Theorem seq3f1oleml 10955

Description: Lemma for seq3f1o 10956. 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 4133	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝑎 ≤ 𝑁 ↔ 𝑥 ≤ 𝑁))
9		2fveq3 5700	. . . . 5 ⊢ (𝑎 = 𝑥 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑥)))
10	8, 9	ifbieq1d 3663	. . . 4 ⊢ (𝑎 = 𝑥 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
11	10	cbvmptv 4227	. . 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 10954	. 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 5695	. . . . . 6 ⊢ (𝑏 = 𝑥 → (𝑓‘𝑏) = (𝑓‘𝑥))
14		id 19	. . . . . 6 ⊢ (𝑏 = 𝑥 → 𝑏 = 𝑥)
15	13, 14	eqeq12d 2253	. . . . 5 ⊢ (𝑏 = 𝑥 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑥) = 𝑥))
16	15	cbvralv 2786	. . . 4 ⊢ (∀𝑏 ∈ (𝑀...𝑁)(𝑓‘𝑏) = 𝑏 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥)
17	16	3anbi2i 1222	. . 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 1036	. . . 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 10426	. . . . . . . 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 10434	. . . . . . . . . 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 3647	. . . . . . . 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 5695	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → (𝑓‘𝑏) = (𝑓‘𝑘))
26		id 19	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑘 → 𝑏 = 𝑘)
27	25, 26	eqeq12d 2253	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑘 → ((𝑓‘𝑏) = 𝑏 ↔ (𝑓‘𝑘) = 𝑘))
28		simplr2 1071	. . . . . . . . . . 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 2934	. . . . . . . . . 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 5699	. . . . . . . . 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 5695	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑘 → (𝐺‘𝑥) = (𝐺‘𝑘))
33	32	eleq1d 2307	. . . . . . . . . 10 ⊢ (𝑥 = 𝑘 → ((𝐺‘𝑥) ∈ 𝑆 ↔ (𝐺‘𝑘) ∈ 𝑆))
34	6	ralrimiva 2623	. . . . . . . . . . 11 ⊢ (𝜑 → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
35	34	ad2antrr 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))) ∧ 𝑘 ∈ (𝑀...𝑁)) → ∀𝑥 ∈ (ℤ≥‘𝑀)(𝐺‘𝑥) ∈ 𝑆)
36	33, 35, 21	rspcdva 2934	. . . . . . . . 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 2315	. . . . . . . 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 2315	. . . . . . 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 4133	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝑎 ≤ 𝑁 ↔ 𝑘 ≤ 𝑁))
40		2fveq3 5700	. . . . . . . . 9 ⊢ (𝑎 = 𝑘 → (𝐺‘(𝑓‘𝑎)) = (𝐺‘(𝑓‘𝑘)))
41	39, 40	ifbieq1d 3663	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → if(𝑎 ≤ 𝑁, (𝐺‘(𝑓‘𝑎)), (𝐺‘𝑀)) = if(𝑘 ≤ 𝑁, (𝐺‘(𝑓‘𝑘)), (𝐺‘𝑀)))
42		eqid 2238	. . . . . . . 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 5781	. . . . . . 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 415	. . . . . 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 2275	. . . . 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 5695	. . . . . . . . . 10 ⊢ (𝑎 = (𝑓‘𝑥) → (𝐺‘𝑎) = (𝐺‘(𝑓‘𝑥)))
48	47	eleq1d 2307	. . . . . . . . 9 ⊢ (𝑎 = (𝑓‘𝑥) → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘(𝑓‘𝑥)) ∈ 𝑆))
49	34	ad3antrrr 496	. . . . . . . . . 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 5695	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝐺‘𝑎) = (𝐺‘𝑥))
51	50	eleq1d 2307	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → ((𝐺‘𝑎) ∈ 𝑆 ↔ (𝐺‘𝑥) ∈ 𝑆))
52	51	cbvralv 2786	. . . . . . . . . 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 1034	. . . . . . . . . . . . 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 492	. . . . . . . . . . . 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 5639	. . . . . . . . . . . 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 9933	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ ℤ)
61	4, 60	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ ℤ)
62	61	ad3antrrr 496	. . . . . . . . . . . . 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 10422	. . . . . . . . . . . . 13 ⊢ ((𝑥 ∈ (ℤ≥‘𝑀) ∧ 𝑁 ∈ ℤ) → (𝑥 ∈ (𝑀...𝑁) ↔ 𝑥 ≤ 𝑁))
64	59, 62, 63	syl2anc 415	. . . . . . . . . . . 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 5844	. . . . . . . . . 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 10426	. . . . . . . . . 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 2934	. . . . . . . 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 5695	. . . . . . . . . 10 ⊢ (𝑎 = 𝑀 → (𝐺‘𝑎) = (𝐺‘𝑀))
71	70	eleq1d 2307	. . . . . . . . 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 496	. . . . . . . . 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 9928	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ ℤ)
75	4, 74	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℤ)
76	75	ad3antrrr 496	. . . . . . . . . 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 9938	. . . . . . . . . 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 2934	. . . . . . . 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 9933	. . . . . . . . . 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 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)) → 𝑁 ∈ ℤ)
83		zdcle 9723	. . . . . . . . 9 ⊢ ((𝑥 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑥 ≤ 𝑁)
84	81, 82, 83	syl2anc 415	. . . . . . . 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 3670	. . . . . . 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 5781	. . . . . . 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 415	. . . . . 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 2315	. . . . 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 481	. . . . 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 481	. . . . 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 10918	. . . 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 2273	. . 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 1954	1 ⊢ (𝜑 → (seq𝑀( + , 𝐿)‘𝑁) = (seq𝑀( + , 𝐺)‘𝑁))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  DECID wdc 846   ∧ w3a 1009   = wceq 1402   ∈ wcel 2209  ∀wral 2528  ifcif 3638   class class class wbr 4130   ↦ cmpt 4192  ⟶wf 5373  –1-1-onto→wf1o 5376  ‘cfv 5377  (class class class)co 6085   ≤ cle 8361  ℤcz 9646  ℤ_≥cuz 9923  ...cfz 10413  seqcseq 10886

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9306  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-fzo 10552  df-seqfrec 10887

This theorem is used by:  seq3f1o  10956