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Theorem seq3f1olemstep 10777

Description: Lemma for seq3f1o 10780. Given a permutation which is constant up to a point, supply a new one which is constant for one more position. (Contributed by Jim Kingdon, 19-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)
iseqf1olemstep.k	⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
iseqf1olemstep.j	⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemstep.const	⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
seq3f1olemstep.jp	⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
seq3f1olemstep.p	⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))

**Assertion**
Ref	Expression
seq3f1olemstep	⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

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

Proof of Theorem seq3f1olemstep Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iseqf1olemstep.j	. . . . . 6 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
2		f1of 5583	. . . . . 6 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
3	1, 2	syl 14	. . . . 5 ⊢ (𝜑 → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
4		iseqf1olemstep.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ (𝑀...𝑁))
5		elfzel1 10259	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
6	4, 5	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		elfzel2 10258	. . . . . . 7 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
8	4, 7	syl 14	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9	6, 8	fzfigd 10694	. . . . 5 ⊢ (𝜑 → (𝑀...𝑁) ∈ Fin)
10		fex 5883	. . . . 5 ⊢ ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
11	3, 9, 10	syl2anc 411	. . . 4 ⊢ (𝜑 → 𝐽 ∈ V)
12	11	adantr 276	. . 3 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → 𝐽 ∈ V)
13	1	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
14		iseqf1olemstep.const	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
15	14	adantr 276	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
16		eqcom 2233	. . . . . . . . . 10 ⊢ (𝐾 = (^◡𝐽‘𝐾) ↔ (^◡𝐽‘𝐾) = 𝐾)
17	16	biimpi 120	. . . . . . . . 9 ⊢ (𝐾 = (^◡𝐽‘𝐾) → (^◡𝐽‘𝐾) = 𝐾)
18	17	adantl 277	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (^◡𝐽‘𝐾) = 𝐾)
19		f1ocnvfvb 5921	. . . . . . . . . 10 ⊢ ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐽‘𝐾) = 𝐾 ↔ (^◡𝐽‘𝐾) = 𝐾))
20	1, 4, 4, 19	syl3anc 1273	. . . . . . . . 9 ⊢ (𝜑 → ((𝐽‘𝐾) = 𝐾 ↔ (^◡𝐽‘𝐾) = 𝐾))
21	20	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ((𝐽‘𝐾) = 𝐾 ↔ (^◡𝐽‘𝐾) = 𝐾))
22	18, 21	mpbird 167	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (𝐽‘𝐾) = 𝐾)
23		elfzelz 10260	. . . . . . . . . 10 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
24	4, 23	syl 14	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℤ)
25	24	adantr 276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → 𝐾 ∈ ℤ)
26		fveq2 5639	. . . . . . . . . 10 ⊢ (𝑥 = 𝐾 → (𝐽‘𝑥) = (𝐽‘𝐾))
27		id 19	. . . . . . . . . 10 ⊢ (𝑥 = 𝐾 → 𝑥 = 𝐾)
28	26, 27	eqeq12d 2246	. . . . . . . . 9 ⊢ (𝑥 = 𝐾 → ((𝐽‘𝑥) = 𝑥 ↔ (𝐽‘𝐾) = 𝐾))
29	28	ralsng 3709	. . . . . . . 8 ⊢ (𝐾 ∈ ℤ → (∀𝑥 ∈ {𝐾} (𝐽‘𝑥) = 𝑥 ↔ (𝐽‘𝐾) = 𝐾))
30	25, 29	syl 14	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (∀𝑥 ∈ {𝐾} (𝐽‘𝑥) = 𝑥 ↔ (𝐽‘𝐾) = 𝐾))
31	22, 30	mpbird 167	. . . . . 6 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ {𝐾} (𝐽‘𝑥) = 𝑥)
32		ralun 3389	. . . . . 6 ⊢ ((∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥 ∧ ∀𝑥 ∈ {𝐾} (𝐽‘𝑥) = 𝑥) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽‘𝑥) = 𝑥)
33	15, 31, 32	syl2anc 411	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽‘𝑥) = 𝑥)
34		elfzuz 10256	. . . . . . . 8 ⊢ (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ_≥‘𝑀))
35		fzisfzounsn 10483	. . . . . . . 8 ⊢ (𝐾 ∈ (ℤ_≥‘𝑀) → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
36	4, 34, 35	3syl 17	. . . . . . 7 ⊢ (𝜑 → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
37	36	raleqdv 2736	. . . . . 6 ⊢ (𝜑 → (∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽‘𝑥) = 𝑥))
38	37	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽‘𝑥) = 𝑥))
