Theorem cyc3genpm 33031

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 767	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2		lencl 14533	. . . . . . . 8 ⊢ (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ0)
3	2	ad2antlr 725	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℕ0)
4	3	nn0zd 12627	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (♯‘𝑣) ∈ ℤ)
5		simpr 483	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝑄 = (𝑆 Σg 𝑣))
6	5	fveq2d 6894	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)))
7		simplll 773	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 𝐷 ∈ Fin)
8		simpllr 774	. . . . . . . . 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 2836	. . . . . . . 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 2726	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2726	. . . . . . . . 9 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
14	11, 12, 13	psgnevpm 21578	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
15	7, 10, 14	syl2anc 582	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16		eqid 2726	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
17	11, 16, 13	psgnvalii 19500	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σg 𝑣)) = (-1↑(♯‘𝑣)))
18	7, 1, 17	syl2anc 582	. . . . . . 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 2775	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20		m1exp1 16370	. . . . . . 7 ⊢ ((♯‘𝑣) ∈ ℤ → ((-1↑(♯‘𝑣)) = 1 ↔ 2 ∥ (♯‘𝑣)))
21	20	biimpa 475	. . . . . 6 ⊢ (((♯‘𝑣) ∈ ℤ ∧ (-1↑(♯‘𝑣)) = 1) → 2 ∥ (♯‘𝑣))
22	4, 19, 21	syl2anc 582	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → 2 ∥ (♯‘𝑣))
23		oveq2 7421	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑆 Σg 𝑥) = (𝑆 Σg ∅))
24	23	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
25	24	rexbidv 3169	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤)))
26	25	imbi2d 339	. . . . . . 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 7421	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑢))
28	27	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
29	28	rexbidv 3169	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
30	29	imbi2d 339	. . . . . . 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 7421	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σg 𝑥) = (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)))
32	31	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤)))
33	32	rexbidv 3169	. . . . . . . 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 339	. . . . . . 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 7421	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑆 Σg 𝑥) = (𝑆 Σg 𝑣))
36	35	eqeq1d 2728	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
37	36	rexbidv 3169	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑥) = (𝑆 Σg 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
38	37	imbi2d 339	. . . . . . 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 14539	. . . . . . . . 9 ⊢ ∅ ∈ Word 𝐶
40	39	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41		simpr 483	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
42	41	oveq2d 7429	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σg 𝑤) = (𝑆 Σg ∅))
43	42	eqeq2d 2737	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σg ∅) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg ∅) = (𝑆 Σg ∅)))
44		eqidd 2727	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆 Σg ∅) = (𝑆 Σg ∅))
45	40, 43, 44	rspcedvd 3609	. . . . . . 7 ⊢ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg ∅) = (𝑆 Σg 𝑤))
46		ccatcl 14574	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word 𝐶 ∧ 𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
47	46	ad5ant24 759	. . . . . . . . . . . . 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 7421	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σg 𝑤) = (𝑆 Σg (𝑣 ++ 𝑐)))
49	48	eqeq2d 2737	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 ++ 𝑐) → ((𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg 𝑤) ↔ (𝑆 Σg (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σg (𝑣 ++ 𝑐))))
50	49	adantl 480	. . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 19391	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55		grpmnd 18927	. . . . . . . . . . . . . . . . . 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 19460	. . . . . . . . . . . . . . . . . . 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 784	. . . . . . . . . . . . . . . . . . 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 724	. . . . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . . . 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 2726	. . . . . . . . . . . . . . . . . 18 ⊢ (+g‘𝑆) = (+g‘𝑆)
65	12, 64	gsumws2 18824	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σg ⟨“𝑖𝑗”⟩) = (𝑖(+g‘𝑆)𝑗))
