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Theorem cyc3genpm 31419
Description: The alternating group 𝐴 is generated by 3-cycles. Property (a) of [Lang] p. 32 . (Contributed by Thierry Arnoux, 27-Sep-2023.)
Hypotheses
Ref Expression
cyc3genpm.t 𝐶 = (𝑀 “ (♯ “ {3}))
cyc3genpm.a 𝐴 = (pmEven‘𝐷)
cyc3genpm.s 𝑆 = (SymGrp‘𝐷)
cyc3genpm.n 𝑁 = (♯‘𝐷)
cyc3genpm.m 𝑀 = (toCyc‘𝐷)
Assertion
Ref Expression
cyc3genpm (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
Distinct variable groups:   𝑤,𝐴   𝑤,𝐶   𝑤,𝐷   𝑤,𝑁   𝑤,𝑄   𝑤,𝑆
Allowed substitution hint:   𝑀(𝑤)

Proof of Theorem cyc3genpm
Dummy variables 𝑖 𝑢 𝑣 𝑐 𝑒 𝑓 𝑔 𝑗 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 766 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2 lencl 14236 . . . . . . . 8 (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ0)
32ad2antlr 724 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℕ0)
43nn0zd 12424 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℤ)
5 simpr 485 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 = (𝑆 Σg 𝑣))
65fveq2d 6778 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)))
7 simplll 772 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
8 simpllr 773 . . . . . . . . 9 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄𝐴)
9 cyc3genpm.a . . . . . . . . 9 𝐴 = (pmEven‘𝐷)
108, 9eleqtrdi 2849 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 ∈ (pmEven‘𝐷))
11 cyc3genpm.s . . . . . . . . 9 𝑆 = (SymGrp‘𝐷)
12 eqid 2738 . . . . . . . . 9 (Base‘𝑆) = (Base‘𝑆)
13 eqid 2738 . . . . . . . . 9 (pmSgn‘𝐷) = (pmSgn‘𝐷)
1411, 12, 13psgnevpm 20794 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
157, 10, 14syl2anc 584 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16 eqid 2738 . . . . . . . . 9 ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
1711, 16, 13psgnvalii 19117 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
187, 1, 17syl2anc 584 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
196, 15, 183eqtr3rd 2787 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20 m1exp1 16085 . . . . . . 7 ((♯‘𝑣) ∈ ℤ → ((-1↑(♯‘𝑣)) = 1 ↔ 2 ∥ (♯‘𝑣)))
2120biimpa 477 . . . . . 6 (((♯‘𝑣) ∈ ℤ ∧ (-1↑(♯‘𝑣)) = 1) → 2 ∥ (♯‘𝑣))
224, 19, 21syl2anc 584 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 2 ∥ (♯‘𝑣))
23 oveq2 7283 . . . . . . . . . 10 (𝑥 = ∅ → (𝑆 Σg 𝑥) = (𝑆 Σg ∅))
2423eqeq1d 2740 . . . . . . . . 9 (𝑥 = ∅ → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
2524rexbidv 3226 . . . . . . . 8 (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
2625imbi2d 341 . . . . . . 7 (𝑥 = ∅ → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))))
27 oveq2 7283 . . . . . . . . . 10 (𝑥 = 𝑢 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑢))
2827eqeq1d 2740 . . . . . . . . 9 (𝑥 = 𝑢 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
2928rexbidv 3226 . . . . . . . 8 (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
3029imbi2d 341 . . . . . . 7 (𝑥 = 𝑢 → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))))
31 oveq2 7283 . . . . . . . . . 10 (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σg 𝑥) = (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)))
3231eqeq1d 2740 . . . . . . . . 9 (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
3332rexbidv 3226 . . . . . . . 8 (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
3433imbi2d 341 . . . . . . 7 (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))))
35 oveq2 7283 . . . . . . . . . 10 (𝑥 = 𝑣 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑣))
3635eqeq1d 2740 . . . . . . . . 9 (𝑥 = 𝑣 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
3736rexbidv 3226 . . . . . . . 8 (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
3837imbi2d 341 . . . . . . 7 (𝑥 = 𝑣 → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))))
39 wrd0 14242 . . . . . . . . 9 ∅ ∈ Word 𝐶
4039a1i 11 . . . . . . . 8 (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41 simpr 485 . . . . . . . . . 10 ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
4241oveq2d 7291 . . . . . . . . 9 ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σg 𝑤) = (𝑆 Σg ∅))
4342eqeq2d 2749 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σg ∅) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg ∅)))
44 eqidd 2739 . . . . . . . 8 (𝐷 ∈ Fin → (𝑆 Σg ∅) = (𝑆 Σg ∅))
4540, 43, 44rspcedvd 3563 . . . . . . 7 (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))
