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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 ∧ (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)
iseqf1o.l 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
iseqf1o.p 𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
Assertion
Ref Expression
seq3f1olemp (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 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.
StepHypRef Expression
1 iseqf1o.4 . . 3 (𝜑𝑁 ∈ (ℤ𝑀))
2 eluzfz2 9988 . . 3 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ (𝑀...𝑁))
31, 2syl 14 . 2 (𝜑𝑁 ∈ (𝑀...𝑁))
4 oveq2 5861 . . . . . . 7 (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
54raleqdv 2671 . . . . . 6 (𝑤 = 𝑀 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥))
653anbi2d 1312 . . . . 5 (𝑤 = 𝑀 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
76exbidv 1818 . . . 4 (𝑤 = 𝑀 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
87imbi2d 229 . . 3 (𝑤 = 𝑀 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
9 oveq2 5861 . . . . . . 7 (𝑤 = 𝑘 → (𝑀...𝑤) = (𝑀...𝑘))
109raleqdv 2671 . . . . . 6 (𝑤 = 𝑘 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥))
11103anbi2d 1312 . . . . 5 (𝑤 = 𝑘 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
1211exbidv 1818 . . . 4 (𝑤 = 𝑘 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
1312imbi2d 229 . . 3 (𝑤 = 𝑘 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
14 oveq2 5861 . . . . . . 7 (𝑤 = (𝑘 + 1) → (𝑀...𝑤) = (𝑀...(𝑘 + 1)))
1514raleqdv 2671 . . . . . 6 (𝑤 = (𝑘 + 1) → (∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥))
16153anbi2d 1312 . . . . 5 (𝑤 = (𝑘 + 1) → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
1716exbidv 1818 . . . 4 (𝑤 = (𝑘 + 1) → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
1817imbi2d 229 . . 3 (𝑤 = (𝑘 + 1) → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
19 oveq2 5861 . . . . . . 7 (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
2019raleqdv 2671 . . . . . 6 (𝑤 = 𝑁 → (∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥))
21203anbi2d 1312 . . . . 5 (𝑤 = 𝑁 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
2221exbidv 1818 . . . 4 (𝑤 = 𝑁 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
2322imbi2d 229 . . 3 (𝑤 = 𝑁 → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑤)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ↔ (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
24 iseqf1o.1 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
25 iseqf1o.2 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
26 iseqf1o.3 . . . . 5 ((𝜑 ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
27 iseqf1o.6 . . . . 5 (𝜑𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
28 iseqf1o.7 . . . . 5 ((𝜑𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
29 eluzfz1 9987 . . . . . 6 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ (𝑀...𝑁))
301, 29syl 14 . . . . 5 (𝜑𝑀 ∈ (𝑀...𝑁))
31 ral0 3516 . . . . . . 7 𝑥 ∈ ∅ (𝐹𝑥) = 𝑥
32 fzo0 10124 . . . . . . . 8 (𝑀..^𝑀) = ∅
3332raleqi 2669 . . . . . . 7 (∀𝑥 ∈ (𝑀..^𝑀)(𝐹𝑥) = 𝑥 ↔ ∀𝑥 ∈ ∅ (𝐹𝑥) = 𝑥)
3431, 33mpbir 145 . . . . . 6 𝑥 ∈ (𝑀..^𝑀)(𝐹𝑥) = 𝑥
3534a1i 9 . . . . 5 (𝜑 → ∀𝑥 ∈ (𝑀..^𝑀)(𝐹𝑥) = 𝑥)
36 f1of 5442 . . . . . . . . . 10 (𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) → 𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
3727, 36syl 14 . . . . . . . . 9 (𝜑𝐹:(𝑀...𝑁)⟶(𝑀...𝑁))
38 eluzel2 9492 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → 𝑀 ∈ ℤ)
391, 38syl 14 . . . . . . . . . 10 (𝜑𝑀 ∈ ℤ)
40 eluzelz 9496 . . . . . . . . . . 11 (𝑁 ∈ (ℤ𝑀) → 𝑁 ∈ ℤ)
411, 40syl 14 . . . . . . . . . 10 (𝜑𝑁 ∈ ℤ)
