	Expression
sumeq2	⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . . . . . . . 12 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
2		simp-4l 541	. . . . . . . . . . . 12 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
3		nfcsb1v 3102	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
4		nfcsb1v 3102	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐶
5	3, 4	nfeq 2337	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶
6		csbeq1a 3078	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
7		csbeq1a 3078	. . . . . . . . . . . . . 14 ⊢ (𝑘 = 𝑛 → 𝐶 = ⦋𝑛 / 𝑘⦌𝐶)
8	6, 7	eqeq12d 2202	. . . . . . . . . . . . 13 ⊢ (𝑘 = 𝑛 → (𝐵 = 𝐶 ↔ ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶))
9	5, 8	rspc 2847	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶))
10	1, 2, 9	sylc 62	. . . . . . . . . . 11 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) ∧ 𝑛 ∈ 𝐴) → ⦋𝑛 / 𝑘⦌𝐵 = ⦋𝑛 / 𝑘⦌𝐶)
11		simpllr 534	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑚 ∈ ℤ)
12		simplrl 535	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑚))
13		simplrr 536	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
14		simpr 110	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → 𝑛 ∈ ℤ)
15	11, 12, 13, 14	sumdc 11379	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → DECID 𝑛 ∈ 𝐴)
16	10, 15	ifeq1dadc 3576	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ ℤ) → if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0) = if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
17	16	mpteq2dva 4105	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
18	17	seqeq3d 10466	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
19	18	breq1d 4025	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
20	19	pm5.32da 452	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
21		df-3an 981	. . . . . 6 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥))
22		df-3an 981	. . . . . 6 ⊢ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ↔ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
23	20, 21, 22	3bitr4g 223	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
24	23	rexbidva 2484	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
25		f1of 5473	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
26	25	ad3antlr 493	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑓:(1...𝑚)⟶𝐴)
27		simplr 528	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ∈ ℕ)
28		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ≤ 𝑚)
29		simp-4r 542	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℕ)
30	29	nnzd 9387	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℤ)
31		fznn 10102	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℤ → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑚)))
32	30, 31	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑚)))
33	27, 28, 32	mpbir2and 945	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ∈ (1...𝑚))
34	26, 33	ffvelcdmd 5665	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → (𝑓‘𝑛) ∈ 𝐴)
35		simp-4l 541	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
36		nfcsb1v 3102	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
37		nfcsb1v 3102	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐶
38	36, 37	nfeq 2337	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
39		csbeq1a 3078	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
40		csbeq1a 3078	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐶 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
41	39, 40	eqeq12d 2202	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑛) → (𝐵 = 𝐶 ↔ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
42	38, 41	rspc 2847	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
43	34, 35, 42	sylc 62	. . . . . . . . . . . 12 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
44		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
45	44	nnzd 9387	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
46		simpllr 534	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
47	46	nnzd 9387	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
48		zdcle 9342	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛 ≤ 𝑚)
49	45, 47, 48	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛 ≤ 𝑚)
50	43, 49	ifeq1dadc 3576	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0))
51	50	mpteq2dva 4105	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))
52	51	seqeq3d 10466	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))) = seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0))))
53	52	fveq1d 5529	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))
54	53	eqeq2d 2199	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))
55	54	pm5.32da 452	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
56	55	exbidv 1835	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
57	56	rexbidva 2484	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
58	24, 57	orbi12d 794	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))))
59	58	iotabidv 5211	. 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚)))))
60		df-sumdc 11375	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)))‘𝑚))))
61		df-sumdc 11375	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 0)))‘𝑚))))
62	59, 60, 61	3eqtr4g 2245	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   ∧ w3a 979   = wceq 1363  ∃wex 1502   ∈ wcel 2158  ∀wral 2465  ∃wrex 2466  ⦋csb 3069   ⊆ wss 3141  ifcif 3546   class class class wbr 4015   ↦ cmpt 4076  ℩cio 5188  ⟶wf 5224  –1-1-onto→wf1o 5227  ‘cfv 5228  (class class class)co 5888  0cc0 7824  1c1 7825   + caddc 7827   ≤ cle 8006  ℕcn 8932  ℤcz 9266  ℤ_≥cuz 9541  ...cfz 10021  seqcseq 10458   ⇝ cli 11299  Σcsu 11374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-ltadd 7940

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-f1o 5235  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-recs 6319  df-frec 6405  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-inn 8933  df-n0 9190  df-z 9267  df-uz 9542  df-fz 10022  df-seqfrec 10459  df-sumdc 11375

This theorem is referenced by:  sumeq2i  11385  sumeq2d  11388  fsum00  11483