Theorem prodeq2 11722

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 2528	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘∀𝑘 ∈ 𝐴 𝐵 = 𝐶
2		nfv 1542	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑘 𝑚 ∈ ℤ
3	1, 2	nfan 1579	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ)
4		nfv 1542	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘(𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)
5	3, 4	nfan 1579	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴))
6		nfv 1542	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘 𝑛 ∈ (ℤ≥‘𝑚)
7	5, 6	nfan 1579	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑘(((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑛 ∈ (ℤ≥‘𝑚))
8		simp-4l 541	. . . . . . . . . . . . . . . 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 2544	. . . . . . . . . . . . . . . 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 534	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → 𝑚 ∈ ℤ)
14		simplrl 535	. . . . . . . . . . . . . . . 16 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℤ) ∧ (𝐴 ⊆ (ℤ≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ≥‘𝑚)DECID 𝑗 ∈ 𝐴)) ∧ 𝑘 ∈ ℤ) → 𝐴 ⊆ (ℤ≥‘𝑚))
15		simplrr 536	. . . . . . . . . . . . . . . 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 11523	. . . . . . . . . . . . . . 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 3591	. . . . . . . . . . . . 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 4122	. . . . . . . . . . . 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 10547	. . . . . . . . . . 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 4043	. . . . . . . . . 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 1839	. . . . . . . 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 2494	. . . . . . 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 3591	. . . . . . . . . 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 4122	. . . . . . . . 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 10547	. . . . . . . 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 4043	. . . . . . 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 2494	. . . 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 5504	. . . . . . . . . . . . . . 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 528	. . . . . . . . . . . . . . 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 542	. . . . . . . . . . . . . . . . 17 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℕ)
38	37	nnzd 9447	. . . . . . . . . . . . . . . 16 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → 𝑚 ∈ ℤ)
39		fznn 10164	. . . . . . . . . . . . . . . 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 946	. . . . . . . . . . . . . 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 5698	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → (𝑓‘𝑛) ∈ 𝐴)
43		simp-4l 541	. . . . . . . . . . . . 13 ⊢ (((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) ∧ 𝑛 ≤ 𝑚) → ∀𝑘 ∈ 𝐴 𝐵 = 𝐶)
44		nfcsb1v 3117	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵
45		nfcsb1v 3117	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐶
46	44, 45	nfeq 2347	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑘⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶
47		csbeq1a 3093	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐵)
48		csbeq1a 3093	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = (𝑓‘𝑛) → 𝐶 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶)
49	47, 48	eqeq12d 2211	. . . . . . . . . . . . . 14 ⊢ (𝑘 = (𝑓‘𝑛) → (𝐵 = 𝐶 ↔ ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑛) / 𝑘⦌𝐶))
50	46, 49	rspc 2862	. . . . . . . . . . . . 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 9447	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑛 ∈ ℤ)
54		simpllr 534	. . . . . . . . . . . . . 14 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℕ)
55	54	nnzd 9447	. . . . . . . . . . . . 13 ⊢ ((((∀𝑘 ∈ 𝐴 𝐵 = 𝐶 ∧ 𝑚 ∈ ℕ) ∧ 𝑓:(1...𝑚)–1-1-onto→𝐴) ∧ 𝑛 ∈ ℕ) → 𝑚 ∈ ℤ)
56		zdcle 9402	. . . . . . . . . . . . 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 3591	. . . . . . . . . . 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 4123	. . . . . . . . . 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 10547	. . . . . . . . 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 5560	. . . . . . . 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 2208	. . . . . . 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 1839	. . . . 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 2494	. . . 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 794	. . 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 5241	. 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 11716	. 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 11716	. 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 2254	1 ⊢ (∀𝑘 ∈ 𝐴 𝐵 = 𝐶 → ∏𝑘 ∈ 𝐴 𝐵 = ∏𝑘 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709  DECID wdc 835   = wceq 1364  ∃wex 1506   ∈ wcel 2167  ∀wral 2475  ∃wrex 2476  ⦋csb 3084   ⊆ wss 3157  ifcif 3561   class class class wbr 4033   ↦ cmpt 4094  ℩cio 5217  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922  0cc0 7879  1c1 7880   · cmul 7884   ≤ cle 8062   # cap 8608  ℕcn 8990  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  seqcseq 10539   ⇝ cli 11443  ∏cprod 11715

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-addcom 7979  ax-addass 7981  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-0id 7987  ax-rnegex 7988  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-ltadd 7995

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-mpt 4096  df-id 4328  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-inn 8991  df-n0 9250  df-z 9327  df-uz 9602  df-fz 10084  df-seqfrec 10540  df-proddc 11716

This theorem is referenced by:  prodeq2i  11727  prodeq2d  11730  gausslemma2dlem5a  15306