Theorem cyc3genpm 33789

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 → (𝑄 ∈ 𝐴 ↔ ∃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 ℎ 𝑗 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		simplr 781	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2		lencl 14708	. . . . . . . 8 ⊢ (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ0)
3	2	ad2antlr 740	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℕ0)
4	3	nn0zd 12748	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℤ)
5		simpr 490	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 = (𝑆 Σg 𝑣))
6	5	fveq2d 6915	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)))
7		simplll 787	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
8		simpllr 788	. . . . . . . . 9 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 ∈ 𝐴)
9		cyc3genpm.a	. . . . . . . . 9 ⊢ 𝐴 = (pmEven‘𝐷)
10	8, 9	eleqtrdi 2874	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 ∈ (pmEven‘𝐷))
11		cyc3genpm.s	. . . . . . . . 9 ⊢ 𝑆 = (SymGrp‘𝐷)
12		eqid 2763	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2763	. . . . . . . . 9 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
14	11, 12, 13	psgnevpm 21928	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
15	7, 10, 14	syl2anc 596	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16		eqid 2763	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
17	11, 16, 13	psgnvalii 19756	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = ( -1↑(♯‘𝑣)))
18	7, 1, 17	syl2anc 596	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = ( -1↑(♯‘𝑣)))
19	6, 15, 18	3eqtr3rd 2807	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ( -1↑(♯‘𝑣)) = 1)
20		m1exp1 16578	. . . . . . 7 ⊢ ((♯‘𝑣) ∈ ℤ → (( -1↑(♯‘𝑣)) = 1 ↔ 2 ∥ (♯‘𝑣)))
21	20	biimpa 482	. . . . . 6 ⊢ (((♯‘𝑣) ∈ ℤ ∧ ( -1↑(♯‘𝑣)) = 1) → 2 ∥ (♯‘𝑣))
22	4, 19, 21	syl2anc 596	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 2 ∥ (♯‘𝑣))
23		oveq2 7454	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑆 Σg 𝑥) = (𝑆 Σg ∅))
24	23	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
25	24	rexbidv 3190	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
26	25	imbi2d 343	. . . . . . 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 7454	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑢))
28	27	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
29	28	rexbidv 3190	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
30	29	imbi2d 343	. . . . . . 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 7454	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σg 𝑥) = (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)))
32	31	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
33	32	rexbidv 3190	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
34	33	imbi2d 343	. . . . . . 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 7454	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑣))
36	35	eqeq1d 2765	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
37	36	rexbidv 3190	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
38	37	imbi2d 343	. . . . . . 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 14714	. . . . . . . . 9 ⊢ ∅ ∈ Word 𝐶
40	39	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41		simpr 490	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
42	41	oveq2d 7462	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σg 𝑤) = (𝑆 Σg ∅))
43	42	eqeq2d 2774	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σg ∅) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg ∅)))
44		eqidd 2764	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆 Σg ∅) = (𝑆 Σg ∅))
45	40, 43, 44	rspcedvd 3582	. . . . . . 7 ⊢ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))
46		ccatcl 14749	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word 𝐶 ∧ 𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
47	46	ad5ant24 773	. . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
48		oveq2 7454	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑣 ++ 𝑐)))
49	48	eqeq2d 2774	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 ++ 𝑐) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
50	49	adantl 487	. . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) ∧ 𝑤 = (𝑣 ++ 𝑐)) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
