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Theorem prodeq2 11333
Description: Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)
Assertion
Ref Expression
prodeq2 (∀𝑘𝐴 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
Distinct variable group:   𝐴,𝑘
Allowed substitution hints:   𝐵(𝑘)   𝐶(𝑘)

Proof of Theorem prodeq2
Dummy variables 𝑓 𝑗 𝑚 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfra1 2466 . . . . . . . . . . . . . . . 16 𝑘𝑘𝐴 𝐵 = 𝐶
2 nfv 1508 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ ℤ
31, 2nfan 1544 . . . . . . . . . . . . . . 15 𝑘(∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ)
4 nfv 1508 . . . . . . . . . . . . . . 15 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
53, 4nfan 1544 . . . . . . . . . . . . . 14 𝑘((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴))
6 nfv 1508 . . . . . . . . . . . . . 14 𝑘 𝑛 ∈ (ℤ𝑚)
75, 6nfan 1544 . . . . . . . . . . . . 13 𝑘(((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚))
8 simp-4l 530 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → ∀𝑘𝐴 𝐵 = 𝐶)
9 simpr 109 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → 𝑘𝐴)
10 rsp 2480 . . . . . . . . . . . . . . . 16 (∀𝑘𝐴 𝐵 = 𝐶 → (𝑘𝐴𝐵 = 𝐶))
118, 9, 10sylc 62 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → 𝐵 = 𝐶)
1211adantllr 472 . . . . . . . . . . . . . 14 ((((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → 𝐵 = 𝐶)
13 simpllr 523 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑚 ∈ ℤ)
14 simplrl 524 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝐴 ⊆ (ℤ𝑚))
15 simplrr 525 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
16 simpr 109 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
1713, 14, 15, 16sumdc 11134 . . . . . . . . . . . . . . 15 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘𝐴)
1817adantlr 468 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘𝐴)
1912, 18ifeq1dadc 3502 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
207, 19mpteq2da 4017 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) → (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
2120seqeq3d 10233 . . . . . . . . . . 11 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2221breq1d 3939 . . . . . . . . . 10 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) → (seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦))
2322anbi2d 459 . . . . . . . . 9 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) → ((𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
2423exbidv 1797 . . . . . . . 8 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦)))
2524rexbidva 2434 . . . . . . 7 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (∃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)))
2611, 17ifeq1dadc 3502 . . . . . . . . . 10 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
275, 26mpteq2da 4017 . . . . . . . . 9 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
2827seqeq3d 10233 . . . . . . . 8 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))))
2928breq1d 3939 . . . . . . 7 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))
3025, 29anbi12d 464 . . . . . 6 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → ((∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥)))
3130pm5.32da 447 . . . . 5 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))))
3231rexbidva 2434 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃𝑛 ∈ (ℤ𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1))) ⇝ 𝑥))))
33 f1of 5367 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
3433ad3antlr 484 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑓:(1...𝑚)⟶𝐴)
35 simplr 519 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ ℕ)
36 simpr 109 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛𝑚)
37 simp-4r 531 . . . . . . . . . . . . . . . . 17 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℕ)
3837nnzd 9179 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℤ)
39 fznn 9876 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℤ → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
4038, 39syl 14 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛𝑚)))
4135, 36, 40mpbir2and 928 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ (1...𝑚))
4234, 41ffvelrnd 5556 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) ∈ 𝐴)
43 simp-4l 530 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → ∀𝑘𝐴 𝐵 = 𝐶)
44 nfcsb1v 3035 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐵
45 nfcsb1v 3035 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐶
4644, 45nfeq 2289 . . . . . . . . . . . . . 14 𝑘(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶
47 csbeq1a 3012 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐵 = (𝑓𝑛) / 𝑘𝐵)
48 csbeq1a 3012 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐶 = (𝑓𝑛) / 𝑘𝐶)
4947, 48eqeq12d 2154 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑛) → (𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
5046, 49rspc 2783 . . . . . . . . . . . . 13 ((𝑓𝑛) ∈ 𝐴 → (∀𝑘𝐴 𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
5142, 43, 50sylc 62 . . . . . . . . . . . 12 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶)
52 simpr 109 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5352nnzd 9179 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
54 simpllr 523 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
5554nnzd 9179 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
56 zdcle 9134 . . . . . . . . . . . . 13 ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛𝑚)
5753, 55, 56syl2anc 408 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛𝑚)
5851, 57ifeq1dadc 3502 . . . . . . . . . . 11 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))
5958mpteq2dva 4018 . . . . . . . . . 10 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))
6059seqeq3d 10233 . . . . . . . . 9 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))))
6160fveq1d 5423 . . . . . . . 8 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))
6261eqeq2d 2151 . . . . . . 7 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))
6362pm5.32da 447 . . . . . 6 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
6463exbidv 1797 . . . . 5 ((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
6564rexbidva 2434 . . . 4 (∀𝑘𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto𝐴𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
6632, 65orbi12d 782 . . 3 (∀𝑘𝐴 𝐵 = 𝐶 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))))
6766iotabidv 5109 . 2 (∀𝑘𝐴 𝐵 = 𝐶 → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚)))))
68 df-proddc 11327 . 2 𝑘𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1)))‘𝑚))))
69 df-proddc 11327 . 2 𝑘𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴) ∧ (∃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 ↦ if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1)))‘𝑚))))
7067, 68, 693eqtr4g 2197 1 (∀𝑘𝐴 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wo 697  DECID wdc 819   = wceq 1331  wex 1468  wcel 1480  wral 2416  wrex 2417  csb 3003  wss 3071  ifcif 3474   class class class wbr 3929  cmpt 3989  cio 5086  wf 5119  1-1-ontowf1o 5122  cfv 5123  (class class class)co 5774  0cc0 7627  1c1 7628   · cmul 7632  cle 7808   # cap 8350  cn 8727  cz 9061  cuz 9333  ...cfz 9797  seqcseq 10225  cli 11054  cprod 11326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-addcom 7727  ax-addass 7729  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-0id 7735  ax-rnegex 7736  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-ltadd 7743
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-recs 6202  df-frec 6288  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-inn 8728  df-n0 8985  df-z 9062  df-uz 9334  df-fz 9798  df-seqfrec 10226  df-proddc 11327
This theorem is referenced by:  prodeq2i  11338  prodeq2d  11341
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