39	33, 38	mpbird 167	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥)
40		seq3f1olemstep.jp	. . . . 5 ⊢ (𝜑 → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
41	40	adantr 276	. . . 4 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
42	13, 39, 41	3jca 1203	. . 3 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
43		nfcv 2374	. . . 4 ⊢ Ⅎ𝑓𝐽
44		nfv 1576	. . . . 5 ⊢ Ⅎ𝑓 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
45		nfv 1576	. . . . 5 ⊢ Ⅎ𝑓∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥
46		nfcv 2374	. . . . . . . 8 ⊢ Ⅎ𝑓𝑀
47		nfcv 2374	. . . . . . . 8 ⊢ Ⅎ𝑓 +
48		nfcsb1v 3160	. . . . . . . 8 ⊢ Ⅎ𝑓⦋𝐽 / 𝑓⦌𝑃
49	46, 47, 48	nfseq 10720	. . . . . . 7 ⊢ Ⅎ𝑓seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)
50		nfcv 2374	. . . . . . 7 ⊢ Ⅎ𝑓𝑁
51	49, 50	nffv 5649	. . . . . 6 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁)
52	51	nfeq1 2384	. . . . 5 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
53	44, 45, 52	nf3an 1614	. . . 4 ⊢ Ⅎ𝑓(𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
54		f1oeq1 5571	. . . . 5 ⊢ (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
55		fveq1 5638	. . . . . . 7 ⊢ (𝑓 = 𝐽 → (𝑓‘𝑥) = (𝐽‘𝑥))
56	55	eqeq1d 2240	. . . . . 6 ⊢ (𝑓 = 𝐽 → ((𝑓‘𝑥) = 𝑥 ↔ (𝐽‘𝑥) = 𝑥))
57	56	ralbidv 2532	. . . . 5 ⊢ (𝑓 = 𝐽 → (∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥))
58		csbeq1a 3136	. . . . . . . 8 ⊢ (𝑓 = 𝐽 → 𝑃 = ⦋𝐽 / 𝑓⦌𝑃)
59	58	seqeq3d 10718	. . . . . . 7 ⊢ (𝑓 = 𝐽 → seq𝑀( + , 𝑃) = seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃))
60	59	fveq1d 5641	. . . . . 6 ⊢ (𝑓 = 𝐽 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁))
61	60	eqeq1d 2240	. . . . 5 ⊢ (𝑓 = 𝐽 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
62	54, 57, 61	3anbi123d 1348	. . . 4 ⊢ (𝑓 = 𝐽 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
63	43, 53, 62	spcegf 2889	. . 3 ⊢ (𝐽 ∈ V → ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋𝐽 / 𝑓⦌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))))
64	12, 42, 63	sylc 62	. 2 ⊢ ((𝜑 ∧ 𝐾 = (^◡𝐽‘𝐾)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
65	4	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝐾 ∈ (𝑀...𝑁))
66	1	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
67		eqid 2231	. . . 4 ⊢ (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
68	65, 66, 67	iseqf1olemqf1o 10769	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
69	14	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽‘𝑥) = 𝑥)
70	65, 66, 67, 69	iseqf1olemqk 10770	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥)
71		iseqf1o.1	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
72	71	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
73		iseqf1o.2	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
74	73	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
75		iseqf1o.3	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
76	75	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
77		iseqf1o.4	. . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
78	77	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝑁 ∈ (ℤ_≥‘𝑀))
79		iseqf1o.6	. . . . . 6 ⊢ (𝜑 → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
80	79	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
81		iseqf1o.7	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
82	81	adantlr 477	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) ∧ 𝑥 ∈ (ℤ_≥‘𝑀)) → (𝐺‘𝑥) ∈ 𝑆)
83		neqne 2410	. . . . . 6 ⊢ (¬ 𝐾 = (^◡𝐽‘𝐾) → 𝐾 ≠ (^◡𝐽‘𝐾))
84	83	adantl 277	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝐾 ≠ (^◡𝐽‘𝐾))
85		seq3f1olemstep.p	. . . . 5 ⊢ 𝑃 = (𝑥 ∈ (ℤ_≥‘𝑀) ↦ if(𝑥 ≤ 𝑁, (𝐺‘(𝑓‘𝑥)), (𝐺‘𝑀)))
86	72, 74, 76, 78, 80, 82, 65, 66, 69, 84, 67, 85	seq3f1olemqsum 10776	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁))
87	40	adantr 276	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (seq𝑀( + , ⦋𝐽 / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
88	86, 87	eqtr3d 2266	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
89	65, 5	syl 14	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝑀 ∈ ℤ)