46 ccatcl 14277 . . . . . . . . . . . . . 14 ((𝑣 ∈ Word 𝐶𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
4746ad5ant24 758 . . . . . . . . . . . . 13 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
48 oveq2 7283 . . . . . . . . . . . . . . 15 (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑣 ++ 𝑐)))
4948eqeq2d 2749 . . . . . . . . . . . . . 14 (𝑤 = (𝑣 ++ 𝑐) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
5049adantl 482 . . . . . . . . . . . . 13 (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) ∧ 𝑤 = (𝑣 ++ 𝑐)) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
51 simpllr 773 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣))
52 simpllr 773 . . . . . . . . . . . . . . . . . . 19 ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
5352ad2antrr 723 . . . . . . . . . . . . . . . . . 18 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝐷 ∈ Fin)
5411symggrp 19008 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55 grpmnd 18584 . . . . . . . . . . . . . . . . . 18 (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
5653, 54, 553syl 18 . . . . . . . . . . . . . . . . 17 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑆 ∈ Mnd)
5716, 11, 12symgtrf 19077 . . . . . . . . . . . . . . . . . . 19 ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
5857a1i 11 . . . . . . . . . . . . . . . . . 18 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆))
59 simp-5r 783 . . . . . . . . . . . . . . . . . . 19 ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → 𝑖 ∈ ran (pmTrsp‘𝐷))
6059ad2antrr 723 . . . . . . . . . . . . . . . . . 18 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑖 ∈ ran (pmTrsp‘𝐷))
6158, 60sseldd 3922 . . . . . . . . . . . . . . . . 17 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑖 ∈ (Base‘𝑆))
62 simp-6r 785 . . . . . . . . . . . . . . . . . 18 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ ran (pmTrsp‘𝐷))
6358, 62sseldd 3922 . . . . . . . . . . . . . . . . 17 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ (Base‘𝑆))
64 eqid 2738 . . . . . . . . . . . . . . . . . 18 (+g𝑆) = (+g𝑆)
6512, 64gsumws2 18481 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g𝑆)𝑗))
6656, 61, 63, 65syl3anc 1370 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g𝑆)𝑗))
67 simpr 485 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐))
6866, 67eqtrd 2778 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑆 Σg 𝑐))
6951, 68oveq12d 7293 . . . . . . . . . . . . . 14 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → ((𝑆 Σg 𝑢)(+g𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑣)(+g𝑆)(𝑆 Σg 𝑐)))
70 sswrd 14225 . . . . . . . . . . . . . . . . 17 (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
7158, 70syl 17 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
72 simp-7l 786 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word ran (pmTrsp‘𝐷))
7371, 72sseldd 3922 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word (Base‘𝑆))
7461, 63s2cld 14584 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆))
7512, 64gsumccat 18480 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Mnd ∧ 𝑢 ∈ Word (Base‘𝑆) ∧ ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑢)(+g𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)))
7656, 73, 74, 75syl3anc 1370 . . . . . . . . . . . . . 14 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑢)(+g𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)))
77 cyc3genpm.t . . . . . . . . . . . . . . . . . . . 20 𝐶 = (𝑀 “ (♯ “ {3}))
78 cyc3genpm.m . . . . . . . . . . . . . . . . . . . . 21 𝑀 = (toCyc‘𝐷)
7978imaeq1i 5966 . . . . . . . . . . . . . . . . . . . 20 (𝑀 “ (♯ “ {3})) = ((toCyc‘𝐷) “ (♯ “ {3}))
8077, 79eqtri 2766 . . . . . . . . . . . . . . . . . . 19 𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))
8180, 9cyc3evpm 31417 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin → 𝐶𝐴)
8211, 12evpmss 20791 . . . . . . . . . . . . . . . . . . 19 (pmEven‘𝐷) ⊆ (Base‘𝑆)
839, 82eqsstri 3955 . . . . . . . . . . . . . . . . . 18 𝐴 ⊆ (Base‘𝑆)
8481, 83sstrdi 3933 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85 sswrd 14225 . . . . . . . . . . . . . . . . 17 (𝐶 ⊆ (Base‘𝑆) → Word 𝐶 ⊆ Word (Base‘𝑆))
8653, 84, 853syl 18 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → Word 𝐶 ⊆ Word (Base‘𝑆))
87 simp-4r 781 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑣 ∈ Word 𝐶)
8886, 87sseldd 3922 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑣 ∈ Word (Base‘𝑆))
