Theorem cyc3genpm 33213

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 769	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2		lencl 14495	. . . . . . . 8 ⊢ (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ0)
3	2	ad2antlr 728	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℕ0)
4	3	nn0zd 12549	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℤ)
5		simpr 484	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 = (𝑆 Σg 𝑣))
6	5	fveq2d 6844	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)))
7		simplll 775	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
8		simpllr 776	. . . . . . . . 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 2846	. . . . . . . 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 2736	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2736	. . . . . . . . 9 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
14	11, 12, 13	psgnevpm 21569	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
15	7, 10, 14	syl2anc 585	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16		eqid 2736	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
17	11, 16, 13	psgnvalii 19484	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
18	7, 1, 17	syl2anc 585	. . . . . . 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 2780	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20		m1exp1 16345	. . . . . . 7 ⊢ ((♯‘𝑣) ∈ ℤ → ((-1↑(♯‘𝑣)) = 1 ↔ 2 ∥ (♯‘𝑣)))
21	20	biimpa 476	. . . . . 6 ⊢ (((♯‘𝑣) ∈ ℤ ∧ (-1↑(♯‘𝑣)) = 1) → 2 ∥ (♯‘𝑣))
22	4, 19, 21	syl2anc 585	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 2 ∥ (♯‘𝑣))
23		oveq2 7375	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑆 Σg 𝑥) = (𝑆 Σg ∅))
24	23	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
25	24	rexbidv 3161	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
26	25	imbi2d 340	. . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑢))
28	27	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
29	28	rexbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
30	29	imbi2d 340	. . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σg 𝑥) = (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)))
32	31	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
33	32	rexbidv 3161	. . . . . . . 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 340	. . . . . . 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 7375	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑣))
36	35	eqeq1d 2738	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
37	36	rexbidv 3161	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
38	37	imbi2d 340	. . . . . . 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 14501	. . . . . . . . 9 ⊢ ∅ ∈ Word 𝐶
40	39	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41		simpr 484	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
42	41	oveq2d 7383	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σg 𝑤) = (𝑆 Σg ∅))
43	42	eqeq2d 2747	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σg ∅) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg ∅)))
44		eqidd 2737	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆 Σg ∅) = (𝑆 Σg ∅))
45	40, 43, 44	rspcedvd 3566	. . . . . . 7 ⊢ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))
46		ccatcl 14536	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word 𝐶 ∧ 𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
47	46	ad5ant24 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
48		oveq2 7375	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑣 ++ 𝑐)))
49	48	eqeq2d 2747	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 ++ 𝑐) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
50	49	adantl 481	. . . . . . . . . . . . 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 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(+g‘𝑆)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)
53	52	ad2antrr 727	. . . . . . . . . . . . . . . . . 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 19375	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55		grpmnd 18916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑆 ∈ Grp → 𝑆 ∈ Mnd)
56	53, 54, 55	3syl 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑆 ∈ Mnd)
57	16, 11, 12	symgtrf 19444	. . . . . . . . . . . . . . . . . . 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 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))
60	59	ad2antrr 727	. . . . . . . . . . . . . . . . . 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 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(+g‘𝑆)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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ ran (pmTrsp‘𝐷))
63	58, 62	sseldd 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑗 ∈ (Base‘𝑆))
64		eqid 2736	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘𝑆) = (+g‘𝑆)
65	12, 64	gsumws2 18810	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g‘𝑆)𝑗))
66	56, 61, 63, 65	syl3anc 1374	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 7385	. . . . . . . . . . . . . 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 14484	. . . . . . . . . . . . . . . . 17 ⊢ (ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
71	58, 70	syl 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(+g‘𝑆)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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word ran (pmTrsp‘𝐷))
73	71, 72	sseldd 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑢 ∈ Word (Base‘𝑆))
74	61, 63	s2cld 14833	. . . . . . . . . . . . . . 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 18809	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 6022	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 “ (◡♯ “ {3})) = ((toCyc‘𝐷) “ (◡♯ “ {3}))
80	77, 79	eqtri 2759	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3}))
81	80, 9	cyc3evpm 33211	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
82	11, 12	evpmss 21566	. . . . . . . . . . . . . . . . . . 19 ⊢ (pmEven‘𝐷) ⊆ (Base‘𝑆)
83	9, 82	eqsstri 3968	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 ⊆ (Base‘𝑆)
84	81, 83	sstrdi 3934	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85		sswrd 14484	. . . . . . . . . . . . . . . . 17 ⊢ (𝐶 ⊆ (Base‘𝑆) → Word 𝐶 ⊆ Word (Base‘𝑆))
86	53, 84, 85	3syl 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 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑣 ∈ Word 𝐶)
88	86, 87	sseldd 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(+g‘𝑆)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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word 𝐶)
90	86, 89	sseldd 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(+g‘𝑆)𝑗) = (𝑆 Σg 𝑐)) → 𝑐 ∈ Word (Base‘𝑆))
91	12, 64	gsumccat 18809	. . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . 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 3566	. . . . . . . . . . . 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 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”⟩))) → 𝑒 ∈ 𝐷)
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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑓 ∈ 𝐷)
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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑔 ∈ 𝐷)
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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → ℎ ∈ 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)) ∧ 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 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))
103	52	ad6antr 737	. . . . . . . . . . . . . . 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 494	. . . . . . . . . . . . . . 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 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)) ∧ 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 33212	. . . . . . . . . . . . . 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 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”⟩))) → 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)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝑗 ∈ ran (pmTrsp‘𝐷))
109	16, 78	trsp2cyc 33184	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
110	107, 108, 109	syl2anc 585	. . . . . . . . . . . . . 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 3197	. . . . . . . . . . . . 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 33184	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
113	52, 59, 112	syl2anc 585	. . . . . . . . . . . . 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 3197	. . . . . . . . . . . 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 3145	. . . . . . . . . . 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 751	. . . . . . . . . 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 484	. . . . . . . . . . . 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)))
119	117, 118	mpd 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 7375	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
121	120	eqeq2d 2747	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
122	121	cbvrexvw 3216	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
123	119, 122	sylibr 234	. . . . . . . . . 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 3145	. . . . . . . . 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 412	. . . . . . . 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 1348	. . . . . . 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 33013	. . . . . 6 ⊢ ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
128	127	imp 406	. . . . 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 838	. . . 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 2738	. . . . 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 3161	. . . 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 257	. . 3 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
133	83	sseli 3917	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝑆))
134	11, 12, 16	psgnfitr 19492	. . . . 5 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣)))
135	134	biimpa 476	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
136	133, 135	sylan2 594	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
137	132, 136	r19.29a 3145	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
138		simpr 484	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 = (𝑆 Σg 𝑤))
139	11	altgnsg 33210	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
140	9, 139	eqeltrid 2840	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141		nsgsubg 19133	. . . . . . . 8 ⊢ (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142		subgsubm 19124	. . . . . . . 8 ⊢ (𝐴 ∈ (SubGrp‘𝑆) → 𝐴 ∈ (SubMnd‘𝑆))
143	140, 141, 142	3syl 18	. . . . . . 7 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubMnd‘𝑆))
144	143	adantr 480	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝐴 ∈ (SubMnd‘𝑆))
145		sswrd 14484	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → Word 𝐶 ⊆ Word 𝐴)
146	81, 145	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147	146	sselda 3921	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148		gsumwsubmcl 18805	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σg 𝑤) ∈ 𝐴)
149	144, 147, 148	syl2anc 585	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σg 𝑤) ∈ 𝐴)
150	149	adantr 480	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → (𝑆 Σg 𝑤) ∈ 𝐴)
151	138, 150	eqeltrd 2836	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
152	151	r19.29an 3141	. 2 ⊢ ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
153	137, 152	impbida 801	1 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ≠ wne 2932  ∃wrex 3061   ⊆ wss 3889  ∅c0 4273  {csn 4567   class class class wbr 5085  ^◡ccnv 5630  ran crn 5632   “ cima 5634  ‘cfv 6498  (class class class)co 7367  Fincfn 8893  1c1 11039  -cneg 11378  2c2 12236  3c3 12237  ℕ₀cn0 12437  ℤcz 12524  ↑cexp 14023  ♯chash 14292  Word cword 14475   ++ cconcat 14532  ⟨“cs2 14803   ∥ cdvds 16221  Basecbs 17179  +_gcplusg 17220   Σ_g cgsu 17403  Mndcmnd 18702  SubMndcsubmnd 18750  Grpcgrp 18909  SubGrpcsubg 19096  NrmSGrpcnsg 19097  SymGrpcsymg 19344  pmTrspcpmtr 19416  pmSgncpsgn 19464  pmEvencevpm 19465  toCycctocyc 33167

