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Theorem summodc 11526

Description: A sum has at most one limit. (Contributed by Mario Carneiro, 3-Apr-2014.) (Revised by Jim Kingdon, 4-May-2023.)

**Hypotheses**
Ref	Expression
isummo.1	⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
isummo.2	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
summodclem2.g	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
summodc.3	⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))

**Assertion**
Ref	Expression
summodc	⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))

Distinct variable groups: 𝑘,𝑛,𝐴 𝑛,𝐹 𝜑,𝑘,𝑛 𝐴,𝑓,𝑗,𝑚,𝑘,𝑛 𝐵,𝑛 𝑓,𝐹,𝑘,𝑚 𝜑,𝑓,𝑚,𝑥,𝑘,𝑛 𝑥,𝐴,𝑗 𝐵,𝑓,𝑗,𝑚 𝑗,𝐹,𝑥 𝑛,𝐺,𝑥 𝜑,𝑗,𝑥 Allowed substitution hints: 𝐵(𝑥,𝑘) 𝐺(𝑓,𝑗,𝑘,𝑚)

Proof of Theorem summodc Dummy variables 𝑎 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 5554	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → (ℤ_≥‘𝑚) = (ℤ_≥‘𝑛))
2	1	sseq2d 3209	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝐴 ⊆ (ℤ_≥‘𝑚) ↔ 𝐴 ⊆ (ℤ_≥‘𝑛)))
3	1	raleqdv 2696	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ↔ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴))
4		seqeq1 10521	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → seq𝑚( + , 𝐹) = seq𝑛( + , 𝐹))
5	4	breq1d 4039	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (seq𝑚( + , 𝐹) ⇝ 𝑦 ↔ seq𝑛( + , 𝐹) ⇝ 𝑦))
6	2, 3, 5	3anbi123d 1323	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
7	6	cbvrexv 2727	. . . . . . 7 ⊢ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ↔ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))
8		reeanv 2664	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)))
9		simprl3 1046	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑚( + , 𝐹) ⇝ 𝑥)
10		isummo.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑘 ∈ ℤ ↦ if(𝑘 ∈ 𝐴, 𝐵, 0))
11		simpll 527	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝜑)
12		isummo.2	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
13	11, 12	sylan 283	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
14		simplrl 535	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑚 ∈ ℤ)
15		simplrr 536	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑛 ∈ ℤ)
16		simprl1 1044	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ_≥‘𝑚))
17		simprr1 1047	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝐴 ⊆ (ℤ_≥‘𝑛))
18		eleq1w 2254	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = 𝑘 → (𝑗 ∈ 𝐴 ↔ 𝑘 ∈ 𝐴))
19	18	dcbid 839	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = 𝑘 → (DECID 𝑗 ∈ 𝐴 ↔ DECID 𝑘 ∈ 𝐴))
20		simprl2 1045	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴)
21	20	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴)
22		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → 𝑘 ∈ (ℤ_≥‘𝑚))
23	19, 21, 22	rspcdva 2869	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑚)) → DECID 𝑘 ∈ 𝐴)
24		simprr2 1048	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴)
25	24	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴)
26		simpr 110	. . . . . . . . . . . . . . 15 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → 𝑘 ∈ (ℤ_≥‘𝑛))
27	19, 25, 26	rspcdva 2869	. . . . . . . . . . . . . 14 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) ∧ 𝑘 ∈ (ℤ_≥‘𝑛)) → DECID 𝑘 ∈ 𝐴)
28	10, 13, 14, 15, 16, 17, 23, 27	sumrbdc 11522	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑛( + , 𝐹) ⇝ 𝑥))
29	9, 28	mpbid 147	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑥)
30		simprr3 1049	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → seq𝑛( + , 𝐹) ⇝ 𝑦)
31		climuni 11436	. . . . . . . . . . . 12 ⊢ ((seq𝑛( + , 𝐹) ⇝ 𝑥 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦)
32	29, 30, 31	syl2anc 411	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ)) ∧ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦))) → 𝑥 = 𝑦)
33	32	exp31 364	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑚 ∈ ℤ ∧ 𝑛 ∈ ℤ) → (((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦)))
34	33	rexlimdvv 2618	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ ℤ ∃𝑛 ∈ ℤ ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
35	8, 34	biimtrrid 153	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∧ ∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦)) → 𝑥 = 𝑦))
36	35	expdimp 259	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑛 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑛) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑛)DECID 𝑗 ∈ 𝐴 ∧ seq𝑛( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
