Theorem seq3f1oleml 10608

Description: Lemma for seq3f1o 10609. This is more or less the result, but stated in terms of 𝐹 and 𝐺 without 𝐻. 𝐿 and 𝐻 may differ in terms of what happens to terms after 𝑁. The terms after 𝑁 don't matter for the value at 𝑁 but we need some definition given the way our theorems concerning seq work. (Contributed by Jim Kingdon, 17-Aug-2022.)

Hypotheses

Ref	Expression
iseqf1o.1	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4	⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
iseqf1o.6	⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7	⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
iseqf1o.l	⊢ 𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))

Assertion

Ref	Expression
seq3f1oleml	⊢ (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Distinct variable groups:   𝑥, + ,𝑦,𝑧   𝑥,𝐹,𝑦,𝑧   𝑥,𝐺,𝑦,𝑧   𝑥,𝐿,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝑁,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Proof of Theorem seq3f1oleml

Dummy variables 𝑓 𝑘 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1o.1	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
2		iseqf1o.2	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
3		iseqf1o.3	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
4		iseqf1o.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀))
5		iseqf1o.6	. . 3 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
6		iseqf1o.7	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
7		iseqf1o.l	. . 3 ⊢ 𝐿 = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
8		breq1 4036	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
9		2fveq3 5563	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑥)))
10	8, 9	ifbieq1d 3583	. . . 4 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
11	10	cbvmptv 4129	. . 3 ⊢ (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
12	1, 2, 3, 4, 5, 6, 7, 11	seq3f1olemp 10607	. 2 ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
13		fveq2 5558	. . . . . 6 ⊢ (𝑏 = 𝑥 → (𝑓‘𝑏) = (𝑓‘𝑥))
14		id 19	. . . . . 6 ⊢ (𝑏 = 𝑥 → 𝑏 = 𝑥)
15	13, 14	eqeq12d 2211	. . . . 5 ⊢ (𝑏 = 𝑥 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑥) = 𝑥))
16	15	cbvralv 2729	. . . 4 ⊢ (∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥)
17	16	3anbi2i 1193	. . 3 ⊢ ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
18		simpr3 1007	. . . 4 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
19	4	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑁 ∈ (ℤ≥‘𝑀))
20		elfzuz 10096	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
21	20	adantl 277	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (ℤ≥‘𝑀))
22		elfzle2 10103	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
23	22	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ≤ 𝑁)
24	23	iftrued 3568	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)) = (𝐺‘(𝑓‘𝑘)))
25		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → (𝑓‘𝑏) = (𝑓‘𝑘))
26		id 19	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → 𝑏 = 𝑘)
27	25, 26	eqeq12d 2211	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑘 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑘) = 𝑘))
28		simplr2 1042	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏)
29		simpr 110	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → 𝑘 ∈ (𝑀...𝑁))
30	27, 28, 29	rspcdva 2873	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝑓‘𝑘) = 𝑘)
31	30	fveq2d 5562	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓‘𝑘)) = (𝐺‘𝑘))
32		fveq2 5558	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
33	32	eleq1d 2265	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑘) ∈ 𝑆))
34	6	ralrimiva 2570	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
35	34	ad2antrr 488	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
36	33, 35, 21	rspcdva 2873	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘𝑘) ∈ 𝑆)
37	31, 36	eqeltrd 2273	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐺‘(𝑓‘𝑘)) ∈ 𝑆)
38	24, 37	eqeltrd 2273	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆)
39		breq1 4036	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
40		2fveq3 5563	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑘)))
41	39, 40	ifbieq1d 3583	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)))
42		eqid 2196	. . . . . . . 8 ⊢ (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))) = (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))
43	41, 42	fvmptg 5637	. . . . . . 7 ⊢ ((𝑘 ∈ (ℤ≥‘𝑀) ∧ if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑘) = if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)))
44	21, 38, 43	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑘) = if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)))
45	44, 24, 31	3eqtrd 2233	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑘) = (𝐺‘𝑘))
46		simpr 110	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ (ℤ≥‘𝑀))
47		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑎 = (𝑓‘𝑥) → (𝐺‘𝑎) = (𝐺‘(𝑓‘𝑥)))
48	47	eleq1d 2265	. . . . . . . . 9 ⊢ (𝑎 = (𝑓‘𝑥) → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓‘𝑥)) ∈ 𝑆))
