Theorem prodeq2ii 15877

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 12803	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (ℤ≥‘𝑚) → 𝑛 ∈ ℤ)
2	1	adantl 481	. . . . . . . . . . . 12 ⊢ ((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → 𝑛 ∈ ℤ)
3		nfra1 3261	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4		rsp 3225	. . . . . . . . . . . . . . . . . . . . 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 4492	. . . . . . . . . . . . . . . . . . . 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 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 2767	. . . . . . . . . . . . . . . . . . 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 6874	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐵), ( I ‘1))
13		fvif 6874	. . . . . . . . . . . . . . . . . 18 ⊢ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)) = if(𝑘 ∈ 𝐴, ( I ‘𝐶), ( I ‘1))
14	11, 12, 13	3eqtr4g 2789	. . . . . . . . . . . . . . . . 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 5199	. . . . . . . . . . . . . . . 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 6860	. . . . . . . . . . . . . 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 2729	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))
20		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))
21	19, 20	fvmptex 6982	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐵, 1)))‘𝑥)
22		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))
23		eqid 2729	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))
24	22, 23	fvmptex 6982	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘 ∈ 𝐴, 𝐶, 1)))‘𝑥)
25	18, 21, 24	3eqtr4g 2789	. . . . . . . . . . . 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 13992	. . . . . . . . . . 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 5117	. . . . . . . . . 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 1921	. . . . . . . 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 3155	. . . . . . 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 6860	. . . . . . . . . 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 2789	. . . . . . . . 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 13992	. . . . . . 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 5117	. . . . . 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 1441	. . . . 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 3155	. . . 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 768	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ ℕ)
42		nnuz 12836	. . . . . . . . . 10 ⊢ ℕ = (ℤ≥‘1)
43	41, 42	eleqtrdi 2838	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → 𝑚 ∈ (ℤ≥‘1))
44		f1of 6800	. . . . . . . . . . . . . 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 7053	. . . . . . . . . . . . 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 774	. . . . . . . . . . . 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 3886	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵)
50		nfcsb1v 3886	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
51	49, 50	nfeq 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)
52		csbeq1a 3876	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵))
53		csbeq1a 3876	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑥) → ( I ‘𝐶) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶))
54	52, 53	eqeq12d 2745	. . . . . . . . . . . . 13 ⊢ (𝑘 = (𝑓‘𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐵) = ⦋(𝑓‘𝑥) / 𝑘⦌( I ‘𝐶)))
55	51, 54	rspc 3576	. . . . . . . . . . . 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 6871	. . . . . . . . . . . 12 ⊢ (𝑓‘𝑥) ∈ V
58		csbfv2g 6907	. . . . . . . . . . . 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 6907	. . . . . . . . . . . 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 2787	. . . . . . . . . 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 13514	. . . . . . . . . . . 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 6858	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑥 → (𝑓‘𝑛) = (𝑓‘𝑥))
66	65	csbeq1d 3866	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐵)
67		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
68	66, 67	fvmpti 6967	. . . . . . . . . . 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 3866	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑥 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐶 = ⦋(𝑓‘𝑥) / 𝑘⦌𝐶)
71		eqid 2729	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶) = (𝑛 ∈ ℕ ↦ ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
72	70, 71	fvmpti 6967	. . . . . . . . . . 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 2774	. . . . . . . . 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 13991	. . . . . . . 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 2740	. . . . . . 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 1921	. . . . 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 3155	. . . 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 6495	. 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 15870	. 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 15870	. 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 2789	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 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2925  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ⦋csb 3862   ⊆ wss 3914  ifcif 4488   class class class wbr 5107   ↦ cmpt 5188   I cid 5532  ℩cio 6462  ⟶wf 6507  –1-1-onto→wf1o 6510  ‘cfv 6511  (class class class)co 7387  0cc0 11068  1c1 11069   · cmul 11073  ℕcn 12186  ℤcz 12529  ℤ_≥cuz 12793  ...cfz 13468  seqcseq 13966   ⇝ cli 15450  ∏cprod 15869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125  ax-1cn 11126  ax-icn 11127  ax-addcl 11128  ax-addrcl 11129  ax-mulcl 11130  ax-mulrcl 11131  ax-mulcom 11132  ax-addass 11133  ax-mulass 11134  ax-distr 11135  ax-i2m1 11136  ax-1ne0 11137  ax-1rid 11138  ax-rnegex 11139  ax-rrecex 11140  ax-cnre 11141  ax-pre-lttri 11142  ax-pre-lttrn 11143  ax-pre-ltadd 11144  ax-pre-mulgt0 11145

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-tr 5215  df-id 5533  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6274  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-om 7843  df-1st 7968  df-2nd 7969  df-frecs 8260  df-wrecs 8291  df-recs 8340  df-rdg 8378  df-er 8671  df-en 8919  df-dom 8920  df-sdom 8921  df-pnf 11210  df-mnf 11211  df-xr 11212  df-ltxr 11213  df-le 11214  df-sub 11407  df-neg 11408  df-nn 12187  df-n0 12443  df-z 12530  df-uz 12794  df-fz 13469  df-seq 13967  df-prod 15870

This theorem is referenced by:  prodeq2  15878  prod2id  15894