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Theorem prodeq2ii 14867
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 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 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.
StepHypRef Expression
1 eluzelz 11917 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑚) → 𝑛 ∈ ℤ)
21adantl 469 . . . . . . . . . . . 12 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ ℤ)
3 nfra1 3136 . . . . . . . . . . . . . . . . 17 𝑘𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4 rsp 3124 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
54adantr 468 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
6 ifeq1 4290 . . . . . . . . . . . . . . . . . . . 20 (( I ‘𝐵) = ( I ‘𝐶) → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
75, 6syl6 35 . . . . . . . . . . . . . . . . . . 19 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1))))
8 iffalse 4295 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = ( I ‘1))
9 iffalse 4295 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)) = ( I ‘1))
108, 9eqtr4d 2850 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
117, 10pm2.61d1 172 . . . . . . . . . . . . . . . . . 18 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
12 fvif 6427 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐵, 1)) = if(𝑘𝐴, ( I ‘𝐵), ( I ‘1))
13 fvif 6427 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐶, 1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1))
1411, 12, 133eqtr4g 2872 . . . . . . . . . . . . . . . . 17 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → ( I ‘if(𝑘𝐴, 𝐵, 1)) = ( I ‘if(𝑘𝐴, 𝐶, 1)))
153, 14mpteq2da 4944 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1615adantr 468 . . . . . . . . . . . . . . 15 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑛)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1716fveq1d 6413 . . . . . . . . . . . . . 14 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑛)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥))
1817adantlr 697 . . . . . . . . . . . . 13 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑥 ∈ (ℤ𝑛)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥))
19 eqid 2813 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
20 eqid 2813 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))
2119, 20fvmptex 6518 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥)
22 eqid 2813 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
23 eqid 2813 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))
2422, 23fvmptex 6518 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥)
2518, 21, 243eqtr4g 2872 . . . . . . . . . . . 12 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑥 ∈ (ℤ𝑛)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥))
262, 25seqfeq 13052 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2726breq1d 4861 . . . . . . . . . 10 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → (seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2827anbi2d 616 . . . . . . . . 9 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → ((𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
2928exbidv 2012 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → (∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
3029rexbidva 3244 . . . . . . 7 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
3130adantr 468 . . . . . 6 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
32 simpr 473 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
3315adantr 468 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
3433fveq1d 6413 . . . . . . . . . 10 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥))
3534, 21, 243eqtr4g 2872 . . . . . . . . 9 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥))
3635adantlr 697 . . . . . . . 8 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥))
3732, 36seqfeq 13052 . . . . . . 7 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
3837breq1d 4861 . . . . . 6 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))
3931, 383anbi23d 1556 . . . . 5 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → ((𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)))
4039rexbidva 3244 . . . 4 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)))
41 simplr 776 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
42 nnuz 11944 . . . . . . . . . 10 ℕ = (ℤ‘1)
4341, 42syl6eleq 2902 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ (ℤ‘1))
44 f1of 6356 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
4544ad2antlr 709 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
46 ffvelrn 6582 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)⟶𝐴𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
4745, 46sylancom 578 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
48 simplll 782 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
49 nfcsb1v 3751 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵)
50 nfcsb1v 3751 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐶)
5149, 50nfeq 2967 . . . . . . . . . . . . 13 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)
52 csbeq1a 3744 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐵))
53 csbeq1a 3744 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐶) = (𝑓𝑥) / 𝑘( I ‘𝐶))
5452, 53eqeq12d 2828 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
5551, 54rspc 3503 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ 𝐴 → (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
5647, 48, 55sylc 65 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶))
57 fvex 6424 . . . . . . . . . . . 12 (𝑓𝑥) ∈ V
58 csbfv2g 6455 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
5957, 58ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵)
60 csbfv2g 6455 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐶) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
6157, 60ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐶) = ( I ‘(𝑓𝑥) / 𝑘𝐶)
6256, 59, 613eqtr3g 2870 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ( I ‘(𝑓𝑥) / 𝑘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
63 elfznn 12596 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
6463adantl 469 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
65 fveq2 6411 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → (𝑓𝑛) = (𝑓𝑥))
6665csbeq1d 3742 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑥) / 𝑘𝐵)
67 eqid 2813 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
6866, 67fvmpti 6505 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
6964, 68syl 17 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
7065csbeq1d 3742 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐶 = (𝑓𝑥) / 𝑘𝐶)
71 eqid 2813 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
7270, 71fvmpti 6505 . . . . . . . . . . 11 (𝑥 ∈ ℕ → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
7364, 72syl 17 . . . . . . . . . 10 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥) = ( I ‘(𝑓𝑥) / 𝑘𝐶))
7462, 69, 733eqtr4d 2857 . . . . . . . . 9 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)‘𝑥) = ((𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)‘𝑥))
7543, 74seqfveq 13051 . . . . . . . 8 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))
7675eqeq2d 2823 . . . . . . 7 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))
7776pm5.32da 570 . . . . . 6 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7877exbidv 2012 . . . . 5 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
7978rexbidva 3244 . . . 4 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
8040, 79orbi12d 933 . . 3 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))))
8180iotabidv 6088 . 2 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚)))))
82 df-prod 14860 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵))‘𝑚))))
83 df-prod 14860 . 2 𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ𝑚) ∧ ∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 ≠ 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶))‘𝑚))))
8481, 82, 833eqtr4g 2872 1 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 865  w3a 1100   = wceq 1637  wex 1859  wcel 2157  wne 2985  wral 3103  wrex 3104  Vcvv 3398  csb 3735  wss 3776  ifcif 4286   class class class wbr 4851  cmpt 4930   I cid 5225  cio 6065  wf 6100  1-1-ontowf1o 6103  cfv 6104  (class class class)co 6877  0cc0 10224  1c1 10225   · cmul 10229  cn 11308  cz 11646  cuz 11907  ...cfz 12552  seqcseq 13027  cli 14441  cprod 14859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-fal 1651  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-nn 11309  df-n0 11563  df-z 11647  df-uz 11908  df-fz 12553  df-seq 13028  df-prod 14860
This theorem is referenced by:  prodeq2  14868  prod2id  14882
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