89 simplr 766 . . . . . . . . . . . . . . . 16 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word 𝐶)
9086, 89sseldd 3922 . . . . . . . . . . . . . . 15 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word (Base‘𝑆))
9112, 64gsumccat 18480 . . . . . . . . . . . . . . 15 ((𝑆 ∈ Mnd ∧ 𝑣 ∈ Word (Base‘𝑆) ∧ 𝑐 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑣 ++ 𝑐)) = ((𝑆 Σg 𝑣)(+g𝑆)(𝑆 Σg 𝑐)))
9256, 88, 90, 91syl3anc 1370 . . . . . . . . . . . . . 14 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑣 ++ 𝑐)) = ((𝑆 Σg 𝑣)(+g𝑆)(𝑆 Σg 𝑐)))
9369, 76, 923eqtr4d 2788 . . . . . . . . . . . . 13 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐)))
9447, 50, 93rspcedvd 3563 . . . . . . . . . . . 12 ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))
95 cyc3genpm.n . . . . . . . . . . . . . . 15 𝑁 = (♯‘𝐷)
96 simp-6r 785 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑒𝐷)
97 simp-5r 783 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑓𝐷)
98 simpllr 773 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑔𝐷)
99 simplr 766 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝐷)
100 simp-4r 781 . . . . . . . . . . . . . . . 16 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
101100simprd 496 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))
102 simprr 770 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑗 = (𝑀‘⟨“𝑔”⟩))
10352ad6antr 733 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝐷 ∈ Fin)
104100simpld 495 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑒𝑓)
105 simprl 768 . . . . . . . . . . . . . . 15 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → 𝑔)
10677, 9, 11, 95, 78, 64, 96, 97, 98, 99, 101, 102, 103, 104, 105cyc3genpmlem 31418 . . . . . . . . . . . . . 14 ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔𝐷) ∧ 𝐷) ∧ (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩))) → ∃𝑐 ∈ Word 𝐶(𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐))
107 simp-6r 785 . . . . . . . . . . . . . . 15 (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝐷 ∈ Fin)
108 simp-7r 787 . . . . . . . . . . . . . . 15 (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝑗 ∈ ran (pmTrsp‘𝐷))
10916, 78trsp2cyc 31390 . . . . . . . . . . . . . . 15 ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔𝐷𝐷 (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩)))
110107, 108, 109syl2anc 584 . . . . . . . . . . . . . 14 (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑔𝐷𝐷 (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩)))
111106, 110r19.29vva 3266 . . . . . . . . . . . . 13 (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) ∧ 𝑒𝐷) ∧ 𝑓𝐷) ∧ (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑐 ∈ Word 𝐶(𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐))
11216, 78trsp2cyc 31390 . . . . . . . . . . . . . 14 ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒𝐷𝑓𝐷 (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
11352, 59, 112syl2anc 584 . . . . . . . . . . . . 13 ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → ∃𝑒𝐷𝑓𝐷 (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
114111, 113r19.29vva 3266 . . . . . . . . . . . 12 ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → ∃𝑐 ∈ Word 𝐶(𝑖(+g𝑆)𝑗) = (𝑆 Σg 𝑐))
11594, 114r19.29a 3218 . . . . . . . . . . 11 ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))
116115adantl3r 747 . . . . . . . . . 10 (((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))
117 simpr 485 . . . . . . . . . . . 12 (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) → 𝐷 ∈ Fin)
118 simplr 766 . . . . . . . . . . . 12 (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
119117, 118mpd 15 . . . . . . . . . . 11 (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
120 oveq2 7283 . . . . . . . . . . . . 13 (𝑣 = 𝑤 → (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
121120eqeq2d 2749 . . . . . . . . . . . 12 (𝑣 = 𝑤 → ((𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
122121cbvrexvw 3384 . . . . . . . . . . 11 (∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
123119, 122sylibr 233 . . . . . . . . . 10 (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣))
124116, 123r19.29a 3218 . . . . . . . . 9 (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))
125124ex 413 . . . . . . . 8 ((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
126125ex3 1345 . . . . . . 7 ((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))))
