Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  seq3f1olemstep GIF version

Theorem seq3f1olemstep 10286
 Description: Lemma for seq3f1o 10289. 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 5367 . . . . . 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 9817 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ)
64, 5syl 14 . . . . . 6 (𝜑𝑀 ∈ ℤ)
7 elfzel2 9816 . . . . . . 7 (𝐾 ∈ (𝑀...𝑁) → 𝑁 ∈ ℤ)
84, 7syl 14 . . . . . 6 (𝜑𝑁 ∈ ℤ)
96, 8fzfigd 10216 . . . . 5 (𝜑 → (𝑀...𝑁) ∈ Fin)
10 fex 5647 . . . . 5 ((𝐽:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐽 ∈ V)
113, 9, 10syl2anc 408 . . . 4 (𝜑𝐽 ∈ V)
1211adantr 274 . . 3 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽 ∈ V)
131adantr 274 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
14 iseqf1olemstep.const . . . . . . 7 (𝜑 → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
1514adantr 274 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
16 eqcom 2141 . . . . . . . . . 10 (𝐾 = (𝐽𝐾) ↔ (𝐽𝐾) = 𝐾)
1716biimpi 119 . . . . . . . . 9 (𝐾 = (𝐽𝐾) → (𝐽𝐾) = 𝐾)
1817adantl 275 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
19 f1ocnvfvb 5681 . . . . . . . . . 10 ((𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁) ∧ 𝐾 ∈ (𝑀...𝑁)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
201, 4, 4, 19syl3anc 1216 . . . . . . . . 9 (𝜑 → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2120adantr 274 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → ((𝐽𝐾) = 𝐾 ↔ (𝐽𝐾) = 𝐾))
2218, 21mpbird 166 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (𝐽𝐾) = 𝐾)
23 elfzelz 9818 . . . . . . . . . 10 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ ℤ)
244, 23syl 14 . . . . . . . . 9 (𝜑𝐾 ∈ ℤ)
2524adantr 274 . . . . . . . 8 ((𝜑𝐾 = (𝐽𝐾)) → 𝐾 ∈ ℤ)
26 fveq2 5421 . . . . . . . . . 10 (𝑥 = 𝐾 → (𝐽𝑥) = (𝐽𝐾))
27 id 19 . . . . . . . . . 10 (𝑥 = 𝐾𝑥 = 𝐾)
2826, 27eqeq12d 2154 . . . . . . . . 9 (𝑥 = 𝐾 → ((𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
2928ralsng 3564 . . . . . . . 8 (𝐾 ∈ ℤ → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3025, 29syl 14 . . . . . . 7 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥 ↔ (𝐽𝐾) = 𝐾))
3122, 30mpbird 166 . . . . . 6 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥)
32 ralun 3258 . . . . . 6 ((∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥 ∧ ∀𝑥 ∈ {𝐾} (𝐽𝑥) = 𝑥) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
3315, 31, 32syl2anc 408 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥)
34 elfzuz 9814 . . . . . . . 8 (𝐾 ∈ (𝑀...𝑁) → 𝐾 ∈ (ℤ𝑀))
35 fzisfzounsn 10025 . . . . . . . 8 (𝐾 ∈ (ℤ𝑀) → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
364, 34, 353syl 17 . . . . . . 7 (𝜑 → (𝑀...𝐾) = ((𝑀..^𝐾) ∪ {𝐾}))
3736raleqdv 2632 . . . . . 6 (𝜑 → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3837adantr 274 . . . . 5 ((𝜑𝐾 = (𝐽𝐾)) → (∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ↔ ∀𝑥 ∈ ((𝑀..^𝐾) ∪ {𝐾})(𝐽𝑥) = 𝑥))
3933, 38mpbird 166 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥)
40 seq3f1olemstep.jp . . . . 5 (𝜑 → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4140adantr 274 . . . 4 ((𝜑𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
4213, 39, 413jca 1161 . . 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 2281 . . . 4 𝑓𝐽
44 nfv 1508 . . . . 5 𝑓 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
45 nfv 1508 . . . . 5 𝑓𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥
46 nfcv 2281 . . . . . . . 8 𝑓𝑀
47 nfcv 2281 . . . . . . . 8 𝑓 +
48 nfcsb1v 3035 . . . . . . . 8 𝑓𝐽 / 𝑓𝑃
4946, 47, 48nfseq 10240 . . . . . . 7 𝑓seq𝑀( + , 𝐽 / 𝑓𝑃)
50 nfcv 2281 . . . . . . 7 𝑓𝑁
5149, 50nffv 5431 . . . . . 6 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁)
5251nfeq1 2291 . . . . 5 𝑓(seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
5344, 45, 52nf3an 1545 . . . 4 𝑓(𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥 ∧ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
54 f1oeq1 5356 . . . . 5 (𝑓 = 𝐽 → (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ↔ 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)))
55 fveq1 5420 . . . . . . 7 (𝑓 = 𝐽 → (𝑓𝑥) = (𝐽𝑥))
5655eqeq1d 2148 . . . . . 6 (𝑓 = 𝐽 → ((𝑓𝑥) = 𝑥 ↔ (𝐽𝑥) = 𝑥))
5756ralbidv 2437 . . . . 5 (𝑓 = 𝐽 → (∀𝑥 ∈ (𝑀...𝐾)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝐾)(𝐽𝑥) = 𝑥))
58 csbeq1a 3012 . . . . . . . 8 (𝑓 = 𝐽𝑃 = 𝐽 / 𝑓𝑃)
5958seqeq3d 10238 . . . . . . 7 (𝑓 = 𝐽 → seq𝑀( + , 𝑃) = seq𝑀( + , 𝐽 / 𝑓𝑃))
6059fveq1d 5423 . . . . . 6 (𝑓 = 𝐽 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁))
6160eqeq1d 2148 . . . . 5 (𝑓 = 𝐽 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
6254, 57, 613anbi123d 1290 . . . 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 2769 . . 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 274 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐾 ∈ (𝑀...𝑁))
