Theorem sumeq2ii 15333

Description: Equality theorem for sum, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Mario Carneiro, 13-Jun-2019.)

Assertion

Ref	Expression
sumeq2ii	⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Distinct variable group:   𝐴,𝑘

Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem sumeq2ii

Dummy variables 𝑓 𝑚 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 484	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
2		simpr 484	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
3		simplll 771	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) ∧ 𝑛 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶))
4		nfcv 2906	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 I
5		nfcsb1v 3853	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
6	4, 5	nffv 6766	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐵)
7		nfcsb1v 3853	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐶
8	4, 7	nffv 6766	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐶)
9	6, 8	nfeq 2919	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)
10		csbeq1a 3842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
11	10	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ( I ‘𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐵))
12		csbeq1a 3842	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → 𝐶 = ⦋𝑛 / 𝑘⦌𝐶)
13	12	fveq2d 6760	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ( I ‘𝐶) = ( I ‘⦋𝑛 / 𝑘⦌𝐶))
14	11, 13	eqeq12d 2754	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (( I ‘𝐵) = ( I ‘𝐶) ↔ ( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)))
15	9, 14	rspc 3539	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)))
16	2, 3, 15	sylc 65	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) ∧ 𝑛 ∈ 𝐴) → ( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶))
17	16	ifeq1da 4487	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐵), ( I ‘0)) = if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐶), ( I ‘0)))
18		fvif 6772	. . . . . . . . . . . 12 ⊢ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐵), ( I ‘0))
19		fvif 6772	. . . . . . . . . . . 12 ⊢ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐶), ( I ‘0))
20	17, 18, 19	3eqtr4g 2804	. . . . . . . . . . 11 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
21	20	mpteq2dv 5172	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
22	21	fveq1d 6758	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))‘𝑥))
23		eqid 2738	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
24		eqid 2738	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))
25	23, 24	fvmptex 6871	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))‘𝑥)
26		eqid 2738	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
27		eqid 2738	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) = (𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))
28	26, 27	fvmptex 6871	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)))‘𝑥)
29	22, 25, 28	3eqtr4g 2804	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))‘𝑥))
30	1, 29	seqfeq 13676	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) = seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))))
31	30	breq1d 5080	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥 ↔ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥))
32	31	anbi2d 628	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
33	32	rexbidva 3224	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ seq𝑚( + , (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))) ⇝ 𝑥)))
34		simplr 765	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℕ)
35		nnuz 12550	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
36	34, 35	eleqtrdi 2849	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ (ℤ≥‘1))
37		f1of 6700	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
38	37	ad2antlr 723	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
39		ffvelrn 6941	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)⟶𝐴 ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
40	38, 39	sylancom 587	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
41		simplll 771	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶))
42		nfcsb1v 3853	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵)
43		nfcsb1v 3853	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
44	42, 43	nfeq 2919	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
45		csbeq1a 3842	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵))
46		csbeq1a 3842	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐶) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
47	45, 46	eqeq12d 2754	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
48	44, 47	rspc 3539	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
49	40, 41, 48	sylc 65	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
50		fvex 6769	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
51		csbfv2g 6800	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ V → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
52	50, 51	ax-mp 5	. . . . . . . . . . 11 ⊢ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
53		csbfv2g 6800	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ V → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
54	50, 53	ax-mp 5	. . . . . . . . . . 11 ⊢ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
55	49, 52, 54	3eqtr3g 2802	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
56		elfznn 13214	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
57	56	adantl 481	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
58		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (𝑓‘𝑛) = (𝑓‘𝑥))
59	58	csbeq1d 3832	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
60		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
61	59, 60	fvmpti 6856	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
62	57, 61	syl 17	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
63	58	csbeq1d 3832	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
64		eqid 2738	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
65	63, 64	fvmpti 6856	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
66	57, 65	syl 17	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
67	55, 62, 66	3eqtr4d 2788	. . . . . . . . 9 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥))
68	36, 67	seqfveq 13675	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))
69	68	eqeq2d 2749	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚) ↔ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))
70	69	pm5.32da 578	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
71	70	exbidv 1925	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
72	71	rexbidva 3224	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
73	33, 72	orbi12d 915	. . 3 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
74	73	iotabidv 6402	. 2 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
75		df-sum 15326	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
76		df-sum 15326	. 2 ⊢ Σ𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ 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 ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
77	74, 75, 76	3eqtr4g 2804	1 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422  ⦋csb 3828   ⊆ wss 3883  ifcif 4456   class class class wbr 5070   ↦ cmpt 5153   I cid 5479  ℩cio 6374  ⟶wf 6414  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  0cc0 10802  1c1 10803   + caddc 10805  ℕcn 11903  ℤcz 12249  ℤ_≥cuz 12511  ...cfz 13168  seqcseq 13649   ⇝ cli 15121  Σcsu 15325

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-om 7688  df-1st 7804  df-2nd 7805  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-er 8456  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-nn 11904  df-n0 12164  df-z 12250  df-uz 12512  df-fz 13169  df-seq 13650  df-sum 15326

This theorem is referenced by:  sumeq2  15334  sum2id  15348