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Theorem prodeq2 11549
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 2508 . . . . . . . . . . . . . . . 16 𝑘𝑘𝐴 𝐵 = 𝐶
2 nfv 1528 . . . . . . . . . . . . . . . 16 𝑘 𝑚 ∈ ℤ
31, 2nfan 1565 . . . . . . . . . . . . . . 15 𝑘(∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ)
4 nfv 1528 . . . . . . . . . . . . . . 15 𝑘(𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
53, 4nfan 1565 . . . . . . . . . . . . . 14 𝑘((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴))
6 nfv 1528 . . . . . . . . . . . . . 14 𝑘 𝑛 ∈ (ℤ𝑚)
75, 6nfan 1565 . . . . . . . . . . . . 13 𝑘(((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚))
8 simp-4l 541 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → ∀𝑘𝐴 𝐵 = 𝐶)
9 simpr 110 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → 𝑘𝐴)
10 rsp 2524 . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . 14 ((((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘𝐴) → 𝐵 = 𝐶)
13 simpllr 534 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑚 ∈ ℤ)
14 simplrl 535 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝐴 ⊆ (ℤ𝑚))
15 simplrr 536 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)
16 simpr 110 . . . . . . . . . . . . . . . 16 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
1713, 14, 15, 16sumdc 11350 . . . . . . . . . . . . . . 15 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘𝐴)
1817adantlr 477 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑛 ∈ (ℤ𝑚)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘𝐴)
1912, 18ifeq1dadc 3564 . . . . . . . . . . . . 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 4089 . . . . . . . . . . . 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 10439 . . . . . . . . . . 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 4010 . . . . . . . . . 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 464 . . . . . . . . 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 1825 . . . . . . . 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 2474 . . . . . . 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 3564 . . . . . . . . . 10 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) ∧ 𝑘 ∈ ℤ) → if(𝑘𝐴, 𝐵, 1) = if(𝑘𝐴, 𝐶, 1))
275, 26mpteq2da 4089 . . . . . . . . 9 (((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ𝑚) ∧ ∀𝑗 ∈ (ℤ𝑚)DECID 𝑗𝐴)) → (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘𝐴, 𝐶, 1)))
2827seqeq3d 10439 . . . . . . . 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 4010 . . . . . . 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 473 . . . . . 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 452 . . . . 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 2474 . . . 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 5457 . . . . . . . . . . . . . . 15 (𝑓:(1...𝑚)–1-1-onto𝐴𝑓:(1...𝑚)⟶𝐴)
3433ad3antlr 493 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑓:(1...𝑚)⟶𝐴)
35 simplr 528 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ ℕ)
36 simpr 110 . . . . . . . . . . . . . . 15 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛𝑚)
37 simp-4r 542 . . . . . . . . . . . . . . . . 17 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℕ)
3837nnzd 9363 . . . . . . . . . . . . . . . 16 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑚 ∈ ℤ)
39 fznn 10075 . . . . . . . . . . . . . . . 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 944 . . . . . . . . . . . . . 14 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → 𝑛 ∈ (1...𝑚))
4234, 41ffvelcdmd 5648 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → (𝑓𝑛) ∈ 𝐴)
43 simp-4l 541 . . . . . . . . . . . . 13 (((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛𝑚) → ∀𝑘𝐴 𝐵 = 𝐶)
44 nfcsb1v 3090 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐵
45 nfcsb1v 3090 . . . . . . . . . . . . . . 15 𝑘(𝑓𝑛) / 𝑘𝐶
4644, 45nfeq 2327 . . . . . . . . . . . . . 14 𝑘(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶
47 csbeq1a 3066 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐵 = (𝑓𝑛) / 𝑘𝐵)
48 csbeq1a 3066 . . . . . . . . . . . . . . 15 (𝑘 = (𝑓𝑛) → 𝐶 = (𝑓𝑛) / 𝑘𝐶)
4947, 48eqeq12d 2192 . . . . . . . . . . . . . 14 (𝑘 = (𝑓𝑛) → (𝐵 = 𝐶(𝑓𝑛) / 𝑘𝐵 = (𝑓𝑛) / 𝑘𝐶))
5046, 49rspc 2835 . . . . . . . . . . . . 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 110 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
5352nnzd 9363 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
54 simpllr 534 . . . . . . . . . . . . . 14 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
5554nnzd 9363 . . . . . . . . . . . . 13 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
56 zdcle 9318 . . . . . . . . . . . . 13 ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛𝑚)
5753, 55, 56syl2anc 411 . . . . . . . . . . . 12 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛𝑚)
5851, 57ifeq1dadc 3564 . . . . . . . . . . 11 ((((∀𝑘𝐴 𝐵 = 𝐶𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐵, 1) = if(𝑛𝑚, (𝑓𝑛) / 𝑘𝐶, 1))
5958mpteq2dva 4090 . . . . . . . . . 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 10439 . . . . . . . . 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 5513 . . . . . . . 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 2189 . . . . . . 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 452 . . . . . 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 1825 . . . . 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 2474 . . . 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 793 . . 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 5195 . 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 11543 . 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 11543 . 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 2235 1 (∀𝑘𝐴 𝐵 = 𝐶 → ∏𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wo 708  DECID wdc 834   = wceq 1353  wex 1492  wcel 2148  wral 2455  wrex 2456  csb 3057  wss 3129  ifcif 3534   class class class wbr 4000  cmpt 4061  cio 5172  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  0cc0 7802  1c1 7803   · cmul 7807  cle 7983   # cap 8528  cn 8908  cz 9242  cuz 9517  ...cfz 9995  seqcseq 10431  cli 11270  cprod 11542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-seqfrec 10432  df-proddc 11543
This theorem is referenced by:  prodeq2i  11554  prodeq2d  11557
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