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Theorem seq3f1olemp 10783

Description: Lemma for seq3f1o 10785. Existence of a constant permutation of (𝑀...𝑁) which leads to the same sum as the permutation 𝐹 itself. (Contributed by Jim Kingdon, 18-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)))
iseqf1o.p	⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))

**Assertion**
Ref	Expression
seq3f1olemp	⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

Distinct variable groups: + ,𝑓,𝑥,𝑦,𝑧 𝑓,𝐹,𝑥,𝑦,𝑧 𝑓,𝐿,𝑥,𝑦,𝑧 𝑓,𝑀,𝑥,𝑦,𝑧 𝑓,𝑁,𝑥,𝑦,𝑧 𝑥,𝑃,𝑦,𝑧 𝑆,𝑓,𝑥,𝑦,𝑧 𝜑,𝑓,𝑥,𝑦,𝑧 𝑓,𝐺,𝑥 Allowed substitution hints: 𝑃(𝑓) 𝐺(𝑦,𝑧)

Proof of Theorem seq3f1olemp Dummy variables 𝑔 𝑘 𝑤 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1o.4	. . 3 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
2		eluzfz2 10272	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		oveq2 6031	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
5	4	raleqdv 2735	. . . . . 6 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥))
6	5	3anbi2d 1353	. . . . 5 ⊢ (𝑤 = 𝑀 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
7	6	exbidv 1872	. . . 4 ⊢ (𝑤 = 𝑀 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
8	7	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑀 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
9		oveq2 6031	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
10	9	raleqdv 2735	. . . . . 6 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥))
11	10	3anbi2d 1353	. . . . 5 ⊢ (𝑤 = 𝑘 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
12	11	exbidv 1872	. . . 4 ⊢ (𝑤 = 𝑘 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
13	12	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑘 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
14		oveq2 6031	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
15	14	raleqdv 2735	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥))
16	15	3anbi2d 1353	. . . . 5 ⊢ (𝑤 = (𝑘 + 1) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
17	16	exbidv 1872	. . . 4 ⊢ (𝑤 = (𝑘 + 1) → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
18	17	imbi2d 230	. . 3 ⊢ (𝑤 = (𝑘 + 1) → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
19		oveq2 6031	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
20	19	raleqdv 2735	. . . . . 6 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥))
21	20	3anbi2d 1353	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
22	21	exbidv 1872	. . . 4 ⊢ (𝑤 = 𝑁 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
23	22	imbi2d 230	. . 3 ⊢ (𝑤 = 𝑁 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
24		iseqf1o.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
25		iseqf1o.2	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
26		iseqf1o.3	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
27		iseqf1o.6	. . . . 5 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
28		iseqf1o.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
29		eluzfz1 10271	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
30	1, 29	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
31		ral0 3595	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ (𝐹‘𝑥) = 𝑥
32		fzo0 10410	. . . . . . . 8 ⊢ (𝑀..^𝑀) = ∅
33	32	raleqi 2733	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ ∅ (𝐹‘𝑥) = 𝑥)
34	31, 33	mpbir 146	. . . . . 6 ⊢ ∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥
35	34	a1i 9	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥)
36		f1of 5586	. . . . . . . . . 10 ⊢ (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
37	27, 36	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
38		eluzel2 9765	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
39	1, 38	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
40		eluzelz 9770	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
41	1, 40	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42	39, 41	fzfigd 10699	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
43		fex 5888	. . . . . . . . 9 ⊢ ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐹 ∈ V)
44	37, 42, 43	syl2anc 411	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
45		fveq1 5641	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
46	45	fveq2d 5646	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝐺‘(𝑓‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
47	46	ifeq1d 3624	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
48	47	mpteq2dv 4181	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀))) = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀))))
49		iseqf1o.p	. . . . . . . . . 10 ⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
50		iseqf1o.l	. . . . . . . . . 10 ⊢ 𝐿 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
51	48, 49, 50	3eqtr4g 2288	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → 𝑃 = 𝐿)
52	51	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑃 = 𝐿)
53	44, 52	csbied 3173	. . . . . . 7 ⊢ (𝜑 → ⦋𝐹 / 𝑓⦌𝑃 = 𝐿)
54	53	seqeq3d 10723	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , ⦋𝐹 / 𝑓⦌𝑃) = seq𝑀( + , 𝐿))
55	54	fveq1d 5644	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , ⦋𝐹 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
