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Theorem seq3f1olemp 10458

Description: Lemma for seq3f1o 10460. 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 9988	. . 3 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
3	1, 2	syl 14	. 2 ⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁))
4		oveq2 5861	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
5	4	raleqdv 2671	. . . . . 6 ⊢ (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑀)(𝑓‘𝑥) = 𝑥))
6	5	3anbi2d 1312	. . . . 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 1818	. . . 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 229	. . 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 5861	. . . . . . 7 ⊢ (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
10	9	raleqdv 2671	. . . . . 6 ⊢ (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥))
11	10	3anbi2d 1312	. . . . 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 1818	. . . 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 229	. . 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 5861	. . . . . . 7 ⊢ (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
15	14	raleqdv 2671	. . . . . 6 ⊢ (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓‘𝑥) = 𝑥))
16	15	3anbi2d 1312	. . . . 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 1818	. . . 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 229	. . 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 5861	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
20	19	raleqdv 2671	. . . . . 6 ⊢ (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓‘𝑥) = 𝑥))
21	20	3anbi2d 1312	. . . . 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 1818	. . . 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 229	. . 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 9987	. . . . . 6 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
30	1, 29	syl 14	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝑀...𝑁))
31		ral0 3516	. . . . . . 7 ⊢ ∀𝑥 ∈ ∅ (𝐹‘𝑥) = 𝑥
32		fzo0 10124	. . . . . . . 8 ⊢ (𝑀..^𝑀) = ∅
33	32	raleqi 2669	. . . . . . 7 ⊢ (∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ ∅ (𝐹‘𝑥) = 𝑥)
34	31, 33	mpbir 145	. . . . . 6 ⊢ ∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥
35	34	a1i 9	. . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝑀)(𝐹‘𝑥) = 𝑥)
36		f1of 5442	. . . . . . . . . 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 9492	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
39	1, 38	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
40		eluzelz 9496	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
41	1, 40	syl 14	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
42	39, 41	fzfigd 10387	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
43		fex 5725	. . . . . . . . 9 ⊢ ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐹 ∈ V)
44	37, 42, 43	syl2anc 409	. . . . . . . 8 ⊢ (𝜑 → 𝐹 ∈ V)
45		fveq1 5495	. . . . . . . . . . . . 13 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
46	45	fveq2d 5500	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝐹 → (𝐺‘(𝑓‘𝑥)) = (𝐺‘(𝐹‘𝑥)))
47	46	ifeq1d 3543	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)) = if(𝑥 ≤ 𝑁, (𝐺‘(𝐹‘𝑥)), (𝐺‘𝑀)))
48	47	mpteq2dv 4080	. . . . . . . . . 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 2228	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → 𝑃 = 𝐿)
52	51	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 = 𝐹) → 𝑃 = 𝐿)
53	44, 52	csbied 3095	. . . . . . 7 ⊢ (𝜑 → ⦋𝐹 / 𝑓⦌𝑃 = 𝐿)
54	53	seqeq3d 10409	. . . . . 6 ⊢ (𝜑 → seq𝑀( + , ⦋𝐹 / 𝑓⦌𝑃) = seq𝑀( + , 𝐿))
55	54	fveq1d 5498	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , ⦋𝐹 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
56	24, 25, 26, 1, 27, 28, 30, 27, 35, 55, 49	seq3f1olemstep 10457	. . . 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 1521	. . . . . . . 8 ⊢ Ⅎ𝑔(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
59		nfv 1521	. . . . . . . . 9 ⊢ Ⅎ𝑓 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
60		nfv 1521	. . . . . . . . 9 ⊢ Ⅎ𝑓∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥
61		nfcv 2312	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓𝑀
62		nfcv 2312	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓 +
63		nfcsb1v 3082	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓⦋𝑔 / 𝑓⦌𝑃
64	61, 62, 63	nfseq 10411	. . . . . . . . . . 11 ⊢ Ⅎ𝑓seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)
65		nfcv 2312	. . . . . . . . . . 11 ⊢ Ⅎ𝑓𝑁
66	64, 65	nffv 5506	. . . . . . . . . 10 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁)
67	66	nfeq1 2322	. . . . . . . . 9 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