37	7, 36	biimtrid 152	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
38		summodc.3	. . . . . . 7 ⊢ 𝐺 = (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0))
39	10, 12, 38	summodclem2 11525	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
40	37, 39	jaod 718	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥)) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
41	10, 12, 38	summodclem2 11525	. . . . . . . 8 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑦 = 𝑥))
42		equcom 1717	. . . . . . . 8 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
43	41, 42	imbitrdi 161	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
44	43	impancom 260	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) → 𝑥 = 𝑦))
45		oveq2 5926	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (1...𝑚) = (1...𝑛))
46		f1oeq2 5489	. . . . . . . . . . . 12 ⊢ ((1...𝑚) = (1...𝑛) → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑛)–1-1-onto→𝐴))
47	45, 46	syl 14	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...𝑛)–1-1-onto→𝐴))
48		fveq2 5554	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘𝑛))
49	48	eqeq2d 2205	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (𝑦 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑛)))
50	47, 49	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛))))
51	50	exbidv 1836	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛))))
52		f1oeq1 5488	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓:(1...𝑛)–1-1-onto→𝐴 ↔ 𝑔:(1...𝑛)–1-1-onto→𝐴))
53		breq1 4032	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑎 ≤ (♯‘𝐴)))
54		fveq2 5554	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 = 𝑎 → (𝑓‘𝑛) = (𝑓‘𝑎))
55	54	csbeq1d 3087	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 = 𝑎 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑎) / 𝑘⦌𝐵)
56	53, 55	ifbieq1d 3579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 = 𝑎 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑎 ≤ (♯‘𝐴), ⦋(𝑓‘𝑎) / 𝑘⦌𝐵, 0))
57	56	cbvmptv 4125	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑓‘𝑎) / 𝑘⦌𝐵, 0))
58		fveq1 5553	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑎) = (𝑔‘𝑎))
59	58	csbeq1d 3087	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑓 = 𝑔 → ⦋(𝑓‘𝑎) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑎) / 𝑘⦌𝐵)
60	59	ifeq1d 3574	. . . . . . . . . . . . . . . . 17 ⊢ (𝑓 = 𝑔 → if(𝑎 ≤ (♯‘𝐴), ⦋(𝑓‘𝑎) / 𝑘⦌𝐵, 0) = if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0))
61	60	mpteq2dv 4120	. . . . . . . . . . . . . . . 16 ⊢ (𝑓 = 𝑔 → (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑓‘𝑎) / 𝑘⦌𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))
62	57, 61	eqtrid 2238	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = 𝑔 → (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))
63	38, 62	eqtrid 2238	. . . . . . . . . . . . . 14 ⊢ (𝑓 = 𝑔 → 𝐺 = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))
64	63	seqeq3d 10526	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝑔 → seq1( + , 𝐺) = seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0))))
65	64	fveq1d 5556	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → (seq1( + , 𝐺)‘𝑛) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))
66	65	eqeq2d 2205	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑦 = (seq1( + , 𝐺)‘𝑛) ↔ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)))
67	52, 66	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
68	67	cbvexv 1930	. . . . . . . . 9 ⊢ (∃𝑓(𝑓:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑛)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)))
69	51, 68	bitrdi 196	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
70	69	cbvrexv 2727	. . . . . . 7 ⊢ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)))
71		reeanv 2664	. . . . . . . . 9 ⊢ (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) ↔ (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
72		eeanv 1948	. . . . . . . . . . 11 ⊢ (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) ↔ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
73		an4 586	. . . . . . . . . . . . 13 ⊢ (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) ↔ ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
74		1zzd 9344	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 1 ∈ ℤ)
75		simplrr 536	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑛 ∈ ℕ)