56	24, 25, 26, 1, 27, 28, 30, 27, 35, 55, 49	seq3f1olemstep 10782	. . . 4 ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
57	56	a1i 9	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
58		nfv 1576	. . . . . . . 8 ⊢ Ⅎ𝑔(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
59		nfv 1576	. . . . . . . . 9 ⊢ Ⅎ𝑓 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
60		nfv 1576	. . . . . . . . 9 ⊢ Ⅎ𝑓∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥
61		nfcv 2373	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝑀
62		nfcv 2373	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓 +
63		nfcsb1v 3159	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓⦋𝑔 / 𝑓⦌𝑃
64	61, 62, 63	nfseq 10725	. . . . . . . . . . 11 ⊢ Ⅎ𝑓seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)
65		nfcv 2373	. . . . . . . . . . 11 ⊢ Ⅎ𝑓𝑁
66	64, 65	nffv 5652	. . . . . . . . . 10 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁)
67	66	nfeq1 2383	. . . . . . . . 9 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
68	59, 60, 67	nf3an 1614	. . . . . . . 8 ⊢ Ⅎ𝑓(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
69		f1oeq1 5574	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
70		fveq1 5641	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
71	70	eqeq1d 2239	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = 𝑥 ↔ (𝑔‘𝑥) = 𝑥))
72	71	ralbidv 2531	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥))
73		csbeq1a 3135	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → 𝑃 = ⦋𝑔 / 𝑓⦌𝑃)
74	73	seqeq3d 10723	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → seq𝑀( + , 𝑃) = seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃))
75	74	fveq1d 5644	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁))
76	75	eqeq1d 2239	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
77	69, 72, 76	3anbi123d 1348	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
78	58, 68, 77	cbvex 1803	. . . . . . 7 ⊢ (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
79		fveq2 5642	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑔‘𝑥) = (𝑔‘𝑎))
80		id 19	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
81	79, 80	eqeq12d 2245	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑔‘𝑥) = 𝑥 ↔ (𝑔‘𝑎) = 𝑎))
82	81	cbvralv 2766	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ↔ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎)
83	82	3anbi2i 1217	. . . . . . . 8 ⊢ ((𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
84	83	exbii 1653	. . . . . . 7 ⊢ (∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
85	78, 84	bitri 184	. . . . . 6 ⊢ (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
86		simpll 527	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝜑)
87	86, 24	sylan 283	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
88	86, 25	sylan 283	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
89	86, 26	sylan 283	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
90	1	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑁 ∈ (ℤ_≥‘𝑀))
91	27	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
92	86, 28	sylan 283	. . . . . . . . 9 ⊢ ((((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
93		fzofzp1 10478	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
94	93	ad2antlr 489	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (𝑘 + 1) ∈ (𝑀...𝑁))
95		simpr1 1029	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
96		simpr2 1030	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎)
97	96, 82	sylibr 134	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥)
98		elfzoelz 10387	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
99	98	ad2antlr 489	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑘 ∈ ℤ)
100		fzval3 10455	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤ → (𝑀...𝑘) = (𝑀..^(𝑘 + 1)))
101	100	raleqdv 2735	. . . . . . . . . . 11 ⊢ (𝑘 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔‘𝑥) = 𝑥))
102	99, 101	syl 14	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔‘𝑥) = 𝑥))
103	97, 102	mpbid 147	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔‘𝑥) = 𝑥)
104		simpr3 1031	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
105	87, 88, 89, 90, 91, 92, 94, 95, 103, 104, 49	seq3f1olemstep 10782	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
106	105	ex 115	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
107	106	exlimdv 1866	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎 ∧ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
108	85, 107	biimtrid 152	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
109	108	expcom 116	. . . 4 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
110	109	a2d 26	. . 3 ⊢ (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
111	8, 13, 18, 23, 57, 110	fzind2 10491	. 2 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
112	3, 111	mpcom 36	1 ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 1004 = wceq 1397 ∃wex 1540 ∈ wcel 2201 ∀wral 2509 Vcvv 2801 ⦋csb 3126 ∅c0 3493 ifcif 3604 class class class wbr 4089 ↦ cmpt 4151 ⟶wf 5324 –1-1-onto→wf1o 5327 ‘cfv 5328 (class class class)co 6023 Fincfn 6914 1c1 8038 + caddc 8040 ≤ cle 8220 ℤcz 9484 ℤ_≥cuz 9760 ...cfz 10248 ..^cfzo 10382 seqcseq 10715