51		simpllr 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)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg 𝑢) = (𝑆 Σg 𝑣))
52		simpllr 788	. . . . . . . . . . . . . . . . . . 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)
53	52	ad2antrr 739	. . . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝐷 ∈ Fin)
54	11	symggrp 19647	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55		grpmnd 19183	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
56	53, 54, 55	3syl 19	. . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑆 ∈ Mnd)
57	16, 11, 12	symgtrf 19716	. . . . . . . . . . . . . . . . . . 19 ⊢ ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆)
58	57	a1i 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆))
59		simp-5r 798	. . . . . . . . . . . . . . . . . . 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))
60	59	ad2antrr 739	. . . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑖 ∈ ran (pmTrsp‘𝐷))
61	58, 60	sseldd 3914	. . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑖 ∈ (Base‘𝑆))
62		simp-6r 800	. . . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ ran (pmTrsp‘𝐷))
63	58, 62	sseldd 3914	. . . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ (Base‘𝑆))
64		eqid 2763	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘𝑆) = (+g‘𝑆)
65	12, 64	gsumws2 19070	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g‘𝑆)𝑗))
66	56, 61, 63, 65	syl3anc 1399	. . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g‘𝑆)𝑗))
67		simpr 490	. . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑖(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐))
68	66, 67	eqtrd 2798	. . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑆 Σg 𝑐))
69	51, 68	oveq12d 7464	. . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → ((𝑆 Σg 𝑢)(+g‘𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑣)(+g‘𝑆)(𝑆 Σg 𝑐)))
70		sswrd 14697	. . . . . . . . . . . . . . . . 17 ⊢ (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
71	58, 70	syl 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
72		simp-7l 801	. . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word ran (pmTrsp‘𝐷))
73	71, 72	sseldd 3914	. . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word (Base‘𝑆))
74	61, 63	s2cld 15053	. . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆))
75	12, 64	gsumccat 19069	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Mnd ∧ 𝑢 ∈ Word (Base‘𝑆) ∧ ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑢)(+g‘𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)))
76	56, 73, 74, 75	syl3anc 1399	. . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σg 𝑢)(+g‘𝑆)(𝑆 Σg ⟨“𝑖𝑗”⟩)))
77		cyc3genpm.t	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = (𝑀 “ (◡♯ “ {3}))
78		cyc3genpm.m	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑀 = (toCyc‘𝐷)
79	78	imaeq1i 6062	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 “ (◡♯ “ {3})) = ((toCyc‘𝐷) “ (◡♯ “ {3}))
80	77, 79	eqtri 2786	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3}))
81	80, 9	cyc3evpm 33787	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
82	11, 12	evpmss 21925	. . . . . . . . . . . . . . . . . . 19 ⊢ (pmEven‘𝐷) ⊆ (Base‘𝑆)
83	9, 82	eqsstri 3959	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 ⊆ (Base‘𝑆)
84	81, 83	sstrdi 3925	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85		sswrd 14697	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ⊆ (Base‘𝑆) → Word 𝐶 ⊆ Word (Base‘𝑆))
86	53, 84, 85	3syl 19	. . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → Word 𝐶 ⊆ Word (Base‘𝑆))
87		simp-4r 796	. . . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑣 ∈ Word 𝐶)
88	86, 87	sseldd 3914	. . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑣 ∈ Word (Base‘𝑆))
89		simplr 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word 𝐶)
90	86, 89	sseldd 3914	. . . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word (Base‘𝑆))
91	12, 64	gsumccat 19069	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Mnd ∧ 𝑣 ∈ Word (Base‘𝑆) ∧ 𝑐 ∈ Word (Base‘𝑆)) → (𝑆 Σg (𝑣 ++ 𝑐)) = ((𝑆 Σg 𝑣)(+g‘𝑆)(𝑆 Σg 𝑐)))
92	56, 88, 90, 91	syl3anc 1399	. . . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑣 ++ 𝑐)) = ((𝑆 Σg 𝑣)(+g‘𝑆)(𝑆 Σg 𝑐)))