76	75	nnzd 9438	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑛 ∈ ℤ)
77	74, 76	fzfigd 10502	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (1...𝑛) ∈ Fin)
78		simprr 531	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑔:(1...𝑛)–1-1-onto→𝐴)
79	77, 78	fihasheqf1od 10860	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘(1...𝑛)) = (♯‘𝐴))
80	75	nnnn0d 9293	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑛 ∈ ℕ₀)
81		hashfz1 10854	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑛 ∈ ℕ₀ → (♯‘(1...𝑛)) = 𝑛)
82	80, 81	syl 14	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘(1...𝑛)) = 𝑛)
83	79, 82	eqtr3d 2228	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘𝐴) = 𝑛)
84	83	breq2d 4041	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (𝑎 ≤ (♯‘𝐴) ↔ 𝑎 ≤ 𝑛))
85	84	ifbid 3578	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0) = if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0))
86	85	mpteq2dv 4120	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)) = (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))
87	86	seqeq3d 10526	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0))) = seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0))))
88	87	fveq1d 5556	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))
89	88	eqeq2d 2205	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛) ↔ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)))
90	89	anbi2d 464	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)) ↔ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))))
91		simplrl 535	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ)
92	91	nnnn0d 9293	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑚 ∈ ℕ₀)
93		hashfz1 10854	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑚 ∈ ℕ₀ → (♯‘(1...𝑚)) = 𝑚)
94	92, 93	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘(1...𝑚)) = 𝑚)
95	91	nnzd 9438	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑚 ∈ ℤ)
96	74, 95	fzfigd 10502	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (1...𝑚) ∈ Fin)
97		simprl 529	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑓:(1...𝑚)–1-1-onto→𝐴)
98	96, 97	fihasheqf1od 10860	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘(1...𝑚)) = (♯‘𝐴))
99	94, 98	eqtr3d 2228	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑚 = (♯‘𝐴))
100	99	fveq2d 5558	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , 𝐺)‘(♯‘𝐴)))
101		simpll 527	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝜑)
102	101, 12	sylan 283	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ)
103	99, 91	eqeltrrd 2271	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (♯‘𝐴) ∈ ℕ)
104	103, 75	jca 306	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → ((♯‘𝐴) ∈ ℕ ∧ 𝑛 ∈ ℕ))
105	99	oveq2d 5934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (1...𝑚) = (1...(♯‘𝐴)))
106		f1oeq2 5489	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((1...𝑚) = (1...(♯‘𝐴)) → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))
107	105, 106	syl 14	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (𝑓:(1...𝑚)–1-1-onto→𝐴 ↔ 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴))
108	97, 107	mpbid 147	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → 𝑓:(1...(♯‘𝐴))–1-1-onto→𝐴)
109		breq1 4032	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → (𝑛 ≤ (♯‘𝐴) ↔ 𝑗 ≤ (♯‘𝐴)))
110		fveq2 5554	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑛 = 𝑗 → (𝑓‘𝑛) = (𝑓‘𝑗))
111	110	csbeq1d 3087	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑛 = 𝑗 → ⦋(𝑓‘𝑛) / 𝑘⦌𝐵 = ⦋(𝑓‘𝑗) / 𝑘⦌𝐵)
112	109, 111	ifbieq1d 3579	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑛 = 𝑗 → if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0) = if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 0))
113	112	cbvmptv 4125	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ ↦ if(𝑛 ≤ (♯‘𝐴), ⦋(𝑓‘𝑛) / 𝑘⦌𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 0))
114	38, 113	eqtri 2214	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐺 = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ (♯‘𝐴), ⦋(𝑓‘𝑗) / 𝑘⦌𝐵, 0))
115		breq1 4032	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑗 → (𝑎 ≤ 𝑛 ↔ 𝑗 ≤ 𝑛))
116		fveq2 5554	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑎 = 𝑗 → (𝑔‘𝑎) = (𝑔‘𝑗))
117	116	csbeq1d 3087	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑎 = 𝑗 → ⦋(𝑔‘𝑎) / 𝑘⦌𝐵 = ⦋(𝑔‘𝑗) / 𝑘⦌𝐵)
