Theorem sumeq2ii 15740

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 489	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
2		simpr 489	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) ∧ 𝑛 ∈ 𝐴) → 𝑛 ∈ 𝐴)
3		simplll 786	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) ∧ 𝑛 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶))
4		nfcv 2925	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘 I
5		nfcsb1v 3877	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐵
6	4, 5	nffv 6891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐵)
7		nfcsb1v 3877	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘⦋𝑛 / 𝑘⦌𝐶
8	4, 7	nffv 6891	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐶)
9	6, 8	nfeq 2938	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)
10		csbeq1a 3867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → 𝐵 = ⦋𝑛 / 𝑘⦌𝐵)
11	10	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ( I ‘𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐵))
12		csbeq1a 3867	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑛 → 𝐶 = ⦋𝑛 / 𝑘⦌𝐶)
13	12	fveq2d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑛 → ( I ‘𝐶) = ( I ‘⦋𝑛 / 𝑘⦌𝐶))
14	11, 13	eqeq12d 2779	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑛 → (( I ‘𝐵) = ( I ‘𝐶) ↔ ( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)))
15	9, 14	rspc 3569	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ( I ‘⦋𝑛 / 𝑘⦌𝐵) = ( I ‘⦋𝑛 / 𝑘⦌𝐶)))
16	2, 3, 15	sylc 66	. . . . . . . . . . . . 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 4519	. . . . . . . . . . . 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 6897	. . . . . . . . . . . 12 ⊢ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐵), ( I ‘0))
19		fvif 6897	. . . . . . . . . . . 12 ⊢ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = if(𝑛 ∈ 𝐴, ( I ‘⦋𝑛 / 𝑘⦌𝐶), ( I ‘0))
20	17, 18, 19	3eqtr4g 2823	. . . . . . . . . . 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 5205	. . . . . . . . . 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 6883	. . . . . . . . 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 2763	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))
24		eqid 2763	. . . . . . . . . 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 7004	. . . . . . . . 9 ⊢ ((𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0))‘𝑥) = ((𝑛 ∈ ℤ ↦ ( I ‘if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐵, 0)))‘𝑥)
26		eqid 2763	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0)) = (𝑛 ∈ ℤ ↦ if(𝑛 ∈ 𝐴, ⦋𝑛 / 𝑘⦌𝐶, 0))
27		eqid 2763	. . . . . . . . . 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 7004	. . . . . . . . 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 2823	. . . . . . . 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 14059	. . . . . . 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 5119	. . . . . 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 641	. . . . 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 3187	. . . 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 780	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℕ)
35		nnuz 12896	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
36	34, 35	eleqtrdi 2873	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ (ℤ≥‘1))
37		f1of 6820	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
38	37	ad2antlr 739	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
39		ffvelcdm 7076	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)⟶𝐴 ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
40	38, 39	sylancom 599	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
41		simplll 786	. . . . . . . . . . . 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 3877	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵)
43		nfcsb1v 3877	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
44	42, 43	nfeq 2938	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
45		csbeq1a 3867	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵))
46		csbeq1a 3867	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐶) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
47	45, 46	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
48	44, 47	rspc 3569	. . . . . . . . . . . 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 66	. . . . . . . . . . 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 6894	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
51		csbfv2g 6927	. . . . . . . . . . . 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 6927	. . . . . . . . . . . 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 2821	. . . . . . . . . 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 13577	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
57	56	adantl 486	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
58		fveq2 6881	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (𝑓‘𝑛) = (𝑓‘𝑥))
59	58	csbeq1d 3857	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
60		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
61	59, 60	fvmpti 6988	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
62	57, 61	syl 18	. . . . . . . . . 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 3857	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
64		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
65	63, 64	fvmpti 6988	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
66	57, 65	syl 18	. . . . . . . . . 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 2808	. . . . . . . . 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 14058	. . . . . . . 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 2774	. . . . . . 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 589	. . . . . 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 1951	. . . . 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 3187	. . . 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 931	. . 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 6520	. 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 15734	. 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 15734	. 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 2823	1 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → Σ𝑘 ∈ 𝐴 𝐵 = Σ𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570  ∃wex 1809   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3853   ⊆ wss 3905  ifcif 4487   class class class wbr 5109   ↦ cmpt 5192   I cid 5555  ℩cio 6490  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7410  0cc0 11095  1c1 11096   + caddc 11098  ℕcn 12228  ℤcz 12586  ℤ_≥cuz 12857  ...cfz 13530  seqcseq 14033   ⇝ cli 15531  Σcsu 15733

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-seq 14034  df-sum 15734

This theorem is referenced by:  sumeq2  15741  sum2id  15755