4239, 41fzfigd 10387 . . . . . . . . 9 (𝜑 → (𝑀...𝑁) ∈ Fin)
43 fex 5725 . . . . . . . . 9 ((𝐹:(𝑀...𝑁)⟶(𝑀...𝑁) ∧ (𝑀...𝑁) ∈ Fin) → 𝐹 ∈ V)
4437, 42, 43syl2anc 409 . . . . . . . 8 (𝜑𝐹 ∈ V)
45 fveq1 5495 . . . . . . . . . . . . 13 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
4645fveq2d 5500 . . . . . . . . . . . 12 (𝑓 = 𝐹 → (𝐺‘(𝑓𝑥)) = (𝐺‘(𝐹𝑥)))
4746ifeq1d 3543 . . . . . . . . . . 11 (𝑓 = 𝐹 → if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)) = if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
4847mpteq2dv 4080 . . . . . . . . . 10 (𝑓 = 𝐹 → (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀))) = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀))))
49 iseqf1o.p . . . . . . . . . 10 𝑃 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝑓𝑥)), (𝐺𝑀)))
50 iseqf1o.l . . . . . . . . . 10 𝐿 = (𝑥 ∈ (ℤ𝑀) ↦ if(𝑥𝑁, (𝐺‘(𝐹𝑥)), (𝐺𝑀)))
5148, 49, 503eqtr4g 2228 . . . . . . . . 9 (𝑓 = 𝐹𝑃 = 𝐿)
5251adantl 275 . . . . . . . 8 ((𝜑𝑓 = 𝐹) → 𝑃 = 𝐿)
5344, 52csbied 3095 . . . . . . 7 (𝜑𝐹 / 𝑓𝑃 = 𝐿)
5453seqeq3d 10409 . . . . . 6 (𝜑 → seq𝑀( + , 𝐹 / 𝑓𝑃) = seq𝑀( + , 𝐿))
5554fveq1d 5498 . . . . 5 (𝜑 → (seq𝑀( + , 𝐹 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
5624, 25, 26, 1, 27, 28, 30, 27, 35, 55, 49seq3f1olemstep 10457 . . . 4 (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
5756a1i 9 . . 3 (𝑁 ∈ (ℤ𝑀) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑀)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
58 nfv 1521 . . . . . . . 8 𝑔(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
59 nfv 1521 . . . . . . . . 9 𝑓 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁)
60 nfv 1521 . . . . . . . . 9 𝑓𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥
61 nfcv 2312 . . . . . . . . . . . 12 𝑓𝑀
62 nfcv 2312 . . . . . . . . . . . 12 𝑓 +
63 nfcsb1v 3082 . . . . . . . . . . . 12 𝑓𝑔 / 𝑓𝑃
6461, 62, 63nfseq 10411 . . . . . . . . . . 11 𝑓seq𝑀( + , 𝑔 / 𝑓𝑃)
65 nfcv 2312 . . . . . . . . . . 11 𝑓𝑁
6664, 65nffv 5506 . . . . . . . . . 10 𝑓(seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁)
6766nfeq1 2322 . . . . . . . . 9 𝑓(seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)
6859, 60, 67nf3an 1559 . . . . . . . 8 𝑓(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘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 → (𝑓𝑥) = (𝑔𝑥))
7170eqeq1d 2179 . . . . . . . . . 10 (𝑓 = 𝑔 → ((𝑓𝑥) = 𝑥 ↔ (𝑔𝑥) = 𝑥))
7271ralbidv 2470 . . . . . . . . 9 (𝑓 = 𝑔 → (∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥))
73 csbeq1a 3058 . . . . . . . . . . . 12 (𝑓 = 𝑔𝑃 = 𝑔 / 𝑓𝑃)
7473seqeq3d 10409 . . . . . . . . . . 11 (𝑓 = 𝑔 → seq𝑀( + , 𝑃) = seq𝑀( + , 𝑔 / 𝑓𝑃))
7574fveq1d 5498 . . . . . . . . . 10 (𝑓 = 𝑔 → (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁))
7675eqeq1d 2179 . . . . . . . . 9 (𝑓 = 𝑔 → ((seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁) ↔ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
7769, 72, 763anbi123d 1307 . . . . . . . 8 (𝑓 = 𝑔 → ((𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
7858, 68, 77cbvex 1749 . . . . . . 7 (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
79 fveq2 5496 . . . . . . . . . . 11 (𝑥 = 𝑎 → (𝑔𝑥) = (𝑔𝑎))
80 id 19 . . . . . . . . . . 11 (𝑥 = 𝑎𝑥 = 𝑎)
8179, 80eqeq12d 2185 . . . . . . . . . 10 (𝑥 = 𝑎 → ((𝑔𝑥) = 𝑥 ↔ (𝑔𝑎) = 𝑎))
8281cbvralv 2696 . . . . . . . . 9 (∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ↔ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎)
83823anbi2i 1186 . . . . . . . 8 ((𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
8483exbii 1598 . . . . . . 7 (∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
