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Theorem seq3f1olemstep 9918
Description: Lemma for seq3f1o 9921. 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 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
iseqf1o.2 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
iseqf1o.3 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
iseqf1o.4 (𝜑𝑁 ∈ (ℤ𝑀))
iseqf1o.6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1o.7 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
iseqf1olemstep.k (𝜑𝐾 ∈ (𝑀...𝑁))
iseqf1olemstep.j (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
iseqf1olemstep.const (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
seq3f1olemstep.jp (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
seq3f1olemstep.p 𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
Assertion
Ref Expression
seq3f1olemstep (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 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.
StepHypRef Expression
1 iseqf1olemstep.j . . . . . 6 (𝜑𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
2 f1of 5247 . . . . . 6 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
31, 2syl 14 . . . . 5 (𝜑𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
4 iseqf1olemstep.k . . . . . . 7 (𝜑𝐾 ∈ (𝑀...𝑁))
5 elfzel1 9429 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
64, 5syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
7 elfzel2 9428 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
84, 7syl 14 . . . . . 6 (𝜑𝑁 ∈ ℤ)
96, 8fzfigd 9826 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ Fin)
10 fex 5516 . . . . 5 ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
113, 9, 10syl2anc 403 . . . 4 (𝜑𝐽 ∈ V)
1211adantr 270 . . 3 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽 ∈ V)
131adantr 270 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
14 iseqf1olemstep.const . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
1514adantr 270 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
16 eqcom 2090 . . . . . . . . . 10 (𝐾 = (𝐽𝐾) ↔ (𝐽𝐾) = 𝐾)
1716biimpi 118 . . . . . . . . 9 (𝐾 = (𝐽𝐾) → (𝐽𝐾) = 𝐾)
1817adantl 271 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
19 f1ocnvfvb 5551 . . . . . . . . . 10 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
201, 4, 4, 19syl3anc 1174 . . . . . . . . 9 (𝜑 → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2120adantr 270 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2218, 21mpbird 165 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
23 elfzelz 9430 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
244, 23syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ ℤ)
2524adantr 270 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → 𝐾 ∈ ℤ)
26 fveq2 5299 . . . . . . . . . 10 (𝑥 = 𝐾 → (𝐽𝑥) = (𝐽𝐾))
27 id 19 . . . . . . . . . 10 (𝑥 = 𝐾𝑥 = 𝐾)
2826, 27eqeq12d 2102 . . . . . . . . 9 (𝑥 = 𝐾 → ((𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
2928ralsng 3481 . . . . . . . 8 (𝐾 ∈ ℤ → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3025, 29syl 14 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3122, 30mpbird 165 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥)
32 ralun 3182 . . . . . 6 ((∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥 ∧ ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
3315, 31, 32syl2anc 403 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
34 elfzuz 9426 . . . . . . . 8 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
35 fzisfzounsn 9635 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
364, 34, 353syl 17 . . . . . . 7 (𝜑 → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
3736raleqdv 2568 . . . . . 6 (𝜑 → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3837adantr 270 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3933, 38mpbird 165 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥)
40 seq3f1olemstep.jp . . . . 5 (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4140adantr 270 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4213, 39, 413jca 1123 . . 3 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
43 nfcv 2228 . . . 4 𝑓𝐽
44 nfv 1466 . . . . 5 𝑓 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
