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Theorem seq3f1olemstep 10487
Description: Lemma for seq3f1o 10490. 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 5457 . . . . . 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 10010 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
64, 5syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
7 elfzel2 10009 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
84, 7syl 14 . . . . . 6 (𝜑𝑁 ∈ ℤ)
96, 8fzfigd 10417 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ Fin)
10 fex 5741 . . . . 5 ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
113, 9, 10syl2anc 411 . . . 4 (𝜑𝐽 ∈ V)
1211adantr 276 . . 3 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽 ∈ V)
131adantr 276 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
14 iseqf1olemstep.const . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
1514adantr 276 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
16 eqcom 2179 . . . . . . . . . 10 (𝐾 = (𝐽𝐾) ↔ (𝐽𝐾) = 𝐾)
1716biimpi 120 . . . . . . . . 9 (𝐾 = (𝐽𝐾) → (𝐽𝐾) = 𝐾)
1817adantl 277 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
19 f1ocnvfvb 5775 . . . . . . . . . 10 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
201, 4, 4, 19syl3anc 1238 . . . . . . . . 9 (𝜑 → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2120adantr 276 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2218, 21mpbird 167 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
23 elfzelz 10011 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
244, 23syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ ℤ)
2524adantr 276 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → 𝐾 ∈ ℤ)
26 fveq2 5511 . . . . . . . . . 10 (𝑥 = 𝐾 → (𝐽𝑥) = (𝐽𝐾))
27 id 19 . . . . . . . . . 10 (𝑥 = 𝐾𝑥 = 𝐾)
2826, 27eqeq12d 2192 . . . . . . . . 9 (𝑥 = 𝐾 → ((𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
2928ralsng 3631 . . . . . . . 8 (𝐾 ∈ ℤ → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3025, 29syl 14 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3122, 30mpbird 167 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥)
32 ralun 3317 . . . . . 6 ((∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥 ∧ ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
3315, 31, 32syl2anc 411 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
34 elfzuz 10007 . . . . . . . 8 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
35 fzisfzounsn 10222 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
364, 34, 353syl 17 . . . . . . 7 (𝜑 → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
3736raleqdv 2678 . . . . . 6 (𝜑 → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3837adantr 276 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3933, 38mpbird 167 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥)
40 seq3f1olemstep.jp . . . . 5 (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4140adantr 276 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4213, 39, 413jca 1177 . . 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 2319 . . . 4 𝑓𝐽
44 nfv 1528 . . . . 5 𝑓 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
45 nfv 1528 . . . . 5 𝑓𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥
46 nfcv 2319 . . . . . . . 8 𝑓𝑀
47 nfcv 2319 . . . . . . . 8 𝑓 +
48 nfcsb1v 3090 . . . . . . . 8 𝑓𝐽 / 𝑓𝑃
4946, 47, 48nfseq 10441 . . . . . . 7 𝑓seq𝑀( + , 𝐽 / 𝑓𝑃)
50 nfcv 2319 . . . . . . 7 𝑓𝑁
5149, 50nffv 5521 . . . . . 6 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁)
5251nfeq1 2329 . . . . 5 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
5344, 45, 52nf3an 1566 . . . 4 𝑓(𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
54 f1oeq1 5445 . . . . 5 (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
55 fveq1 5510 . . . . . . 7 (𝑓 = 𝐽 → (𝑓𝑥) = (𝐽𝑥))
5655eqeq1d 2186 . . . . . 6 (𝑓 = 𝐽 → ((𝑓𝑥) = 𝑥 ↔ (𝐽𝑥) = 𝑥))
5756ralbidv 2477 . . . . 5 (𝑓 = 𝐽 → (∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥))
58 csbeq1a 3066 . . . . . . . 8 (𝑓 = 𝐽𝑃 = 𝐽 / 𝑓𝑃)
5958seqeq3d 10439 . . . . . . 7 (𝑓 = 𝐽 → seq𝑀( + , 𝑃) = seq𝑀( + , 𝐽 / 𝑓𝑃))
6059fveq1d 5513 . . . . . 6 (𝑓 = 𝐽 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁))
6160eqeq1d 2186 . . . . 5 (𝑓 = 𝐽 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
6254, 57, 613anbi123d 1312 . . . 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 2820 . . 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 62 . 2 ((𝜑𝐾 = (𝐽𝐾)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
654adantr 276 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐾 ∈ (𝑀...𝑁))
661adantr 276 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
67 eqid 2177 . . . 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 10479 . . 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 276 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
7065, 66, 67, 69iseqf1olemqk 10480 . . 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 477 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
73 iseqf1o.2 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7473adantlr 477 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
75 iseqf1o.3 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7675adantlr 477 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
77 iseqf1o.4 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
7877adantr 276 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝑁 ∈ (ℤ𝑀))
79 iseqf1o.6 . . . . . 6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
8079adantr 276 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
81 iseqf1o.7 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
8281adantlr 477 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
83 neqne 2355 . . . . . 6 𝐾 = (𝐽𝐾) → 𝐾 ≠ (𝐽𝐾))
8483adantl 277 . . . . 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 10486 . . . 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 276 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
8886, 87eqtr3d 2212 . . 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 10417 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (𝑀...𝑁) ∈ Fin)
92 mptexg 5737 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) ∈ V)
93 nfcv 2319 . . . . 5 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
94 nfv 1528 . . . . . 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 1528 . . . . . 6 𝑓𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥
96 nfcsb1v 3090 . . . . . . . . 9 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃
9746, 47, 96nfseq 10441 . . . . . . . 8 𝑓seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)
9897, 50nffv 5521 . . . . . . 7 𝑓(seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁)
9998nfeq1 2329 . . . . . 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 1566 . . . . 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 5445 . . . . . 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 5510 . . . . . . . 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 2186 . . . . . . 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 2477 . . . . . 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 3066 . . . . . . . . 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 10439 . . . . . . . 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 5513 . . . . . . 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 2186 . . . . . 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 1312 . . . . 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 2820 . . . 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 1340 . 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 5470 . . . . . . 7 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
114 f1of 5457 . . . . . . 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, 4ffvelcdmd 5648 . . . . 5 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
117 elfzelz 10011 . . . . 5 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
118116, 117syl 14 . . . 4 (𝜑 → (𝐽𝐾) ∈ ℤ)
119 zdceq 9317 . . . 4 ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐾 = (𝐽𝐾))
12024, 118, 119syl2anc 411 . . 3 (𝜑DECID 𝐾 = (𝐽𝐾))
121 exmiddc 836 . . 3 (DECID 𝐾 = (𝐽𝐾) → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
122120, 121syl 14 . 2 (𝜑 → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
12364, 112, 122mpjaodan 798 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 104  wb 105  wo 708  DECID wdc 834  w3a 978   = wceq 1353  wex 1492  wcel 2148  wne 2347  wral 2455  Vcvv 2737  csb 3057  cun 3127  ifcif 3534  {csn 3591   class class class wbr 4000  cmpt 4061  ccnv 4622  wf 5208  1-1-ontowf1o 5211  cfv 5212  (class class class)co 5869  Fincfn 6734  1c1 7803  cle 7983  cmin 8118  cz 9242  cuz 9517  ...cfz 9995  ..^cfzo 10128  seqcseq 10431
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-addass 7904  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-1o 6411  df-er 6529  df-en 6735  df-fin 6737  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-inn 8909  df-n0 9166  df-z 9243  df-uz 9518  df-fz 9996  df-fzo 10129  df-seqfrec 10432
This theorem is referenced by:  seq3f1olemp  10488
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