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Theorem cyc3genpm 30996
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 769 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2 lencl 13974 . . . . . . . 8 (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ0)
32ad2antlr 727 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℕ0)
43nn0zd 12166 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℤ)
5 simpr 488 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 = (𝑆 Σg 𝑣))
65fveq2d 6678 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)))
7 simplll 775 . . . . . . . 8 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
8 simpllr 776 . . . . . . . . 9 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄𝐴)
9 cyc3genpm.a . . . . . . . . 9 𝐴 = (pmEven‘𝐷)
108, 9eleqtrdi 2843 . . . . . . . 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 20405 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
157, 10, 14syl2anc 587 . . . . . . 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 18755 . . . . . . . 8 ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
187, 1, 17syl2anc 587 . . . . . . 7 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
196, 15, 183eqtr3rd 2782 . . . . . 6 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20 m1exp1 15821 . . . . . . 7 ((♯‘𝑣) ∈ ℤ → ((-1↑(♯‘𝑣)) = 1 ↔ 2 ∥ (♯‘𝑣)))
2120biimpa 480 . . . . . 6 (((♯‘𝑣) ∈ ℤ ∧ (-1↑(♯‘𝑣)) = 1) → 2 ∥ (♯‘𝑣))
224, 19, 21syl2anc 587 . . . . 5 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 2 ∥ (♯‘𝑣))
23 oveq2 7178 . . . . . . . . . 10 (𝑥 = ∅ → (𝑆 Σg 𝑥) = (𝑆 Σg ∅))
2423eqeq1d 2740 . . . . . . . . 9 (𝑥 = ∅ → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
2524rexbidv 3207 . . . . . . . 8 (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
2625imbi2d 344 . . . . . . 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 7178 . . . . . . . . . 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 3207 . . . . . . . 8 (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
3029imbi2d 344 . . . . . . 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 7178 . . . . . . . . . 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 3207 . . . . . . . 8 (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
3433imbi2d 344 . . . . . . 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 7178 . . . . . . . . . 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 3207 . . . . . . . 8 (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
3837imbi2d 344 . . . . . . 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 13980 . . . . . . . . 9 ∅ ∈ Word 𝐶
4039a1i 11 . . . . . . . 8 (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41 simpr 488 . . . . . . . . . 10 ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
4241oveq2d 7186 . . . . . . . . 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 3529 . . . . . . 7 (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))
46 ccatcl 14015 . . . . . . . . . . . . . 14 ((𝑣 ∈ Word 𝐶𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
4746ad5ant24 761 . . . . . . . . . . . . 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 7178 . . . . . . . . . . . . . . 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 485 . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . 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 18646 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55 grpmnd 18226 . . . . . . . . . . . . . . . . . 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 18715 . . . . . . . . . . . . . . . . . . 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 786 . . . . . . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . . . . 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 18123 . . . . . . . . . . . . . . . . 17 ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g𝑆)𝑗))
6656, 61, 63, 65syl3anc 1372 . . . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . . . . 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 2773 . . . . . . . . . . . . . . 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 7188 . . . . . . . . . . . . . 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 13963 . . . . . . . . . . . . . . . . 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 789 . . . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . . 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 14322 . . . . . . . . . . . . . . 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 18122 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 5900 . . . . . . . . . . . . . . . . . . . 20 (𝑀 “ (♯ “ {3})) = ((toCyc‘𝐷) “ (♯ “ {3}))
8077, 79eqtri 2761 . . . . . . . . . . . . . . . . . . 19 𝐶 = ((toCyc‘𝐷) “ (♯ “ {3}))
8180, 9cyc3evpm 30994 . . . . . . . . . . . . . . . . . 18 (𝐷 ∈ Fin → 𝐶𝐴)
8211, 12evpmss 20402 . . . . . . . . . . . . . . . . . . 19 (pmEven‘𝐷) ⊆ (Base‘𝑆)
839, 82eqsstri 3911 . . . . . . . . . . . . . . . . . 18 𝐴 ⊆ (Base‘𝑆)
8481, 83sstrdi 3889 . . . . . . . . . . . . . . . . 17 (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85 sswrd 13963 . . . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . 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 3878 . . . . . . . . . . . . . . 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 18122 . . . . . . . . . . . . . . 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 1372 . . . . . . . . . . . . . 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 2783 . . . . . . . . . . . . 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 3529 . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . 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 786 . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . 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 784 . . . . . . . . . . . . . . . 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 499 . . . . . . . . . . . . . . 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 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”⟩))) → 𝑗 = (𝑀‘⟨“𝑔”⟩))
10352ad6antr 736 . . . . . . . . . . . . . . 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 498 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . 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 30995 . . . . . . . . . . . . . 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 788 . . . . . . . . . . . . . . 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 790 . . . . . . . . . . . . . . 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 30967 . . . . . . . . . . . . . . 15 ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔𝐷𝐷 (𝑔𝑗 = (𝑀‘⟨“𝑔”⟩)))