90	65, 7	syl 14	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → 𝑁 ∈ ℤ)
91	89, 90	fzfigd 10694	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (𝑀...𝑁) ∈ Fin)
92		mptexg 5879	. . . 4 ⊢ ((𝑀...𝑁) ∈ Fin → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ∈ V)
93		nfcv 2374	. . . . 5 ⊢ Ⅎ𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))
94		nfv 1576	. . . . . 6 ⊢ Ⅎ𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
95		nfv 1576	. . . . . 6 ⊢ Ⅎ𝑓∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥
96		nfcsb1v 3160	. . . . . . . . 9 ⊢ Ⅎ𝑓⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃
97	46, 47, 96	nfseq 10720	. . . . . . . 8 ⊢ Ⅎ𝑓seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)
98	97, 50	nffv 5649	. . . . . . 7 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁)
99	98	nfeq1 2384	. . . . . 6 ⊢ Ⅎ𝑓(seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
100	94, 95, 99	nf3an 1614	. . . . 5 ⊢ Ⅎ𝑓((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
101		f1oeq1 5571	. . . . . 6 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
102		fveq1 5638	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → (𝑓‘𝑥) = ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥))
103	102	eqeq1d 2240	. . . . . . 7 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → ((𝑓‘𝑥) = 𝑥 ↔ ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥))
104	103	ralbidv 2532	. . . . . 6 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → (∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥))
105		csbeq1a 3136	. . . . . . . . 9 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → 𝑃 = ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)
106	105	seqeq3d 10718	. . . . . . . 8 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → seq𝑀( + , 𝑃) = seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃))
107	106	fveq1d 5641	. . . . . . 7 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁))
108	107	eqeq1d 2240	. . . . . 6 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
109	101, 104, 108	3anbi123d 1348	. . . . 5 ⊢ (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
110	93, 100, 109	spcegf 2889	. . . 4 ⊢ ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) ∈ V → (((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌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))))
111	91, 92, 110	3syl 17	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → (((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , ⦋(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(^◡𝐽‘𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽‘𝑢))) / 𝑓⦌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))))
112	68, 70, 88, 111	mp3and 1376	. 2 ⊢ ((𝜑 ∧ ¬ 𝐾 = (^◡𝐽‘𝐾)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
113		f1ocnv 5596	. . . . . . 7 ⊢ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
114		f1of 5583	. . . . . . 7 ⊢ (^◡𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
115	1, 113, 114	3syl 17	. . . . . 6 ⊢ (𝜑 → ^◡𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
116	115, 4	ffvelcdmd 5783	. . . . 5 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ (𝑀...𝑁))
117		elfzelz 10260	. . . . 5 ⊢ ((^◡𝐽‘𝐾) ∈ (𝑀...𝑁) → (^◡𝐽‘𝐾) ∈ ℤ)
118	116, 117	syl 14	. . . 4 ⊢ (𝜑 → (^◡𝐽‘𝐾) ∈ ℤ)
119		zdceq 9555	. . . 4 ⊢ ((𝐾 ∈ ℤ ∧ (^◡𝐽‘𝐾) ∈ ℤ) → DECID 𝐾 = (^◡𝐽‘𝐾))
120	24, 118, 119	syl2anc 411	. . 3 ⊢ (𝜑 → DECID 𝐾 = (^◡𝐽‘𝐾))
121		exmiddc 843	. . 3 ⊢ (DECID 𝐾 = (^◡𝐽‘𝐾) → (𝐾 = (^◡𝐽‘𝐾) ∨ ¬ 𝐾 = (^◡𝐽‘𝐾)))
122	120, 121	syl 14	. 2 ⊢ (𝜑 → (𝐾 = (^◡𝐽‘𝐾) ∨ ¬ 𝐾 = (^◡𝐽‘𝐾)))
123	64, 112, 122	mpjaodan 805	1 ⊢ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓‘𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 104 ↔ wb 105 ∨ wo 715 DECID wdc 841 ∧ w3a 1004 = wceq 1397 ∃wex 1540 ∈ wcel 2202 ≠ wne 2402 ∀wral 2510 Vcvv 2802 ⦋csb 3127 ∪ cun 3198 ifcif 3605 {csn 3669 class class class wbr 4088 ↦ cmpt 4150 ^◡ccnv 4724 ⟶wf 5322 –1-1-onto→wf1o 5325 ‘cfv 5326 (class class class)co 6018 Fincfn 6909 1c1 8033 ≤ cle 8215 − cmin 8350 ℤcz 9479 ℤ_≥cuz 9755 ...cfz 10243 ..^cfzo 10377 seqcseq 10710