66	56, 61, 63, 65	syl3anc 1368	. . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . 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 2766	. . . . . . . . . . . . . . 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 7431	. . . . . . . . . . . . . 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 14522	. . . . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . 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 14872	. . . . . . . . . . . . . . 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 18823	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 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 6056	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 “ (◡♯ “ {3})) = ((toCyc‘𝐷) “ (◡♯ “ {3}))
80	77, 79	eqtri 2754	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = ((toCyc‘𝐷) “ (◡♯ “ {3}))
81	80, 9	cyc3evpm 33029	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
82	11, 12	evpmss 21575	. . . . . . . . . . . . . . . . . . 19 ⊢ (pmEven‘𝐷) ⊆ (Base‘𝑆)
83	9, 82	eqsstri 4013	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 ⊆ (Base‘𝑆)
84	81, 83	sstrdi 3991	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85		sswrd 14522	. . . . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . . 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 3979	. . . . . . . . . . . . . . 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 18823	. . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . 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 2776	. . . . . . . . . . . . 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 3609	. . . . . . . . . . . 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 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”⟩))) → 𝑒 ∈ 𝐷)
97		simp-5r 784	. . . . . . . . . . . . . . 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 774	. . . . . . . . . . . . . . 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 767	. . . . . . . . . . . . . . 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 782	. . . . . . . . . . . . . . . 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 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”⟩))) → 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))
102		simprr 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”⟩))) → 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))
103	52	ad6antr 734	. . . . . . . . . . . . . . 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 493	. . . . . . . . . . . . . . 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 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”⟩))) → 𝑔 ≠ ℎ)
106	77, 9, 11, 95, 78, 64, 96, 97, 98, 99, 101, 102, 103, 104, 105	cyc3genpmlem 33030	. . . . . . . . . . . . . 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 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”⟩))) → 𝐷 ∈ Fin)
108		simp-7r 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”⟩))) → 𝑗 ∈ ran (pmTrsp‘𝐷))
109	16, 78	trsp2cyc 33002	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
110	107, 108, 109	syl2anc 582	. . . . . . . . . . . . . 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 3204	. . . . . . . . . . . . 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 33002	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
113	52, 59, 112	syl2anc 582	. . . . . . . . . . . . 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 3204	. . . . . . . . . . . 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 3152	. . . . . . . . . . 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 748	. . . . . . . . . 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 483	. . . . . . . . . . . 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 767	. . . . . . . . . . . 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 7421	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (𝑆 Σg 𝑣) = (𝑆 Σg 𝑤))
121	120	eqeq2d 2737	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ (𝑆 Σg 𝑢) = (𝑆 Σg 𝑤)))
122	121	cbvrexvw 3226	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑢) = (𝑆 Σg 𝑤))
123	119, 122	sylibr 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))
124	116, 123	r19.29a 3152	. . . . . . . . 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 411	. . . . . . . 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 1343	. . . . . . 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 32817	. . . . . 6 ⊢ ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σg 𝑣) = (𝑆 Σg 𝑤)))
128	127	imp 405	. . . . 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 836	. . . 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 2728	. . . . 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 3169	. . . 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 256	. . 3 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σg 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
133	83	sseli 3974	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝑆))