8578, 84bitri 183 . . . . . 6 (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) ↔ ∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
86 simpll 524 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝜑)
8786, 24sylan 281 . . . . . . . . 9 ((((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) ∈ 𝑆)
8886, 25sylan 281 . . . . . . . . 9 ((((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥𝑆𝑦𝑆)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
8986, 26sylan 281 . . . . . . . . 9 ((((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ (𝑥𝑆𝑦𝑆𝑧𝑆)) → ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))
901ad2antrr 485 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑁 ∈ (ℤ𝑀))
9127ad2antrr 485 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝐹:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
9286, 28sylan 281 . . . . . . . . 9 ((((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) ∧ 𝑥 ∈ (ℤ𝑀)) → (𝐺𝑥) ∈ 𝑆)
93 fzofzp1 10183 . . . . . . . . . 10 (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
9493ad2antlr 486 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (𝑘 + 1) ∈ (𝑀...𝑁))
95 simpr1 998 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁))
96 simpr2 999 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎)
9796, 82sylibr 133 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥)
98 elfzoelz 10103 . . . . . . . . . . . 12 (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
9998ad2antlr 486 . . . . . . . . . . 11 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → 𝑘 ∈ ℤ)
100 fzval3 10160 . . . . . . . . . . . 12 (𝑘 ∈ ℤ → (𝑀...𝑘) = (𝑀..^(𝑘 + 1)))
101100raleqdv 2671 . . . . . . . . . . 11 (𝑘 ∈ ℤ → (∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔𝑥) = 𝑥))
10299, 101syl 14 . . . . . . . . . 10 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (∀𝑥 ∈ (𝑀...𝑘)(𝑔𝑥) = 𝑥 ↔ ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔𝑥) = 𝑥))
10397, 102mpbid 146 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∀𝑥 ∈ (𝑀..^(𝑘 + 1))(𝑔𝑥) = 𝑥)
104 simpr3 1000 . . . . . . . . 9 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))
10587, 88, 89, 90, 91, 92, 94, 95, 103, 104, 49seq3f1olemstep 10457 . . . . . . . 8 (((𝜑𝑘 ∈ (𝑀..^𝑁)) ∧ (𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))
106105ex 114 . . . . . . 7 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → ((𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
107106exlimdv 1812 . . . . . 6 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (∃𝑔(𝑔:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑎 ∈ (𝑀...𝑘)(𝑔𝑎) = 𝑎 ∧ (seq𝑀( + , 𝑔 / 𝑓𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
10885, 107syl5bi 151 . . . . 5 ((𝜑𝑘 ∈ (𝑀..^𝑁)) → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
109108expcom 115 . . . 4 (𝑘 ∈ (𝑀..^𝑁) → (𝜑 → (∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)) → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
110109a2d 26 . . 3 (𝑘 ∈ (𝑀..^𝑁) → ((𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑘)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...(𝑘 + 1))(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁)))))
1118, 13, 18, 23, 57, 110fzind2 10195 . 2 (𝑁 ∈ (𝑀...𝑁) → (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 𝑥 ∧ (seq𝑀( + , 𝑃)‘𝑁) = (seq𝑀( + , 𝐿)‘𝑁))))
1123, 111mpcom 36 1 (𝜑 → ∃𝑓(𝑓:(𝑀...𝑁)–1-1-onto→(𝑀...𝑁) ∧ ∀𝑥 ∈ (𝑀...𝑁)(𝑓𝑥) = 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-ontowf1o 5197  cfv 5198  (class class class)co 5853  Fincfn 6718  1c1 7775   + caddc 7777  cle 7955  cz 9212  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
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