Theorem seq3f1oleml 10269

Description: Lemma for seq3f1o 10270. 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 3927	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
9		2fveq3 5419	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑥)))
10	8, 9	ifbieq1d 3489	. . . 4 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
11	10	cbvmptv 4019	. . 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 10268	. 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 5414	. . . . . 6 ⊢ (𝑏 = 𝑥 → (𝑓‘𝑏) = (𝑓‘𝑥))
14		id 19	. . . . . 6 ⊢ (𝑏 = 𝑥 → 𝑏 = 𝑥)
15	13, 14	eqeq12d 2152	. . . . 5 ⊢ (𝑏 = 𝑥 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑥) = 𝑥))
16	15	cbvralv 2652	. . . 4 ⊢ (∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥)
17	16	3anbi2i 1173	. . 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 989	. . . 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 274	. . . . 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 9795	. . . . . . . 8 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ∈ (ℤ≥‘𝑀))
21	20	adantl 275	. . . . . . 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 9801	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀...𝑁) → 𝑘 ≤ 𝑁)
23	22	adantl 275	. . . . . . . . 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 3476	. . . . . . . 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 5414	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → (𝑓‘𝑏) = (𝑓‘𝑘))
26		id 19	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → 𝑏 = 𝑘)
27	25, 26	eqeq12d 2152	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑘 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑘) = 𝑘))
28		simplr2 1024	. . . . . . . . . . 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 109	. . . . . . . . . . 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 2789	. . . . . . . . . 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 5418	. . . . . . . . 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 5414	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
33	32	eleq1d 2206	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑘) ∈ 𝑆))
34	6	ralrimiva 2503	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
35	34	ad2antrr 479	. . . . . . . . . 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 2789	. . . . . . . . 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 2214	. . . . . . . 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 2214	. . . . . . 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 3927	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
40		2fveq3 5419	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑘)))
41	39, 40	ifbieq1d 3489	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)))
42		eqid 2137	. . . . . . . 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 5490	. . . . . . 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 408	. . . . . 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 2174	. . . . 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 109	. . . . . . 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 5414	. . . . . . . . . 10 ⊢ (𝑎 = (𝑓‘𝑥) → (𝐺‘𝑎) = (𝐺‘(𝑓‘𝑥)))
48	47	eleq1d 2206	. . . . . . . . 9 ⊢ (𝑎 = (𝑓‘𝑥) → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓‘𝑥)) ∈ 𝑆))
49	34	ad3antrrr 483	. . . . . . . . . 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 5414	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝐺‘𝑎) = (𝐺‘𝑥))
51	50	eleq1d 2206	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
52	51	cbvralv 2652	. . . . . . . . . 10 ⊢ (∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆 ↔ ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
53	49, 52	sylibr 133	. . . . . . . . 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 987	. . . . . . . . . . . . 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 479	. . . . . . . . . . . 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 5360	. . . . . . . . . . . 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 109	. . . . . . . . . . . 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 274	. . . . . . . . . . . . 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 9328	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
61	4, 60	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	ad3antrrr 483	. . . . . . . . . . . . 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 9791	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ≤ 𝑁))
64	59, 62, 63	syl2anc 408	. . . . . . . . . . . 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 166	. . . . . . . . . . 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	ffvelrnd 5549	. . . . . . . . . 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 9795	. . . . . . . . . 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 2789	. . . . . . . 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 5414	. . . . . . . . . 10 ⊢ (𝑎 = 𝑀 → (𝐺‘𝑎) = (𝐺‘𝑀))
71	70	eleq1d 2206	. . . . . . . . 9 ⊢ (𝑎 = 𝑀 → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘𝑀) ∈ 𝑆))
72	34, 52	sylibr 133	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑎 ∈ (ℤ≥‘𝑀)(𝐺‘𝑎) ∈ 𝑆)
73	72	ad3antrrr 483	. . . . . . . . 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 9324	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
75	4, 74	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
76	75	ad3antrrr 483	. . . . . . . . . 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 9333	. . . . . . . . . 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 2789	. . . . . . . 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 9328	. . . . . . . . . 10 ⊢ (𝑥 ∈ (ℤ≥‘𝑀) → 𝑥 ∈ ℤ)
81	80	adantl 275	. . . . . . . . 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 479	. . . . . . . . 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 9120	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
84	81, 82, 83	syl2anc 408	. . . . . . . 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 3496	. . . . . . 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 5490	. . . . . . 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 408	. . . . . 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 2214	. . . . 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 468	. . . . 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 468	. . . . 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 10237	. . . 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 2172	. . 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 286	. 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 1870	1 ⊢ (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104  DECID wdc 819   ∧ w3a 962   = wceq 1331   ∈ wcel 1480  ∀wral 2414  ifcif 3469   class class class wbr 3924   ↦ cmpt 3984  ⟶wf 5114  –1-1-onto→wf1o 5117  ‘cfv 5118  (class class class)co 5767   ≤ cle 7794  ℤcz 9047  ℤ_≥cuz 9319  ...cfz 9783  seqcseq 10211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-addcom 7713  ax-addass 7715  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-0id 7721  ax-rnegex 7722  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-if 3470  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-ilim 4286  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-frec 6281  df-1o 6306  df-er 6422  df-en 6628  df-fin 6630  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-inn 8714  df-n0 8971  df-z 9048  df-uz 9320  df-fz 9784  df-fzo 9913  df-seqfrec 10212

This theorem is referenced by:  seq3f1o  10270