Theorem prodeq2 12117

Description: Equality theorem for product. (Contributed by Scott Fenton, 4-Dec-2017.)

Assertion

Ref	Expression
prodeq2	⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 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.

Step	Hyp	Ref	Expression
1		nfra1 2563	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 𝐵 = 𝐶
2		nfv 1576	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑚 ∈ ℤ
3	1, 2	nfan 1613	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ)
4		nfv 1576	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
5	3, 4	nfan 1613	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴))
6		nfv 1576	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ≥‘𝑚)
7	5, 6	nfan 1613	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚))
8		simp-4l 543	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
9		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝐴)
10		rsp 2579	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (𝑘 ∈ 𝐴 → 𝐵 = 𝐶))
11	8, 9, 10	sylc 62	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
12	11	adantllr 481	. . . . . . . . . . . . . 14 ⊢ ((((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ 𝑘 ∈ ℤ) ∧ 𝑘 ∈ 𝐴) → 𝐵 = 𝐶)
13		simpllr 536	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑚 ∈ ℤ)
14		simplrl 537	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑚))
15		simplrr 538	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
16		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑘 ∈ ℤ)
17	13, 14, 15, 16	sumdc 11918	. . . . . . . . . . . . . . 15 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘 ∈ 𝐴)
18	17	adantlr 477	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ 𝑘 ∈ ℤ) → DECID 𝑘 ∈ 𝐴)
19	12, 18	ifeq1dadc 3636	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑘 ∈ 𝐴, 𝐶, 1))
20	7, 19	mpteq2da 4178	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
21	20	seqeq3d 10716	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
22	21	breq1d 4098	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦 ↔ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦))
23	22	anbi2d 464	. . . . . . . . 9 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → ((𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ (𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
24	23	exbidv 1873	. . . . . . . 8 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚)) → (∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
25	24	rexbidva 2529	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ↔ ∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦)))
26	11, 17	ifeq1dadc 3636	. . . . . . . . . 10 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → if(𝑘 ∈ 𝐴, 𝐵, 1) = if(𝑘 ∈ 𝐴, 𝐶, 1))
27	5, 26	mpteq2da 4178	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1)) = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1)))
28	27	seqeq3d 10716	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) = seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))))
29	28	breq1d 4098	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → (seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥 ↔ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))
30	25, 29	anbi12d 473	. . . . . 6 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) → ((∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥) ↔ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)))
31	30	pm5.32da 452	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) → (((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))))
32	31	rexbidva 2529	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ↔ ∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥))))
33		f1of 5583	. . . . . . . . . . . . . . 15 ⊢ (𝑓:(1...𝑚)–1-1-onto→𝐴 → 𝑓:(1...𝑚)⟶𝐴)
34	33	ad3antlr 493	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑓:(1...𝑚)⟶𝐴)
35		simplr 529	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ∈ ℕ)
36		simpr 110	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ≤ 𝑚)
37		simp-4r 544	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℕ)
38	37	nnzd 9600	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℤ)
39		fznn 10323	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℤ → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑚)))
40	38, 39	syl 14	. . . . . . . . . . . . . . 15 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → (𝑛 ∈ (1...𝑚) ↔ (𝑛 ∈ ℕ ∧ 𝑛 ≤ 𝑚)))
41	35, 36, 40	mpbir2and 952	. . . . . . . . . . . . . 14 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑛 ∈ (1...𝑚))
42	34, 41	ffvelcdmd 5783	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → (𝑓‘𝑛) ∈ 𝐴)
43		simp-4l 543	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
44		nfcsb1v 3160	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
45		nfcsb1v 3160	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐶
46	44, 45	nfeq 2382	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
47		csbeq1a 3136	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
48		csbeq1a 3136	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐶 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
49	47, 48	eqeq12d 2246	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑛) → (𝐵 = 𝐶 ↔ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
50	46, 49	rspc 2904	. . . . . . . . . . . . 13 ⊢ ((𝑓‘𝑛) ∈ 𝐴 → (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
51	42, 43, 50	sylc 62	. . . . . . . . . . . 12 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
52		simpr 110	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℕ)
53	52	nnzd 9600	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
54		simpllr 536	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
55	54	nnzd 9600	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
56		zdcle 9555	. . . . . . . . . . . . 13 ⊢ ((𝑛 ∈ ℤ ∧ 𝑚 ∈ ℤ) → DECID 𝑛 ≤ 𝑚)
57	53, 55, 56	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → DECID 𝑛 ≤ 𝑚)
58	51, 57	ifeq1dadc 3636	. . . . . . . . . . 11 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1) = if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1))
59	58	mpteq2dva 4179	. . . . . . . . . 10 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)) = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))
60	59	seqeq3d 10716	. . . . . . . . 9 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1))) = seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1))))
61	60	fveq1d 5641	. . . . . . . 8 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))
62	61	eqeq2d 2243	. . . . . . 7 ⊢ (((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) → (𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚) ↔ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))
63	62	pm5.32da 452	. . . . . 6 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
64	63	exbidv 1873	. . . . 5 ⊢ ((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
65	64	rexbidva 2529	. . . 4 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
66	32, 65	orbi12d 800	. . 3 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ((∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))) ↔ (∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))))
67	66	iotabidv 5309	. 2 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚)))) = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚)))))
68		df-proddc 12111	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐵 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 1)))‘𝑚))))
69		df-proddc 12111	. 2 ⊢ ∏𝑘 ∈ 𝐴 𝐶 = (℩𝑥(∃𝑚 ∈ ℤ ((𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴) ∧ (∃𝑛 ∈ (ℤ≥‘𝑚)∃𝑦(𝑦 # 0 ∧ seq𝑛( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑦) ∧ seq𝑚( · , (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐶, 1))) ⇝ 𝑥)) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( · , (𝑛 ∈ ℕ ↦ if(𝑛 ≤ 𝑚, ⦋(𝑓‘𝑛) / 𝑘⦌𝐶, 1)))‘𝑚))))
70	67, 68, 69	3eqtr4g 2289	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  ⦋csb 3127   ⊆ wss 3200  ifcif 3605   class class class wbr 4088   ↦ cmpt 4150  ℩cio 5284  ⟶wf 5322  –1-1-onto→wf1o 5325  ‘cfv 5326  (class class class)co 6017  0cc0 8031  1c1 8032   · cmul 8036   ≤ cle 8214   # cap 8760  ℕcn 9142  ℤcz 9478  ℤ_≥cuz 9754  ...cfz 10242  seqcseq 10708   ⇝ cli 11838  ∏cprod 12110

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-addcom 8131  ax-addass 8133  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-0id 8139  ax-rnegex 8140  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-ltadd 8147

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-f1o 5333  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-recs 6470  df-frec 6556  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-inn 9143  df-n0 9402  df-z 9479  df-uz 9755  df-fz 10243  df-seqfrec 10709  df-proddc 12111

This theorem is referenced by:  prodeq2i  12122  prodeq2d  12125  gausslemma2dlem5a  15793