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 2204 ax-14 2205 ax-ext 2213 ax-coll 4204 ax-sep 4207 ax-nul 4215 ax-pow 4264 ax-pr 4299 ax-un 4530 ax-setind 4635 ax-iinf 4686 ax-cnex 8123 ax-resscn 8124 ax-1cn 8125 ax-1re 8126 ax-icn 8127 ax-addcl 8128 ax-addrcl 8129 ax-mulcl 8130 ax-addcom 8132 ax-addass 8134 ax-distr 8136 ax-i2m1 8137 ax-0lt1 8138 ax-0id 8140 ax-rnegex 8141 ax-cnre 8143 ax-pre-ltirr 8144 ax-pre-ltwlin 8145 ax-pre-lttrn 8146 ax-pre-apti 8147 ax-pre-ltadd 8148

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 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2363 df-ne 2403 df-nel 2498 df-ral 2515 df-rex 2516 df-reu 2517 df-rab 2519 df-v 2804 df-sbc 3032 df-csb 3128 df-dif 3202 df-un 3204 df-in 3206 df-ss 3213 df-nul 3495 df-if 3606 df-pw 3654 df-sn 3675 df-pr 3676 df-op 3678 df-uni 3894 df-int 3929 df-iun 3972 df-br 4089 df-opab 4151 df-mpt 4152 df-tr 4188 df-id 4390 df-iord 4463 df-on 4465 df-ilim 4466 df-suc 4468 df-iom 4689 df-xp 4731 df-rel 4732 df-cnv 4733 df-co 4734 df-dm 4735 df-rn 4736 df-res 4737 df-ima 4738 df-iota 5286 df-fun 5328 df-fn 5329 df-f 5330 df-f1 5331 df-fo 5332 df-f1o 5333 df-fv 5334 df-riota 5971 df-ov 6021 df-oprab 6022 df-mpo 6023 df-1st 6303 df-2nd 6304 df-recs 6471 df-frec 6557 df-1o 6582 df-er 6702 df-en 6910 df-fin 6912 df-pnf 8216 df-mnf 8217 df-xr 8218 df-ltxr 8219 df-le 8220 df-sub 8352 df-neg 8353 df-inn 9144 df-n0 9403 df-z 9480 df-uz 9756 df-fz 10244 df-fzo 10378 df-seqfrec 10711

This theorem is referenced by: seq3f1olemp 10778

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