Theorem prodeq2ii 15967

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 12873	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℤ)
2	1	adantl 486	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ ℤ)
3		nfra1 3289	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4		rsp 3253	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
5	4	adantr 485	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘 ∈ 𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
6		ifeq1 4492	. . . . . . . . . . . . . . . . . . . 20 ⊢ (( I ‘𝐵) = ( I ‘𝐶) → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)))
7	5, 6	syl6 36	. . . . . . . . . . . . . . . . . . 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 4497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = ( I ‘1))
9		iffalse 4497	. . . . . . . . . . . . . . . . . . . 20 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)) = ( I ‘1))
10	8, 9	eqtr4d 2801	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ 𝑘 ∈ 𝐴 → if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1)))
11	7, 10	pm2.61d1 182	. . . . . . . . . . . . . . . . . 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 6899	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1))
13		fvif 6899	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1))
14	11, 12, 13	3eqtr4g 2823	. . . . . . . . . . . . . . . . 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 5204	. . . . . . . . . . . . . . . 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 485	. . . . . . . . . . . . . . 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 6885	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 2763	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
20		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))
21	19, 20	fvmptex 7006	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥)
22		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))
23		eqid 2763	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))
24	22, 23	fvmptex 7006	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥)
25	18, 21, 24	3eqtr4g 2823	. . . . . . . . . . . 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 5120	. . . . . . . . . 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 641	. . . . . . . . 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 1951	. . . . . . . 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 3187	. . . . . . 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 485	. . . . . 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 489	. . . . . . . 8 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
33	15	adantr 485	. . . . . . . . . . 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 6885	. . . . . . . . . 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 2823	. . . . . . . . 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 727	. . . . . . . 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 5120	. . . . . 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 1467	. . . . 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 3187	. . . 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 780	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℕ)
42		nnuz 12902	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
43	41, 42	eleqtrdi 2873	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ (ℤ≥‘1))
44		f1of 6822	. . . . . . . . . . . . . 14 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
45	44	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)
46		ffvelcdm 7078	. . . . . . . . . . . . 13 ⊢ ((𝑓:(1...𝑚)⟶𝐴 ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
47	45, 46	sylancom 599	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓‘𝑥) ∈ 𝐴)
48		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))
49		nfcsb1v 3878	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵)
50		nfcsb1v 3878	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
51	49, 50	nfeq 2938	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
52		csbeq1a 3868	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵))
53		csbeq1a 3868	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐶) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
54	52, 53	eqeq12d 2779	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
55	51, 54	rspc 3570	. . . . . . . . . . . 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 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))
57		fvex 6896	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
58		csbfv2g 6929	. . . . . . . . . . . 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 6929	. . . . . . . . . . . 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 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))
63		elfznn 13583	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
64	63	adantl 486	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
65		fveq2 6883	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (𝑓‘𝑛) = (𝑓‘𝑥))
66	65	csbeq1d 3858	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
67		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
68	66, 67	fvmpti 6990	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐵))
69	64, 68	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))
70	65	csbeq1d 3858	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
71		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
72	70, 71	fvmpti 6990	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)‘𝑥) = ( I ‘⦋(𝑓‘𝑥) / 𝑘⦌𝐶))
73	64, 72	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))
74	62, 69, 73	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))
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 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)))
77	76	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))))
78	77	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))))
79	78	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))))
80	40, 79	orbi12d 931	. . 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 6522	. 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 15960	. 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 15960	. 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 2823	1 ⊢ (∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455  ⦋csb 3854   ⊆ wss 3906  ifcif 4488   class class class wbr 5110   ↦ cmpt 5193   I cid 5557  ℩cio 6492  ⟶wf 6534  –1-1-onto→wf1o 6537  ‘cfv 6538  (class class class)co 7412  0cc0 11101  1c1 11102   · cmul 11106  ℕcn 12234  ℤcz 12592  ℤ_≥cuz 12863  ...cfz 13536  seqcseq 14039   ⇝ cli 15537  ∏cprod 15959

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 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178

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 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-er 8695  df-en 8945  df-dom 8946  df-sdom 8947  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-nn 12235  df-n0 12506  df-z 12593  df-uz 12864  df-fz 13537  df-seq 14040  df-prod 15960

This theorem is referenced by:  prodeq2  15968  prod2id  15984