134	11, 12, 16	psgnfitr 19508	. . . . 5 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣)))
135	134	biimpa 475	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
136	133, 135	sylan2 591	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σg 𝑣))
137	132, 136	r19.29a 3152	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤))
138		simpr 483	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 = (𝑆 Σg 𝑤))
139	11	altgnsg 33028	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
140	9, 139	eqeltrid 2830	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141		nsgsubg 19145	. . . . . . . 8 ⊢ (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142		subgsubm 19135	. . . . . . . 8 ⊢ (𝐴 ∈ (SubGrp‘𝑆) → 𝐴 ∈ (SubMnd‘𝑆))
143	140, 141, 142	3syl 18	. . . . . . 7 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (SubMnd‘𝑆))
144	143	adantr 479	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝐴 ∈ (SubMnd‘𝑆))
145		sswrd 14522	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → Word 𝐶 ⊆ Word 𝐴)
146	81, 145	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147	146	sselda 3978	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148		gsumwsubmcl 18819	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σg 𝑤) ∈ 𝐴)
149	144, 147, 148	syl2anc 582	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σg 𝑤) ∈ 𝐴)
150	149	adantr 479	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → (𝑆 Σg 𝑤) ∈ 𝐴)
151	138, 150	eqeltrd 2826	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
152	151	r19.29an 3148	. 2 ⊢ ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)) → 𝑄 ∈ 𝐴)
153	137, 152	impbida 799	1 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σg 𝑤)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∃wrex 3060   ⊆ wss 3946  ∅c0 4322  {csn 4623   class class class wbr 5143  ^◡ccnv 5671  ran crn 5673   “ cima 5675  ‘cfv 6543  (class class class)co 7413  Fincfn 8963  1c1 11147  -cneg 11483  2c2 12310  3c3 12311  ℕ₀cn0 12515  ℤcz 12601  ↑cexp 14072  ♯chash 14339  Word cword 14514   ++ cconcat 14570  ⟨“cs2 14842   ∥ cdvds 16248  Basecbs 17205  +_gcplusg 17258   Σ_g cgsu 17447  Mndcmnd 18719  SubMndcsubmnd 18764  Grpcgrp 18920  SubGrpcsubg 19107  NrmSGrpcnsg 19108  SymGrpcsymg 19357  pmTrspcpmtr 19432  pmSgncpsgn 19480  pmEvencevpm 19481  toCycctocyc 32985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5280  ax-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7735  ax-reg 9625  ax-ac2 10494  ax-cnex 11202  ax-resscn 11203  ax-1cn 11204  ax-icn 11205  ax-addcl 11206  ax-addrcl 11207  ax-mulcl 11208  ax-mulrcl 11209  ax-mulcom 11210  ax-addass 11211  ax-mulass 11212  ax-distr 11213  ax-i2m1 11214  ax-1ne0 11215  ax-1rid 11216  ax-rnegex 11217  ax-rrecex 11218  ax-cnre 11219  ax-pre-lttri 11220  ax-pre-lttrn 11221  ax-pre-ltadd 11222  ax-pre-mulgt0 11223  ax-pre-sup 11224  ax-addf 11225  ax-mulf 11226

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-xor 1506  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-tp 4628  df-op 4630  df-ot 4632  df-uni 4906  df-int 4947  df-iun 4995  df-iin 4996  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6302  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7992  df-2nd 7993  df-tpos 8230  df-frecs 8285  df-wrecs 8316  df-recs 8390  df-rdg 8429  df-1o 8485  df-2o 8486  df-er 8723  df-map 8846  df-en 8964  df-dom 8965  df-sdom 8966  df-fin 8967  df-sup 9475  df-inf 9476  df-card 9972  df-ac 10149  df-pnf 11288  df-mnf 11289  df-xr 11290  df-ltxr 11291  df-le 11292  df-sub 11484  df-neg 11485  df-div 11910  df-nn 12256  df-2 12318  df-3 12319  df-4 12320  df-5 12321  df-6 12322  df-7 12323  df-8 12324  df-9 12325  df-n0 12516  df-xnn0 12588  df-z 12602  df-dec 12721  df-uz 12866  df-rp 13020  df-fz 13530  df-fzo 13673  df-fl 13803  df-mod 13881  df-seq 14013  df-exp 14073  df-hash 14340  df-word 14515  df-lsw 14563  df-concat 14571  df-s1 14596  df-substr 14641  df-pfx 14671  df-splice 14750  df-reverse 14759  df-csh 14789  df-s2 14849  df-s3 14850  df-dvds 16249  df-struct 17141  df-sets 17158  df-slot 17176  df-ndx 17188  df-base 17206  df-ress 17235  df-plusg 17271  df-mulr 17272  df-starv 17273  df-tset 17277  df-ple 17278  df-ds 17280  df-unif 17281  df-0g 17448  df-gsum 17449  df-mre 17591  df-mrc 17592  df-acs 17594  df-mgm 18625  df-sgrp 18704  df-mnd 18720  df-mhm 18765  df-submnd 18766  df-efmnd 18851  df-grp 18923  df-minusg 18924  df-sbg 18925  df-subg 19110  df-nsg 19111  df-ghm 19200  df-gim 19246  df-oppg 19333  df-symg 19358  df-pmtr 19433  df-psgn 19482  df-evpm 19483  df-cmn 19773  df-abl 19774  df-mgp 20111  df-rng 20129  df-ur 20158  df-ring 20211  df-cring 20212  df-oppr 20309  df-dvdsr 20332  df-unit 20333  df-invr 20363  df-dvr 20376  df-drng 20702  df-cnfld 21337  df-tocyc 32986

This theorem is referenced by: (None)