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 619 ax-in2 620 ax-io 716 ax-5 1495 ax-7 1496 ax-gen 1497 ax-ie1 1541 ax-ie2 1542 ax-8 1552 ax-10 1553 ax-11 1554 ax-i12 1555 ax-bndl 1557 ax-4 1558 ax-17 1574 ax-i9 1578 ax-ial 1582 ax-i5r 1583 ax-13 2203 ax-14 2204 ax-ext 2212 ax-coll 4205 ax-sep 4208 ax-nul 4216 ax-pow 4266 ax-pr 4301 ax-un 4532 ax-setind 4637 ax-iinf 4688 ax-cnex 8128 ax-resscn 8129 ax-1cn 8130 ax-1re 8131 ax-icn 8132 ax-addcl 8133 ax-addrcl 8134 ax-mulcl 8135 ax-addcom 8137 ax-addass 8139 ax-distr 8141 ax-i2m1 8142 ax-0lt1 8143 ax-0id 8145 ax-rnegex 8146 ax-cnre 8148 ax-pre-ltirr 8149 ax-pre-ltwlin 8150 ax-pre-lttrn 8151 ax-pre-apti 8152 ax-pre-ltadd 8153

This theorem depends on definitions: df-bi 117 df-dc 842 df-3or 1005 df-3an 1006 df-tru 1400 df-fal 1403 df-nf 1509 df-sb 1810 df-eu 2081 df-mo 2082 df-clab 2217 df-cleq 2223 df-clel 2226 df-nfc 2362 df-ne 2402 df-nel 2497 df-ral 2514 df-rex 2515 df-reu 2516 df-rab 2518 df-v 2803 df-sbc 3031 df-csb 3127 df-dif 3201 df-un 3203 df-in 3205 df-ss 3212 df-nul 3494 df-if 3605 df-pw 3655 df-sn 3676 df-pr 3677 df-op 3679 df-uni 3895 df-int 3930 df-iun 3973 df-br 4090 df-opab 4152 df-mpt 4153 df-tr 4189 df-id 4392 df-iord 4465 df-on 4467 df-ilim 4468 df-suc 4470 df-iom 4691 df-xp 4733 df-rel 4734 df-cnv 4735 df-co 4736 df-dm 4737 df-rn 4738 df-res 4739 df-ima 4740 df-iota 5288 df-fun 5330 df-fn 5331 df-f 5332 df-f1 5333 df-fo 5334 df-f1o 5335 df-fv 5336 df-riota 5976 df-ov 6026 df-oprab 6027 df-mpo 6028 df-1st 6308 df-2nd 6309 df-recs 6476 df-frec 6562 df-1o 6587 df-er 6707 df-en 6915 df-fin 6917 df-pnf 8221 df-mnf 8222 df-xr 8223 df-ltxr 8224 df-le 8225 df-sub 8357 df-neg 8358 df-inn 9149 df-n0 9408 df-z 9485 df-uz 9761 df-fz 10249 df-fzo 10383 df-seqfrec 10716

This theorem is referenced by: seq3f1oleml 10784

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