68	59, 60, 67	nf3an 1559	. . . . . . . 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 5431	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
70		fveq1 5495	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → (𝑓‘𝑥) = (𝑔‘𝑥))
71	70	eqeq1d 2179	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → ((𝑓‘𝑥) = 𝑥 ↔ (𝑔‘𝑥) = 𝑥))
72	71	ralbidv 2470	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑀...𝑘)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥))
73		csbeq1a 3058	. . . . . . . . . . . 12 ⊢ (𝑓 = 𝑔 → 𝑃 = ⦋𝑔 / 𝑓⦌𝑃)
74	73	seqeq3d 10409	. . . . . . . . . . 11 ⊢ (𝑓 = 𝑔 → seq𝑀( + , 𝑃) = seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃))
75	74	fveq1d 5498	. . . . . . . . . 10 ⊢ (𝑓 = 𝑔 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁))
76	75	eqeq1d 2179	. . . . . . . . 9 ⊢ (𝑓 = 𝑔 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , ⦋𝑔 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
77	69, 72, 76	3anbi123d 1307	. . . . . . . 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 1749	. . . . . . 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 5496	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (𝑔‘𝑥) = (𝑔‘𝑎))
80		id 19	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
81	79, 80	eqeq12d 2185	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → ((𝑔‘𝑥) = 𝑥 ↔ (𝑔‘𝑎) = 𝑎))
82	81	cbvralv 2696	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀...𝑘)(𝑔‘𝑥) = 𝑥 ↔ ∀𝑎 ∈ (𝑀...𝑘)(𝑔‘𝑎) = 𝑎)
83	82	3anbi2i 1186	. . . . . . . 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 1598	. . . . . . 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 183	. . . . . 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 524	. . . . . . . . . 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 281	. . . . . . . . 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 281	. . . . . . . . 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 281	. . . . . . . . 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 485	. . . . . . . . 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 485	. . . . . . . . 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 281	. . . . . . . . 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 10183	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
94	93	ad2antlr 486	. . . . . . . . 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 998	. . . . . . . . 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 999	. . . . . . . . . . 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 133	. . . . . . . . . 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 10103	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
99	98	ad2antlr 486	. . . . . . . . . . 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 10160	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℤ → (𝑀...𝑘) = (𝑀..^(𝑘 + 1)))
101	100	raleqdv 2671	. . . . . . . . . . 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 146	. . . . . . . . 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 1000	. . . . . . . . 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 10457	. . . . . . . 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 114	. . . . . . 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 1812	. . . . . 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	syl5bi 151	. . . . 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 115	. . . 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 10195	. 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 103 ↔ wb 104 ∧ w3a 973 = wceq 1348 ∃wex 1485 ∈ wcel 2141 ∀wral 2448 Vcvv 2730 ⦋csb 3049 ∅c0 3414 ifcif 3526 class class class wbr 3989 ↦ cmpt 4050 ⟶wf 5194 –1-1-onto→wf1o 5197 ‘cfv 5198 (class class class)co 5853 Fincfn 6718 1c1 7775 + caddc 7777 ≤ cle 7955 ℤcz 9212 ℤ_≥cuz 9487 ...cfz 9965 ..^cfzo 10098 seqcseq 10401

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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4104 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 ax-cnex 7865 ax-resscn 7866 ax-1cn 7867 ax-1re 7868 ax-icn 7869 ax-addcl 7870 ax-addrcl 7871 ax-mulcl 7872 ax-addcom 7874 ax-addass 7876 ax-distr 7878 ax-i2m1 7879 ax-0lt1 7880 ax-0id 7882 ax-rnegex 7883 ax-cnre 7885 ax-pre-ltirr 7886 ax-pre-ltwlin 7887 ax-pre-lttrn 7888 ax-pre-apti 7889 ax-pre-ltadd 7890

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-ilim 4354 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-riota 5809 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-recs 6284 df-frec 6370 df-1o 6395 df-er 6513 df-en 6719 df-fin 6721 df-pnf 7956 df-mnf 7957 df-xr 7958 df-ltxr 7959 df-le 7960 df-sub 8092 df-neg 8093 df-inn 8879 df-n0 9136 df-z 9213 df-uz 9488 df-fz 9966 df-fzo 10099 df-seqfrec 10402

This theorem is referenced by: seq3f1oleml 10459

W3C validator