93	69, 76, 92	3eqtr4d 2808	. . . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐)))
94	47, 50, 93	rspcedvd 3582	. . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤))
95		cyc3genpm.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (♯‘𝐷)
96		simp-6r 800	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑒 ∈ 𝐷)
97		simp-5r 798	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑓 ∈ 𝐷)
98		simpllr 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑔 ∈ 𝐷)
99		simplr 781	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → ℎ ∈ 𝐷)
100		simp-4r 796	. . . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
101	100	simprd 501	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))
102		simprr 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))
103	52	ad6antr 749	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝐷 ∈ Fin)
104	100	simpld 500	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑒 ≠ 𝑓)
105		simprl 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑔 ≠ ℎ)
106	77, 9, 11, 95, 78, 64, 96, 97, 98, 99, 101, 102, 103, 104, 105	cyc3genpmlem 33788	. . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → ∃𝑐 ∈ Word 𝐶(𝑖(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐))
107		simp-6r 800	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝐷 ∈ Fin)
108		simp-7r 802	. . . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝑗 ∈ ran (pmTrsp‘𝐷))
109	16, 78	trsp2cyc 33760	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
110	107, 108, 109	syl2anc 596	. . . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
111	106, 110	r19.29vva 3226	. . . . . . . . . . . . 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑐 ∈ Word 𝐶(𝑖(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐))
112	16, 78	trsp2cyc 33760	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
113	52, 59, 112	syl2anc 596	. . . . . . . . . . . . 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)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
114	111, 113	r19.29vva 3226	. . . . . . . . . . . 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐))
115	94, 114	r19.29a 3174	. . . . . . . . . . 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))
116	115	adantl3r 763	. . . . . . . . . 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 490	. . . . . . . . . . . 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 781	. . . . . . . . . . . 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)))
119	117, 118	mpd 16	. . . . . . . . . . 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 7454	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
121	120	eqeq2d 2774	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
122	121	cbvrexvw 3245	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
123	119, 122	sylibr 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))
124	116, 123	r19.29a 3174	. . . . . . . . 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))
125	124	ex 418	. . . . . . . 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)))
126	125	ex3 1365	. . . . . . 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))))
127	26, 30, 34, 38, 45, 126	wrdt2ind 33589	. . . . . 6 ⊢ ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
128	127	imp 412	. . . . 5 ⊢ (((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
129	1, 22, 7, 128	syl21anc 851	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
130	5	eqeq1d 2765	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (𝑄 = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
131	130	rexbidv 3190	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
132	129, 131	mpbird 260	. . 3 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
133	83	sseli 3909	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝑆))
134	11, 12, 16	psgnfitr 19764	. . . . 5 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣)))
135	134	biimpa 482	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
136	133, 135	sylan2 605	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
137	132, 136	r19.29a 3174	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
138		simpr 490	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 = (𝑆 Σg 𝑤))
139	11	altgnsg 33786	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