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-reg 9507  ax-ac2 10385  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116  ax-addf 11117  ax-mulf 11118

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1514  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-ot 4576  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-1st 7942  df-2nd 7943  df-tpos 8176  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-sup 9355  df-inf 9356  df-card 9863  df-ac 10038  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-xnn0 12511  df-z 12525  df-dec 12645  df-uz 12789  df-rp 12943  df-fz 13462  df-fzo 13609  df-fl 13751  df-mod 13829  df-seq 13964  df-exp 14024  df-hash 14293  df-word 14476  df-lsw 14525  df-concat 14533  df-s1 14559  df-substr 14604  df-pfx 14634  df-splice 14712  df-reverse 14721  df-csh 14751  df-s2 14810  df-s3 14811  df-dvds 16222  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-0g 17404  df-gsum 17405  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-mhm 18751  df-submnd 18752  df-efmnd 18837  df-grp 18912  df-minusg 18913  df-sbg 18914  df-subg 19099  df-nsg 19100  df-ghm 19188  df-gim 19234  df-oppg 19321  df-symg 19345  df-pmtr 19417  df-psgn 19466  df-evpm 19467  df-cmn 19757  df-abl 19758  df-mgp 20122  df-rng 20134  df-ur 20163  df-ring 20216  df-cring 20217  df-oppr 20317  df-dvdsr 20337  df-unit 20338  df-invr 20368  df-dvr 20381  df-drng 20708  df-cnfld 21353  df-tocyc 33168

This theorem is referenced by: (None)