49	34	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
50		fveq2 5558	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝐺‘𝑎) = (𝐺‘𝑥))
51	50	eleq1d 2265	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
52	51	cbvralv 2729	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆 ↔ ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
53	49, 52	sylibr 134	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
54		simpr1 1005	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
55	54	ad2antrr 488	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
56		f1of 5504	. . . . . . . . . . . 12 ⊢ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
57	55, 56	syl 14	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑓:(𝑀...𝑁)⟶(𝑀...𝑁))
58		simpr 110	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ≤ 𝑁)
59	46	adantr 276	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (ℤ≥‘𝑀))
60		eluzelz 9610	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
61	4, 60	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	ad3antrrr 492	. . . . . . . . . . . . 13 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑁 ∈ ℤ)
63		elfz5 10092	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ≤ 𝑁))
64	59, 62, 63	syl2anc 411	. . . . . . . . . . . 12 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ≤ 𝑁))
65	58, 64	mpbird 167	. . . . . . . . . . 11 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → 𝑥 ∈ (𝑀...𝑁))
66	57, 65	ffvelcdmd 5698	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝑓‘𝑥) ∈ (𝑀...𝑁))
67		elfzuz 10096	. . . . . . . . . 10 ⊢ ((𝑓‘𝑥) ∈ (𝑀...𝑁) → (𝑓‘𝑥) ∈ (ℤ≥‘𝑀))
68	66, 67	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝑓‘𝑥) ∈ (ℤ≥‘𝑀))
69	48, 53, 68	rspcdva 2873	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ 𝑥 ≤ 𝑁) → (𝐺‘(𝑓‘𝑥)) ∈ 𝑆)
70		fveq2 5558	. . . . . . . . . 10 ⊢ (𝑎 = 𝑀 → (𝐺‘𝑎) = (𝐺‘𝑀))
71	70	eleq1d 2265	. . . . . . . . 9 ⊢ (𝑎 = 𝑀 → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
72	34, 52	sylibr 134	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
73	72	ad3antrrr 492	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
74		eluzel2 9606	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
75	4, 74	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
76	75	ad3antrrr 492	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → 𝑀 ∈ ℤ)
77		uzid 9615	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ (ℤ≥‘𝑀))
78	76, 77	syl 14	. . . . . . . . 9 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → 𝑀 ∈ (ℤ≥‘𝑀))
79	71, 73, 78	rspcdva 2873	. . . . . . . 8 ⊢ ((((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) ∧ ¬ 𝑥 ≤ 𝑁) → (𝐺‘𝑀) ∈ 𝑆)
80		eluzelz 9610	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
81	80	adantl 277	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑥 ∈ ℤ)
82	61	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → 𝑁 ∈ ℤ)
83		zdcle 9402	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
84	81, 82, 83	syl2anc 411	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → DECID 𝑥 ≤ 𝑁)
85	69, 79, 84	ifcldadc 3590	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆)
86	10, 42	fvmptg 5637	. . . . . . 7 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) ∈ 𝑆) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑥) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
87	46, 85, 86	syl2anc 411	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑥) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
88	87, 85	eqeltrd 2273	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → ((𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)))‘𝑥) ∈ 𝑆)
89	6	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
90	1	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
91	19, 45, 88, 89, 90	seq3fveq 10571	. . . 4 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
92	18, 91	eqtr3d 2231	. . 3 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
93	17, 92	sylan2br 288	. 2 ⊢ ((𝜑 ∧ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑎 ∈ (ℤ≥‘𝑀) ↦ if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀))))‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))
94	12, 93	exlimddv 1913	1 ⊢ (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 835   ∧ w3a 980   = wceq 1364   ∈ wcel 2167  ∀wral 2475  ifcif 3561   class class class wbr 4033   ↦ cmpt 4094  ⟶wf 5254  –1-1-onto→wf1o 5257  ‘cfv 5258  (class class class)co 5922   ≤ cle 8062  ℤcz 9326  ℤ_≥cuz 9601  ...cfz 10083  seqcseq 10539

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-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  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-apti 7994  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-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  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-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-1o 6474  df-er 6592  df-en 6800  df-fin 6802  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-fzo 10218  df-seqfrec 10540

This theorem is referenced by:  seq3f1o  10609