45 nfv 1466 . . . . 5 𝑓𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥
46 nfcv 2228 . . . . . . . 8 𝑓𝑀
47 nfcv 2228 . . . . . . . 8 𝑓 +
48 nfcsb1v 2963 . . . . . . . 8 𝑓𝐽 / 𝑓𝑃
4946, 47, 48nfseq 9857 . . . . . . 7 𝑓seq𝑀( + , 𝐽 / 𝑓𝑃)
50 nfcv 2228 . . . . . . 7 𝑓𝑁
5149, 50nffv 5309 . . . . . 6 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁)
5251nfeq1 2238 . . . . 5 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
5344, 45, 52nf3an 1503 . . . 4 𝑓(𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
54 f1oeq1 5238 . . . . 5 (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
55 fveq1 5298 . . . . . . 7 (𝑓 = 𝐽 → (𝑓𝑥) = (𝐽𝑥))
5655eqeq1d 2096 . . . . . 6 (𝑓 = 𝐽 → ((𝑓𝑥) = 𝑥 ↔ (𝐽𝑥) = 𝑥))
5756ralbidv 2380 . . . . 5 (𝑓 = 𝐽 → (∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥))
58 csbeq1a 2941 . . . . . . . 8 (𝑓 = 𝐽𝑃 = 𝐽 / 𝑓𝑃)
5958seqeq3d 9854 . . . . . . 7 (𝑓 = 𝐽 → seq𝑀( + , 𝑃) = seq𝑀( + , 𝐽 / 𝑓𝑃))
6059fveq1d 5301 . . . . . 6 (𝑓 = 𝐽 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁))
6160eqeq1d 2096 . . . . 5 (𝑓 = 𝐽 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
6254, 57, 613anbi123d 1248 . . . 4 (𝑓 = 𝐽 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
6343, 53, 62spcegf 2702 . . 3 (𝐽 ∈ V → ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
6412, 42, 63sylc 61 . 2 ((𝜑𝐾 = (𝐽𝐾)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
654adantr 270 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐾 ∈ (𝑀...𝑁))
661adantr 270 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
67 eqid 2088 . . . 4 (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
6865, 66, 67iseqf1olemqf1o 9910 . . 3 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
6914adantr 270 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
7065, 66, 67, 69iseqf1olemqk 9911 . . 3 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥)
71 iseqf1o.1 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
7271adantlr 461 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
73 iseqf1o.2 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7473adantlr 461 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
75 iseqf1o.3 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7675adantlr 461 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
77 iseqf1o.4 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
7877adantr 270 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝑁 ∈ (ℤ𝑀))
79 iseqf1o.6 . . . . . 6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
8079adantr 270 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
81 iseqf1o.7 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
8281adantlr 461 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
83 neqne 2263 . . . . . 6 𝐾 = (𝐽𝐾) → 𝐾 ≠ (𝐽𝐾))
8483adantl 271 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐾 ≠ (𝐽𝐾))
85 seq3f1olemstep.p . . . . 5 𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
8672, 74, 76, 78, 80, 82, 65, 66, 69, 84, 67, 85seq3f1olemqsum 9917 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁))
8740adantr 270 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
8886, 87eqtr3d 2122 . . 3 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
8965, 5syl 14 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝑀 ∈ ℤ)
9065, 7syl 14 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝑁 ∈ ℤ)
9189, 90fzfigd 9826 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (𝑀...𝑁) ∈ Fin)
92 mptexg 5514 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) ∈ V)
93 nfcv 2228 . . . . 5 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
94 nfv 1466 . . . . . 6 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
95 nfv 1466 . . . . . 6 𝑓𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥
96 nfcsb1v 2963 . . . . . . . . 9 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃
9746, 47, 96nfseq 9857 . . . . . . . 8 𝑓seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)
9897, 50nffv 5309 . . . . . . 7 𝑓(seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁)
9998nfeq1 2238 . . . . . 6 𝑓(seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
10094, 95, 99nf3an 1503 . . . . 5 𝑓((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
101 f1oeq1 5238 . . . . . 6 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