118	115, 117	ifbieq1d 3579	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑎 = 𝑗 → if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0) = if(𝑗 ≤ 𝑛, ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 0))
119	118	cbvmptv 4125	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)) = (𝑗 ∈ ℕ ↦ if(𝑗 ≤ 𝑛, ⦋(𝑔‘𝑗) / 𝑘⦌𝐵, 0))
120	10, 102, 104, 108, 78, 114, 119	summodclem3 11523	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (seq1( + , 𝐺)‘(♯‘𝐴)) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))
121	100, 120	eqtrd 2226	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))
122		eqeq12 2206	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)) → (𝑥 = 𝑦 ↔ (seq1( + , 𝐺)‘𝑚) = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)))
123	121, 122	syl5ibrcom 157	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ 𝑛, ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
124	90, 123	sylbid 150	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) ∧ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴)) → ((𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
125	124	expimpd 363	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑔:(1...𝑛)–1-1-onto→𝐴) ∧ (𝑥 = (seq1( + , 𝐺)‘𝑚) ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
126	73, 125	biimtrid 152	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
127	126	exlimdvv 1909	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → (∃𝑓∃𝑔((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ (𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
128	72, 127	biimtrrid 153	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑚 ∈ ℕ ∧ 𝑛 ∈ ℕ)) → ((∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
129	128	rexlimdvva 2619	. . . . . . . . 9 ⊢ (𝜑 → (∃𝑚 ∈ ℕ ∃𝑛 ∈ ℕ (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
130	71, 129	biimtrrid 153	. . . . . . . 8 ⊢ (𝜑 → ((∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ∧ ∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛))) → 𝑥 = 𝑦))
131	130	expdimp 259	. . . . . . 7 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑛 ∈ ℕ ∃𝑔(𝑔:(1...𝑛)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , (𝑎 ∈ ℕ ↦ if(𝑎 ≤ (♯‘𝐴), ⦋(𝑔‘𝑎) / 𝑘⦌𝐵, 0)))‘𝑛)) → 𝑥 = 𝑦))
132	70, 131	biimtrid 152	. . . . . 6 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)) → 𝑥 = 𝑦))
133	44, 132	jaod 718	. . . . 5 ⊢ ((𝜑 ∧ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
134	40, 133	jaodan 798	. . . 4 ⊢ ((𝜑 ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)))) → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))) → 𝑥 = 𝑦))
135	134	expimpd 363	. . 3 ⊢ (𝜑 → (((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
136	135	alrimivv 1886	. 2 ⊢ (𝜑 → ∀𝑥∀𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
137		breq2 4033	. . . . . 6 ⊢ (𝑥 = 𝑦 → (seq𝑚( + , 𝐹) ⇝ 𝑥 ↔ seq𝑚( + , 𝐹) ⇝ 𝑦))
138	137	3anbi3d 1329	. . . . 5 ⊢ (𝑥 = 𝑦 → ((𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
139	138	rexbidv 2495	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ↔ ∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦)))
140		eqeq1 2200	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 = (seq1( + , 𝐺)‘𝑚) ↔ 𝑦 = (seq1( + , 𝐺)‘𝑚)))
141	140	anbi2d 464	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ (𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
142	141	exbidv 1836	. . . . 5 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
143	142	rexbidv 2495	. . . 4 ⊢ (𝑥 = 𝑦 → (∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚)) ↔ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚))))
144	139, 143	orbi12d 794	. . 3 ⊢ (𝑥 = 𝑦 → ((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))))
145	144	mo4 2103	. 2 ⊢ (∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ↔ ∀𝑥∀𝑦(((∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))) ∧ (∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑦) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑦 = (seq1( + , 𝐺)‘𝑚)))) → 𝑥 = 𝑦))
146	136, 145	sylibr 134	1 ⊢ (𝜑 → ∃*𝑥(∃𝑚 ∈ ℤ (𝐴 ⊆ (ℤ_≥‘𝑚) ∧ ∀𝑗 ∈ (ℤ_≥‘𝑚)DECID 𝑗 ∈ 𝐴 ∧ seq𝑚( + , 𝐹) ⇝ 𝑥) ∨ ∃𝑚 ∈ ℕ ∃𝑓(𝑓:(1...𝑚)–1-1-onto→𝐴 ∧ 𝑥 = (seq1( + , 𝐺)‘𝑚))))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 709 DECID wdc 835 ∧ w3a 980 ∀wal 1362 = wceq 1364 ∃wex 1503 ∃*wmo 2043 ∈ wcel 2164 ∀wral 2472 ∃wrex 2473 ⦋csb 3080 ⊆ wss 3153 ifcif 3557 class class class wbr 4029 ↦ cmpt 4090 –1-1-onto→wf1o 5253 ‘cfv 5254 (class class class)co 5918 ℂcc 7870 0cc0 7872 1c1 7873 + caddc 7875 ≤ cle 8055 ℕcn 8982 ℕ₀cn0 9240 ℤcz 9317 ℤ_≥cuz 9592 ...cfz 10074 seqcseq 10518 ♯chash 10846 ⇝ cli 11421