661adantr 274 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
67 eqid 2139 . . . 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 10278 . . 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 274 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → ∀𝑥 ∈ (𝑀..^𝐾)(𝐽𝑥) = 𝑥)
7065, 66, 67, 69iseqf1olemqk 10279 . . 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 468 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
73 iseqf1o.2 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7473adantlr 468 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
75 iseqf1o.3 . . . . . 6 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
7675adantlr 468 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
77 iseqf1o.4 . . . . . 6 (𝜑𝑁 ∈ (ℤ𝑀))
7877adantr 274 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝑁 ∈ (ℤ𝑀))
79 iseqf1o.6 . . . . . 6 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
8079adantr 274 . . . . 5 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
81 iseqf1o.7 . . . . . 6 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
8281adantlr 468 . . . . 5 (((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
83 neqne 2316 . . . . . 6 𝐾 = (𝐽𝐾) → 𝐾 ≠ (𝐽𝐾))
8483adantl 275 . . . . 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 10285 . . . 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 274 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (seq𝑀( + , 𝐽 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
8886, 87eqtr3d 2174 . . 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 10216 . . . 4 ((𝜑 ∧ ¬ 𝐾 = (𝐽𝐾)) → (𝑀...𝑁) ∈ Fin)
92 mptexg 5645 . . . 4 ((𝑀...𝑁) ∈ Fin → (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) ∈ V)
93 nfcv 2281 . . . . 5 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))
94 nfv 1508 . . . . . 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 1508 . . . . . 6 𝑓𝑥 ∈ (𝑀...𝐾)((𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢)))‘𝑥) = 𝑥
96 nfcsb1v 3035 . . . . . . . . 9 𝑓(𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃
9746, 47, 96nfseq 10240 . . . . . . . 8 𝑓seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)
9897, 50nffv 5431 . . . . . . 7 𝑓(seq𝑀( + , (𝑢 ∈ (𝑀...𝑁) ↦ if(𝑢 ∈ (𝐾...(𝐽𝐾)), if(𝑢 = 𝐾, 𝐾, (𝐽‘(𝑢 − 1))), (𝐽𝑢))) / 𝑓𝑃)‘𝑁)
9998nfeq1 2291 . . . . . 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 1545 . . . . 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 5356 . . . . . 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 5420 . . . . . . . 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 2148 . . . . . . 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 2437 . . . . . 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 3012 . . . . . . . . 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 10238 . . . . . . . 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 5423 . . . . . . 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 2148 . . . . . 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 1290 . . . . 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 2769 . . . 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 1318 . 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 5380 . . . . . . 7 (𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐽:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
114 f1of 5367 . . . . . . 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 5556 . . . . 5 (𝜑 → (𝐽𝐾) ∈ (𝑀...𝑁))
117 elfzelz 9818 . . . . 5 ((𝐽𝐾) ∈ (𝑀...𝑁) → (𝐽𝐾) ∈ ℤ)
118116, 117syl 14 . . . 4 (𝜑 → (𝐽𝐾) ∈ ℤ)
119 zdceq 9138 . . . 4 ((𝐾 ∈ ℤ ∧ (𝐽𝐾) ∈ ℤ) → DECID 𝐾 = (𝐽𝐾))
12024, 118, 119syl2anc 408 . . 3 (𝜑DECID 𝐾 = (𝐽𝐾))
121 exmiddc 821 . . 3 (DECID 𝐾 = (𝐽𝐾) → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
122120, 121syl 14 . 2 (𝜑 → (𝐾 = (𝐽𝐾) ∨ ¬ 𝐾 = (𝐽𝐾)))
12364, 112, 122mpjaodan 787 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 103   ↔ wb 104   ∨ wo 697  DECID wdc 819   ∧ w3a 962   = wceq 1331  ∃wex 1468   ∈ wcel 1480   ≠ wne 2308  ∀wral 2416  Vcvv 2686  ⦋csb 3003   ∪ cun 3069  ifcif 3474  {csn 3527   class class class wbr 3929   ↦ cmpt 3989  ◡ccnv 4538  ⟶wf 5119  –1-1-onto→wf1o 5122  ‘cfv 5123  (class class class)co 5774  Fincfn 6634  1c1 7633   ≤ cle 7813   − cmin 7945  ℤcz 9066  ℤ≥cuz 9338  ...cfz 9802  ..^cfzo 9931  seqcseq 10230 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-addcom 7732  ax-addass 7734  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-0id 7740  ax-rnegex 7741  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748 This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-1o 6313  df-er 6429  df-en 6635  df-fin 6637  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-inn 8733  df-n0 8990  df-z 9067  df-uz 9339  df-fz 9803  df-fzo 9932  df-seqfrec 10231 This theorem is referenced by:  seq3f1olemp  10287
 Copyright terms: Public domain W3C validator