Theorem prodeq2ii 15944

Description: Equality theorem for product, with the class expressions 𝐵 and 𝐶 guarded by I to be always sets. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
prodeq2ii	⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)

Distinct variable group:   𝐴,𝑘

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

Proof of Theorem prodeq2ii

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

Step	Hyp	Ref	Expression
1		eluzelz 12886	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℤ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ ℤ)
3		nfra1 3282	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4		rsp 3245	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
5	4	adantr 480	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
6		ifeq1 4535	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( I ‘𝐵) = ( I ‘𝐶) → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)))
7	5, 6	syl6 35	. . . . . . . . . . . . . . . . . . 19 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1))))
8		iffalse 4540	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = ( I ‘1))
9		iffalse 4540	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)) = ( I ‘1))
10	8, 9	eqtr4d 2778	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)))
11	7, 10	pm2.61d1 180	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)))
12		fvif 6923	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1))
13		fvif 6923	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1))
14	11, 12, 13	3eqtr4g 2800	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)) = ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))
15	3, 14	mpteq2da 5246	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))))
16	15	adantr 480	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ≥‘𝑛)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))))
17	16	fveq1d 6909	. . . . . . . . . . . . . 14 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ≥‘𝑛)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥))
18	17	adantlr 715	. . . . . . . . . . . . 13 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ 𝑥 ∈ (ℤ≥‘𝑛)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥))
19		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
20		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))
21	19, 20	fvmptex 7030	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥)
22		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))
23		eqid 2735	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))
24	22, 23	fvmptex 7030	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥)
25	18, 21, 24	3eqtr4g 2800	. . . . . . . . . . . 12 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ 𝑥 ∈ (ℤ≥‘𝑛)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥))
26	2, 25	seqfeq 14065	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
27	26	breq1d 5158	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
28	27	anbi2d 630	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
29	28	exbidv 1919	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
30	29	rexbidva 3175	. . . . . . 7 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
31	30	adantr 480	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
32		simpr 484	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
33	15	adantr 480	. . . . . . . . . . 11 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))))
34	33	fveq1d 6909	. . . . . . . . . 10 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥))
35	34, 21, 24	3eqtr4g 2800	. . . . . . . . 9 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥))
36	35	adantlr 715	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ≥‘𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥))
37	32, 36	seqfeq 14065	. . . . . . 7 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
38	37	breq1d 5158	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))
39	31, 38	3anbi23d 1438	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)))
40	39	rexbidva 3175	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)))
41		simplr 769	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℕ)
42		nnuz 12919	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
43	41, 42	eleqtrdi 2849	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ (ℤ≥‘1))
44		f1of 6849	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
45	44	ad2antlr 727	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
46		ffvelcdm 7101	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)⟶𝐴 ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
47	45, 46	sylancom 588	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
48		simplll 775	. . . . . . . . . . . 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))
49		nfcsb1v 3933	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵)
50		nfcsb1v 3933	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
51	49, 50	nfeq 2917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
52		csbeq1a 3922	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵))
53		csbeq1a 3922	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐶) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
54	52, 53	eqeq12d 2751	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
55	51, 54	rspc 3610	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
56	47, 48, 55	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))
57		fvex 6920	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
58		csbfv2g 6956	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ V → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
59	57, 58	ax-mp 5	. . . . . . . . . . 11 ⊢ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
60		csbfv2g 6956	. . . . . . . . . . . 12 ⊢ ((𝑓‘𝑥) ∈ V → ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
61	57, 60	ax-mp 5	. . . . . . . . . . 11 ⊢ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
62	56, 59, 61	3eqtr3g 2798	. . . . . . . . . 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))
63		elfznn 13590	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
64	63	adantl 481	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
65		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (𝑓‘𝑛) = (𝑓‘𝑥))
66	65	csbeq1d 3912	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
67		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
68	66, 67	fvmpti 7015	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
69	64, 68	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))
70	65	csbeq1d 3912	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
71		eqid 2735	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
72	70, 71	fvmpti 7015	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
73	64, 72	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))
74	62, 69, 73	3eqtr4d 2785	. . . . . . . . 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))
75	43, 74	seqfveq 14064	. . . . . . . 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))
76	75	eqeq2d 2746	. . . . . . 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)))
77	76	pm5.32da 579	. . . . . 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))))
78	77	exbidv 1919	. . . . 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))))
79	78	rexbidva 3175	. . . 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))))
80	40, 79	orbi12d 918	. . 3 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 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) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
81	80	iotabidv 6547	. 2 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 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) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚)))))
82		df-prod 15937	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵))‘𝑚))))
83		df-prod 15937	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))‘𝑚))))
84	81, 82, 83	3eqtr4g 2800	1 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  Vcvv 3478  ⦋csb 3908   ⊆ wss 3963  ifcif 4531   class class class wbr 5148   ↦ cmpt 5231   I cid 5582  ℩cio 6514  ⟶wf 6559  –1-1-onto→wf1o 6562  ‘cfv 6563  (class class class)co 7431  0cc0 11153  1c1 11154   · cmul 11158  ℕcn 12264  ℤcz 12611  ℤ_≥cuz 12876  ...cfz 13544  seqcseq 14039   ⇝ cli 15517  ∏cprod 15936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-prod 15937

This theorem is referenced by:  prodeq2  15945  prod2id  15961