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 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-coll 4144 ax-sep 4147 ax-nul 4155 ax-pow 4203 ax-pr 4238 ax-un 4464 ax-setind 4569 ax-iinf 4620 ax-cnex 7963 ax-resscn 7964 ax-1cn 7965 ax-1re 7966 ax-icn 7967 ax-addcl 7968 ax-addrcl 7969 ax-mulcl 7970 ax-mulrcl 7971 ax-addcom 7972 ax-mulcom 7973 ax-addass 7974 ax-mulass 7975 ax-distr 7976 ax-i2m1 7977 ax-0lt1 7978 ax-1rid 7979 ax-0id 7980 ax-rnegex 7981 ax-precex 7982 ax-cnre 7983 ax-pre-ltirr 7984 ax-pre-ltwlin 7985 ax-pre-lttrn 7986 ax-pre-apti 7987 ax-pre-ltadd 7988 ax-pre-mulgt0 7989 ax-pre-mulext 7990 ax-arch 7991 ax-caucvg 7992

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 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-ral 2477 df-rex 2478 df-reu 2479 df-rmo 2480 df-rab 2481 df-v 2762 df-sbc 2986 df-csb 3081 df-dif 3155 df-un 3157 df-in 3159 df-ss 3166 df-nul 3447 df-if 3558 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-uni 3836 df-int 3871 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-tr 4128 df-id 4324 df-po 4327 df-iso 4328 df-iord 4397 df-on 4399 df-ilim 4400 df-suc 4402 df-iom 4623 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-res 4671 df-ima 4672 df-iota 5215 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 df-fv 5262 df-isom 5263 df-riota 5873 df-ov 5921 df-oprab 5922 df-mpo 5923 df-1st 6193 df-2nd 6194 df-recs 6358 df-irdg 6423 df-frec 6444 df-1o 6469 df-oadd 6473 df-er 6587 df-en 6795 df-dom 6796 df-fin 6797 df-pnf 8056 df-mnf 8057 df-xr 8058 df-ltxr 8059 df-le 8060 df-sub 8192 df-neg 8193 df-reap 8594 df-ap 8601 df-div 8692 df-inn 8983 df-2 9041 df-3 9042 df-4 9043 df-n0 9241 df-z 9318 df-uz 9593 df-q 9685 df-rp 9720 df-fz 10075 df-fzo 10209 df-seqfrec 10519 df-exp 10610 df-ihash 10847 df-cj 10986 df-re 10987 df-im 10988 df-rsqrt 11142 df-abs 11143 df-clim 11422

This theorem is referenced by: fsum3 11530

W3C validator