102 fveq1 5298 . . . . . . . 8 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → (𝑓𝑥) = ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥))
103102eqeq1d 2096 . . . . . . 7 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → ((𝑓𝑥) = 𝑥 ↔ ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥))
104103ralbidv 2380 . . . . . 6 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → (∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥))
105 csbeq1a 2941 . . . . . . . . 9 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → 𝑃 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)
106105seqeq3d 9854 . . . . . . . 8 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → seq𝑀( + , 𝑃) = seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃))
107106fveq1d 5301 . . . . . . 7 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁))
108107eqeq1d 2096 . . . . . 6 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
109101, 104, 1083anbi123d 1248 . . . . 5 (𝑓 = (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
11093, 100, 109spcegf 2702 . . . 4 ((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) ∈ V → (((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
11191, 92, 1103syl 17 . . 3 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))):(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥 ∧ (seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
11268, 70, 88, 111mp3and 1276 . 2 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
113 f1ocnv 5260 . . . . . . 7 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
114 f1of 5247 . . . . . . 7 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
1151, 113, 1143syl 17 . . . . . 6 (𝜑𝐽:(𝑀...𝑁)⟶(𝑀...𝑁))
116115, 4ffvelrnd 5429 . . . . 5 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
117 elfzelz 9430 . . . . 5 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
118116, 117syl 14 . . . 4 (𝜑 → (𝐽𝐾) ∈ ℤ)
119 zdceq 8812 . . . 4 ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐾 = (𝐽𝐾))
12024, 118, 119syl2anc 403 . . 3 (𝜑DECID 𝐾 = (𝐽𝐾))
121 exmiddc 782 . . 3 (DECID 𝐾 = (𝐽𝐾) → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
122120, 121syl 14 . 2 (𝜑 → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
12364, 112, 122mpjaodan 747 1 (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 664  DECID wdc 780  w3a 924   = wceq 1289  wex 1426  wcel 1438  wne 2255  wral 2359  Vcvv 2619  csb 2933  cun 2997  ifcif 3391  {csn 3444   class class class wbr 3843  cmpt 3897  ccnv 4435  wf 5006  1-1-ontowf1o 5009  cfv 5010  (class class class)co 5644  Fincfn 6447  1c1 7341  cle 7513  cmin 7643  cz 8740  cuz 9009  ...cfz 9414  ..^cfzo 9541  seqcseq 9840
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3952  ax-sep 3955  ax-nul 3963  ax-pow 4007  ax-pr 4034  ax-un 4258  ax-setind 4351  ax-iinf 4401  ax-cnex 7426  ax-resscn 7427  ax-1cn 7428  ax-1re 7429  ax-icn 7430  ax-addcl 7431  ax-addrcl 7432  ax-mulcl 7433  ax-addcom 7435  ax-addass 7437  ax-distr 7439  ax-i2m1 7440  ax-0lt1 7441  ax-0id 7443  ax-rnegex 7444  ax-cnre 7446  ax-pre-ltirr 7447  ax-pre-ltwlin 7448  ax-pre-lttrn 7449  ax-pre-apti 7450  ax-pre-ltadd 7451
This theorem depends on definitions:  df-bi 115  df-dc 781  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-csb 2934  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-nul 3287  df-if 3392  df-pw 3429  df-sn 3450  df-pr 3451  df-op 3453  df-uni 3652  df-int 3687  df-iun 3730  df-br 3844  df-opab 3898  df-mpt 3899  df-tr 3935  df-id 4118  df-iord 4191  df-on 4193  df-ilim 4194  df-suc 4196  df-iom 4404  df-xp 4442  df-rel 4443  df-cnv 4444  df-co 4445  df-dm 4446  df-rn 4447  df-res 4448  df-ima 4449  df-iota 4975  df-fun 5012  df-fn 5013  df-f 5014  df-f1 5015  df-fo 5016  df-f1o 5017  df-fv 5018  df-riota 5600  df-ov 5647  df-oprab 5648  df-mpt2 5649  df-1st 5903  df-2nd 5904  df-recs 6062  df-frec 6148  df-1o 6173  df-er 6282  df-en 6448  df-fin 6450  df-pnf 7514  df-mnf 7515  df-xr 7516  df-ltxr 7517  df-le 7518  df-sub 7645  df-neg 7646  df-inn 8413  df-n0 8664  df-z 8741  df-uz 9010  df-fz 9415  df-fzo 9542  df-iseq 9841  df-seq3 9842
This theorem is referenced by:  seq3f1olemp  9919
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