12726, 30, 34, 38, 45, 126wrdt2ind 31225 . . . . . 6 ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
128127imp 407 . . . . 5 (((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
1291, 22, 7, 128syl21anc 835 . . . 4 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
1305eqeq1d 2740 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (𝑄 = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
131130rexbidv 3226 . . . 4 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
132129, 131mpbird 256 . . 3 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
13383sseli 3917 . . . 4 (𝑄𝐴𝑄 ∈ (Base‘𝑆))
13411, 12, 16psgnfitr 19125 . . . . 5 (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣)))
135134biimpa 477 . . . 4 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
136133, 135sylan2 593 . . 3 ((𝐷 ∈ Fin ∧ 𝑄𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
137132, 136r19.29a 3218 . 2 ((𝐷 ∈ Fin ∧ 𝑄𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
138 simpr 485 . . . 4 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 = (𝑆 Σg 𝑤))
13911altgnsg 31416 . . . . . . . . 9 (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
1409, 139eqeltrid 2843 . . . . . . . 8 (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141 nsgsubg 18786 . . . . . . . 8 (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142 subgsubm 18777 . . . . . . . 8 (𝐴 ∈ (SubGrp‘𝑆) → 𝐴 ∈ (SubMnd‘𝑆))
143140, 141, 1423syl 18 . . . . . . 7 (𝐷 ∈ Fin → 𝐴 ∈ (SubMnd‘𝑆))
144143adantr 481 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝐴 ∈ (SubMnd‘𝑆))
145 sswrd 14225 . . . . . . . 8 (𝐶𝐴 → Word 𝐶 ⊆ Word 𝐴)
14681, 145syl 17 . . . . . . 7 (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147146sselda 3921 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148 gsumwsubmcl 18475 . . . . . 6 ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σg 𝑤) ∈ 𝐴)
149144, 147, 148syl2anc 584 . . . . 5 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σg 𝑤) ∈ 𝐴)
150149adantr 481 . . . 4 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → (𝑆 Σg 𝑤) ∈ 𝐴)
151138, 150eqeltrd 2839 . . 3 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄𝐴)
152151r19.29an 3217 . 2 ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)) → 𝑄𝐴)
153137, 152impbida 798 1 (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wne 2943  wrex 3065  wss 3887  c0 4256  {csn 4561   class class class wbr 5074  ccnv 5588  ran crn 5590  cima 5592  cfv 6433  (class class class)co 7275  Fincfn 8733  1c1 10872  -cneg 11206  2c2 12028  3c3 12029  0cn0 12233  cz 12319  cexp 13782  chash 14044  Word cword 14217   ++ cconcat 14273  ⟨“cs2 14554  cdvds 15963  Basecbs 16912  +gcplusg 16962   Σg cgsu 17151  Mndcmnd 18385  SubMndcsubmnd 18429  Grpcgrp 18577  SubGrpcsubg 18749  NrmSGrpcnsg 18750  SymGrpcsymg 18974  pmTrspcpmtr 19049  pmSgncpsgn 19097  pmEvencevpm 19098  toCycctocyc 31373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-ac2 10219  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948  ax-pre-sup 10949  ax-addf 10950  ax-mulf 10951
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-iin 4927  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-se 5545  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-tpos 8042  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-sup 9201  df-inf 9202  df-card 9697  df-ac 9872  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-dec 12438  df-uz 12583  df-rp 12731  df-fz 13240  df-fzo 13383  df-fl 13512  df-mod 13590  df-seq 13722  df-exp 13783  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-reverse 14472  df-csh 14502  df-s2 14561  df-s3 14562  df-dvds 15964  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-mulr 16976  df-starv 16977  df-tset 16981  df-ple 16982  df-ds 16984  df-unif 16985  df-0g 17152  df-gsum 17153  df-mre 17295  df-mrc 17296  df-acs 17298  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-mhm 18430  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-sbg 18582  df-subg 18752  df-nsg 18753  df-ghm 18832  df-gim 18875  df-oppg 18950  df-symg 18975  df-pmtr 19050  df-psgn 19099  df-evpm 19100  df-cmn 19388  df-abl 19389  df-mgp 19721  df-ur 19738  df-ring 19785  df-cring 19786  df-oppr 19862  df-dvdsr 19883  df-unit 19884  df-invr 19914  df-dvr 19925  df-drng 19993  df-cnfld 20598  df-tocyc 31374
This theorem is referenced by: (None)
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