140	9, 139	eqeltrid 2868	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141		nsgsubg 19400	. . . . . . . 8 ⊢ (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142		subgsubm 19391	. . . . . . . 8 ⊢ (𝐴 ∈ (SubGrp‘𝑆) → 𝐴 ∈ (SubMnd‘𝑆))
143	140, 141, 142	3syl 19	. . . . . . 7 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubMnd‘𝑆))
144	143	adantr 486	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝐴 ∈ (SubMnd‘𝑆))
145		sswrd 14697	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → Word 𝐶 ⊆ Word 𝐴)
146	81, 145	syl 18	. . . . . . 7 ⊢ (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147	146	sselda 3913	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148		gsumwsubmcl 19065	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σg 𝑤) ∈ 𝐴)
149	144, 147, 148	syl2anc 596	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σg 𝑤) ∈ 𝐴)
150	149	adantr 486	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → (𝑆 Σg 𝑤) ∈ 𝐴)
151	138, 150	eqeltrd 2864	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
152	151	r19.29an 3170	. 2 ⊢ ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
153	137, 152	impbida 813	1 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∃wrex 3090   ⊆ wss 3900  ∅c0 4281  {csn 4588   class class class wbr 5106  ^◡ccnv 5655  ran crn 5657   “ cima 5659  ‘cfv 6561  (class class class)co 7446  Fincfn 9004  1c1 11231   -cneg 11572  2c2 12427  3c3 12428  ℕ₀cn0 12636  ℤcz 12723  ↑cexp 14234  ♯chash 14504  Word cword 14688   ++ cconcat 14745  ⟨“cs2 15023   ∥ cdvds 16454  Basecbs 17419  +_gcplusg 17460   Σ_g cgsu 17643  Mndcmnd 18955  SubMndcsubmnd 19009  Grpcgrp 19176  SubGrpcsubg 19362  NrmSGrpcnsg 19363  SymGrpcsymg 19616  pmTrspcpmtr 19688  pmSgncpsgn 19736  pmEvencevpm 19737  toCycctocyc 33743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-ac2 10571  ax-cnex 11286  ax-resscn 11287  ax-1cn 11288  ax-icn 11289  ax-addcl 11290  ax-addrcl 11291  ax-mulcl 11292  ax-mulrcl 11293  ax-mulcom 11294  ax-addass 11295  ax-mulass 11296  ax-distr 11297  ax-i2m1 11298  ax-1ne0 11299  ax-1rid 11300  ax-rnegex 11301  ax-rrecex 11302  ax-cnre 11303  ax-pre-lttri 11304  ax-pre-lttrn 11305  ax-pre-ltadd 11306  ax-pre-mulgt0 11307  ax-pre-sup 11308  ax-addf 11309  ax-mulf 11310

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-xor 1542  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-ot 4862  df-uni 4871  df-int 4911  df-iun 4956  df-iin 4957  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-se 5610  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6325  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-om 7905  df-1st 8028  df-2nd 8029  df-tpos 8274  df-frecs 8330  df-wrecs 8361  df-recs 8410  df-rdg 8449  df-1o 8507  df-2o 8508  df-er 8748  df-map 8880  df-en 9005  df-dom 9006  df-sdom 9007  df-fin 9008  df-sup 9464  df-inf 9465  df-card 10050  df-ac 10225  df-pnf 11375  df-mnf 11376  df-xr 11377  df-ltxr 11378  df-le 11379  df-sub 11573  df-neg 11574  df-div 12004  df-nn 12366  df-2 12435  df-3 12436  df-4 12437  df-5 12438  df-6 12439  df-7 12440  df-8 12441  df-9 12442  df-n0 12637  df-xnn0 12710  df-z 12724  df-dec 12845  df-uz 12996  df-rp 13151  df-fz 13670  df-fzo 13819  df-fl 13962  df-mod 14040  df-seq 14175  df-exp 14235  df-hash 14505  df-word 14689  df-lsw 14738  df-concat 14746  df-s1 14773  df-substr 14820  df-pfx 14852  df-splice 14930  df-reverse 14939  df-csh 14971  df-s2 15030  df-s3 15031  df-dvds 16455  df-struct 17357  df-sets 17374  df-slot 17392  df-ndx 17404  df-base 17420  df-ress 17441  df-plusg 17473  df-mulr 17474  df-starv 17475  df-tset 17479  df-ple 17480  df-ds 17482  df-unif 17483  df-0g 17644  df-gsum 17645  df-mre 17788  df-mrc 17789  df-acs 17791  df-mgm 18848  df-sgrp 18940  df-mnd 18956  df-mhm 19010  df-submnd 19011  df-efmnd 19097  df-grp 19179  df-minusg 19180  df-sbg 19181  df-subg 19365  df-nsg 19366  df-ghm 19460  df-gim 19505  df-oppg 19593  df-symg 19617  df-pmtr 19689  df-psgn 19738  df-evpm 19739  df-cmn 20029  df-abl 20030  df-mgp 20394  df-rng 20408  df-ur 20441  df-ring 20494  df-cring 20495  df-oppr 20600  df-dvdsr 20620  df-unit 20621  df-invr 20651  df-dvr 20664  df-drng 21015  df-cnfld 21712  df-tocyc 33744

This theorem is used by: (None)