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Theorem prodeq2ii 15359
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 12334 . . . . . . . . . . . . 13 (𝑛 ∈ (ℤ𝑚) → 𝑛 ∈ ℤ)
21adantl 485 . . . . . . . . . . . 12 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → 𝑛 ∈ ℤ)
3 nfra1 3131 . . . . . . . . . . . . . . . . 17 𝑘𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶)
4 rsp 3118 . . . . . . . . . . . . . . . . . . . . 21 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
54adantr 484 . . . . . . . . . . . . . . . . . . . 20 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → (𝑘𝐴 → ( I ‘𝐵) = ( I ‘𝐶)))
6 ifeq1 4418 . . . . . . . . . . . . . . . . . . . 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 4423 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = ( I ‘1))
9 iffalse 4423 . . . . . . . . . . . . . . . . . . . 20 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)) = ( I ‘1))
108, 9eqtr4d 2776 . . . . . . . . . . . . . . . . . . 19 𝑘𝐴 → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
117, 10pm2.61d1 183 . . . . . . . . . . . . . . . . . 18 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, ( I ‘𝐵), ( I ‘1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1)))
12 fvif 6690 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐵, 1)) = if(𝑘𝐴, ( I ‘𝐵), ( I ‘1))
13 fvif 6690 . . . . . . . . . . . . . . . . . 18 ( I ‘if(𝑘𝐴, 𝐶, 1)) = if(𝑘𝐴, ( I ‘𝐶), ( I ‘1))
1411, 12, 133eqtr4g 2798 . . . . . . . . . . . . . . . . 17 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑘 ∈ ℤ) → ( I ‘if(𝑘𝐴, 𝐵, 1)) = ( I ‘if(𝑘𝐴, 𝐶, 1)))
153, 14mpteq2da 5124 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1615adantr 484 . . . . . . . . . . . . . . 15 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑛)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
1716fveq1d 6676 . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . 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 2738 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))
20 eqid 2738 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))
2119, 20fvmptex 6789 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1)))‘𝑥)
22 eqid 2738 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))
23 eqid 2738 . . . . . . . . . . . . . 14 (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))
2422, 23fvmptex 6789 . . . . . . . . . . . . 13 ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1)))‘𝑥)
2518, 21, 243eqtr4g 2798 . . . . . . . . . . . 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 13487 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑛 ∈ (ℤ𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2726breq1d 5040 . . . . . . . . . 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 632 . . . . . . . . 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 1928 . . . . . . . 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 3206 . . . . . . 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 484 . . . . . 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 488 . . . . . . . 8 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → 𝑚 ∈ ℤ)
3315adantr 484 . . . . . . . . . . 11 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐵, 1))) = (𝑘 ∈ ℤ ↦ ( I ‘if(𝑘𝐴, 𝐶, 1))))
3433fveq1d 6676 . . . . . . . . . 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 2798 . . . . . . . . 9 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑥 ∈ (ℤ𝑚)) → ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))‘𝑥) = ((𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))‘𝑥))
3635adantlr 715 . . . . . . . 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 13487 . . . . . . 7 ((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℤ) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
3837breq1d 5040 . . . . . 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 1440 . . . . 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 3206 . . . 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 769 . . . . . . . . . 10 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ ℕ)
42 nnuz 12363 . . . . . . . . . 10 ℕ = (ℤ‘1)
4341, 42eleqtrdi 2843 . . . . . . . . 9 (((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → 𝑚 ∈ (ℤ‘1))
44 f1of 6618 . . . . . . . . . . . . . 14 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
4544ad2antlr 727 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑓:(1...𝑚)⟶𝐴)
46 ffvelrn 6859 . . . . . . . . . . . . 13 ((𝑓:(1...𝑚)⟶𝐴𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
4745, 46sylancom 591 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → (𝑓𝑥) ∈ 𝐴)
48 simplll 775 . . . . . . . . . . . 12 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → ∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶))
49 nfcsb1v 3814 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵)
50 nfcsb1v 3814 . . . . . . . . . . . . . 14 𝑘(𝑓𝑥) / 𝑘( I ‘𝐶)
5149, 50nfeq 2912 . . . . . . . . . . . . 13 𝑘(𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)
52 csbeq1a 3804 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐵))
53 csbeq1a 3804 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑥) → ( I ‘𝐶) = (𝑓𝑥) / 𝑘( I ‘𝐶))
5452, 53eqeq12d 2754 . . . . . . . . . . . . 13 (𝑘 = (𝑓𝑥) → (( I ‘𝐵) = ( I ‘𝐶) ↔ (𝑓𝑥) / 𝑘( I ‘𝐵) = (𝑓𝑥) / 𝑘( I ‘𝐶)))
5551, 54rspc 3514 . . . . . . . . . . . 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 6687 . . . . . . . . . . . 12 (𝑓𝑥) ∈ V
58 csbfv2g 6718 . . . . . . . . . . . 12 ((𝑓𝑥) ∈ V → (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵))
5957, 58ax-mp 5 . . . . . . . . . . 11 (𝑓𝑥) / 𝑘( I ‘𝐵) = ( I ‘(𝑓𝑥) / 𝑘𝐵)
60 csbfv2g 6718 . . . . . . . . . . . 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 2796 . . . . . . . . . 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 13027 . . . . . . . . . . . 12 (𝑥 ∈ (1...𝑚) → 𝑥 ∈ ℕ)
6463adantl 485 . . . . . . . . . . 11 ((((∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑥 ∈ (1...𝑚)) → 𝑥 ∈ ℕ)
65 fveq2 6674 . . . . . . . . . . . . 13 (𝑛 = 𝑥 → (𝑓𝑛) = (𝑓𝑥))
6665csbeq1d 3794 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑥) / 𝑘𝐵)
67 eqid 2738 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐵)
6866, 67fvmpti 6774 . . . . . . . . . . 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 3794 . . . . . . . . . . . 12 (𝑛 = 𝑥(𝑓𝑛) / 𝑘𝐶 = (𝑓𝑥) / 𝑘𝐶)
71 eqid 2738 . . . . . . . . . . . 12 (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶) = (𝑛 ∈ ℕ ↦ (𝑓𝑛) / 𝑘𝐶)
7270, 71fvmpti 6774 . . . . . . . . . . 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 2783 . . . . . . . . 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 13486 . . . . . . . 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 2749 . . . . . . 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 582 . . . . . 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 1928 . . . . 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 3206 . . . 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 918 . . 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 6323 . 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 15352 . 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 15352 . 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 2798 1 (∀𝑘𝐴 ( I ‘𝐵) = ( I ‘𝐶) → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3a 1088   = wceq 1542  wex 1786  wcel 2114  wne 2934  wral 3053  wrex 3054  Vcvv 3398  csb 3790  wss 3843  ifcif 4414   class class class wbr 5030  cmpt 5110   I cid 5428  cio 6295  wf 6335  1-1-ontowf1o 6338  cfv 6339  (class class class)co 7170  0cc0 10615  1c1 10616   · cmul 10620  cn 11716  cz 12062  cuz 12324  ...cfz 12981  seqcseq 13460  cli 14931  cprod 15351
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-fz 12982  df-seq 13461  df-prod 15352
This theorem is referenced by:  prodeq2  15360  prod2id  15374
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