110107, 108, 109syl2anc 587 . . . . . . . . . . . . . 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 3243 . . . . . . . . . . . . 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 30967 . . . . . . . . . . . . . 14 ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒𝐷𝑓𝐷 (𝑒𝑓𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
11352, 59, 112syl2anc 587 . . . . . . . . . . . . 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 3243 . . . . . . . . . . . 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 3199 . . . . . . . . . . 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 750 . . . . . . . . . 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 488 . . . . . . . . . . . 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 769 . . . . . . . . . . . 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 7178 . . . . . . . . . . . . 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 3350 . . . . . . . . . . 11 (∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
123119, 122sylibr 237 . . . . . . . . . 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 3199 . . . . . . . . 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 416 . . . . . . . 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 1347 . . . . . . 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 30800 . . . . . 6 ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
128127imp 410 . . . . 5 (((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
1291, 22, 7, 128syl21anc 837 . . . 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 3207 . . . 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 260 . . 3 ((((𝐷 ∈ Fin ∧ 𝑄𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
13383sseli 3873 . . . 4 (𝑄𝐴𝑄 ∈ (Base‘𝑆))
13411, 12, 16psgnfitr 18763 . . . . 5 (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣)))
135134biimpa 480 . . . 4 ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
136133, 135sylan2 596 . . 3 ((𝐷 ∈ Fin ∧ 𝑄𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
137132, 136r19.29a 3199 . 2 ((𝐷 ∈ Fin ∧ 𝑄𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
138 simpr 488 . . . 4 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 = (𝑆 Σg 𝑤))
13911altgnsg 30993 . . . . . . . . 9 (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
1409, 139eqeltrid 2837 . . . . . . . 8 (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141 nsgsubg 18428 . . . . . . . 8 (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142 subgsubm 18419 . . . . . . . 8 (𝐴 ∈ (SubGrp‘𝑆) → 𝐴 ∈ (SubMnd‘𝑆))
143140, 141, 1423syl 18 . . . . . . 7 (𝐷 ∈ Fin → 𝐴 ∈ (SubMnd‘𝑆))
144143adantr 484 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝐴 ∈ (SubMnd‘𝑆))
145 sswrd 13963 . . . . . . . 8 (𝐶𝐴 → Word 𝐶 ⊆ Word 𝐴)
14681, 145syl 17 . . . . . . 7 (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147146sselda 3877 . . . . . 6 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148 gsumwsubmcl 18117 . . . . . 6 ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σg 𝑤) ∈ 𝐴)
149144, 147, 148syl2anc 587 . . . . 5 ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σg 𝑤) ∈ 𝐴)
150149adantr 484 . . . 4 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → (𝑆 Σg 𝑤) ∈ 𝐴)
151138, 150eqeltrd 2833 . . 3 (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄𝐴)
152151r19.29an 3198 . 2 ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)) → 𝑄𝐴)
153137, 152impbida 801 1 (𝐷 ∈ Fin → (𝑄𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1542  wcel 2114  wne 2934  wrex 3054  wss 3843  c0 4211  {csn 4516   class class class wbr 5030  ccnv 5524  ran crn 5526  cima 5528  cfv 6339  (class class class)co 7170  Fincfn 8555  1c1 10616  -cneg 10949  2c2 11771  3c3 11772  0cn0 11976  cz 12062  cexp 13521  chash 13782  Word cword 13955   ++ cconcat 14011  ⟨“cs2 14292  cdvds 15699  Basecbs 16586  +gcplusg 16668   Σg cgsu 16817  Mndcmnd 18027  SubMndcsubmnd 18071  Grpcgrp 18219  SubGrpcsubg 18391  NrmSGrpcnsg 18392  SymGrpcsymg 18613  pmTrspcpmtr 18687  pmSgncpsgn 18735  pmEvencevpm 18736  toCycctocyc 30950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-reg 9129  ax-ac2 9963  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692  ax-pre-sup 10693  ax-addf 10694  ax-mulf 10695
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-xor 1507  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rmo 3061  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-ot 4525  df-uni 4797  df-int 4837  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-se 5484  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-isom 6348  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-1st 7714  df-2nd 7715  df-tpos 7921  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-1o 8131  df-2o 8132  df-er 8320  df-map 8439  df-en 8556  df-dom 8557  df-sdom 8558  df-fin 8559  df-sup 8979  df-inf 8980  df-card 9441  df-ac 9616  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-div 11376  df-nn 11717  df-2 11779  df-3 11780  df-4 11781  df-5 11782  df-6 11783  df-7 11784  df-8 11785  df-9 11786  df-n0 11977  df-xnn0 12049  df-z 12063  df-dec 12180  df-uz 12325  df-rp 12473  df-fz 12982  df-fzo 13125  df-fl 13253  df-mod 13329  df-seq 13461  df-exp 13522  df-hash 13783  df-word 13956  df-lsw 14004  df-concat 14012  df-s1 14039  df-substr 14092  df-pfx 14122  df-splice 14201  df-reverse 14210  df-csh 14240  df-s2 14299  df-s3 14300  df-dvds 15700  df-struct 16588  df-ndx 16589  df-slot 16590  df-base 16592  df-sets 16593  df-ress 16594  df-plusg 16681  df-mulr 16682  df-starv 16683  df-tset 16687  df-ple 16688  df-ds 16690  df-unif 16691  df-0g 16818  df-gsum 16819  df-mre 16960  df-mrc 16961  df-acs 16963  df-mgm 17968  df-sgrp 18017  df-mnd 18028  df-mhm 18072  df-submnd 18073  df-efmnd 18150  df-grp 18222  df-minusg 18223  df-sbg 18224  df-subg 18394  df-nsg 18395  df-ghm 18474  df-gim 18517  df-oppg 18592  df-symg 18614  df-pmtr 18688  df-psgn 18737  df-evpm 18738  df-cmn 19026  df-abl 19027  df-mgp 19359  df-ur 19371  df-ring 19418  df-cring 19419  df-oppr 19495  df-dvdsr 19513  df-unit 19514  df-invr 19544  df-dvr 19555  df-drng 19623  df-cnfld 20218  df-tocyc 30951
This theorem is referenced by: (None)
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