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Theorem prodeq2ii 15016
 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 11978 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑚) → 𝑛 ∈ ℤ)
21adantl 475 . . . . . . . . . . . 12 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ ℤ)
3 nfra1 3150 . . . . . . . . . . . . . . . . 17 𝑘𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4 rsp 3138 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
54adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
6 ifeq1 4310 . . . . . . . . . . . . . . . . . . . 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 4315 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = ( I ‘1))
9 iffalse 4315 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)) = ( I ‘1))
108, 9eqtr4d 2864 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
117, 10pm2.61d1 173 . . . . . . . . . . . . . . . . . 18 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
12 fvif 6449 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐵, 1)) = if(𝑘𝐴, ( I ‘𝐵), ( I ‘1))
13 fvif 6449 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐶, 1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1))
1411, 12, 133eqtr4g 2886 . . . . . . . . . . . . . . . . 17 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → ( I ‘if(𝑘𝐴, 𝐵, 1)) = ( I ‘if(𝑘𝐴, 𝐶, 1)))
153, 14mpteq2da 4966 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1615adantr 474 . . . . . . . . . . . . . . 15 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑛)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1716fveq1d 6435 . . . . . . . . . . . . . 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 708 . . . . . . . . . . . . 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 2825 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
20 eqid 2825 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))
2119, 20fvmptex 6541 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥)
22 eqid 2825 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
23 eqid 2825 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))
2422, 23fvmptex 6541 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥)
2518, 21, 243eqtr4g 2886 . . . . . . . . . . . 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 13120 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2726breq1d 4883 . . . . . . . . . 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 624 . . . . . . . . 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 2022 . . . . . . . 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 3259 . . . . . . 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 474 . . . . . 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 479 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
3315adantr 474 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
3433fveq1d 6435 . . . . . . . . . 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 2886 . . . . . . . . 9 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥))
3635adantlr 708 . . . . . . . 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 13120 . . . . . . 7 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
3837breq1d 4883 . . . . . 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 1569 . . . . 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 3259 . . . 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 787 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
42 nnuz 12005 . . . . . . . . . 10 ℕ = (ℤ‘1)
4341, 42syl6eleq 2916 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ (ℤ‘1))
44 f1of 6378 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
4544ad2antlr 720 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
46 ffvelrn 6606 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)⟶𝐴𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
4745, 46sylancom 584 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
48 simplll 793 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
49 nfcsb1v 3773 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵)
50 nfcsb1v 3773 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐶)
5149, 50nfeq 2981 . . . . . . . . . . . . 13 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)
52 csbeq1a 3766 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐵))
53 csbeq1a 3766 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐶) = (𝑓𝑥) / 𝑘( I ‘𝐶))
5452, 53eqeq12d 2840 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
5551, 54rspc 3520 . . . . . . . . . . . 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 6446 . . . . . . . . . . . 12 (𝑓𝑥) ∈ V
58 csbfv2g 6478 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
5957, 58ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵)
60 csbfv2g 6478 . . . . . . . . . . . 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 2884 . . . . . . . . . 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 12663 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
6463adantl 475 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
65 fveq2 6433 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → (𝑓𝑛) = (𝑓𝑥))
6665csbeq1d 3764 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑥) / 𝑘𝐵)
67 eqid 2825 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
6866, 67fvmpti 6528 . . . . . . . . . . 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 3764 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐶 = (𝑓𝑥) / 𝑘𝐶)
71 eqid 2825 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
7270, 71fvmpti 6528 . . . . . . . . . . 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 2871 . . . . . . . . 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 13119 . . . . . . . 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 2835 . . . . . . 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 576 . . . . . 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 2022 . . . . 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 3259 . . . 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 949 . . 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 6107 . 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 15009 . 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 15009 . 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 2886 1 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∨ wo 880   ∧ w3a 1113   = wceq 1658  ∃wex 1880   ∈ wcel 2166   ≠ wne 2999  ∀wral 3117  ∃wrex 3118  Vcvv 3414  ⦋csb 3757   ⊆ wss 3798  ifcif 4306   class class class wbr 4873   ↦ cmpt 4952   I cid 5249  ℩cio 6084  ⟶wf 6119  –1-1-onto→wf1o 6122  ‘cfv 6123  (class class class)co 6905  0cc0 10252  1c1 10253   · cmul 10257  ℕcn 11350  ℤcz 11704  ℤ≥cuz 11968  ...cfz 12619  seqcseq 13095   ⇝ cli 14592  ∏cprod 15008 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-fal 1672  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-om 7327  df-1st 7428  df-2nd 7429  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-n0 11619  df-z 11705  df-uz 11969  df-fz 12620  df-seq 13096  df-prod 15009 This theorem is referenced by:  prodeq2  15017  prod2id  15031
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