cyc3genpm - Mathbox for Thierry Arnoux

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Theorem cyc3genpm 32581

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 Σ_g 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 765	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2		lencl 14487	. . . . . . . 8 ⊢ (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ₀)
3	2	ad2antlr 723	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (♯‘𝑣) ∈ ℕ₀)
4	3	nn0zd 12588	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (♯‘𝑣) ∈ ℤ)
5		simpr 483	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑄 = (𝑆 Σ_g 𝑣))
6	5	fveq2d 6894	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = ((pmSgn‘𝐷)‘(𝑆 Σ_g 𝑣)))
7		simplll 771	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝐷 ∈ Fin)
8		simpllr 772	. . . . . . . . 9 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑄 ∈ 𝐴)
9		cyc3genpm.a	. . . . . . . . 9 ⊢ 𝐴 = (pmEven‘𝐷)
10	8, 9	eleqtrdi 2841	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑄 ∈ (pmEven‘𝐷))
11		cyc3genpm.s	. . . . . . . . 9 ⊢ 𝑆 = (SymGrp‘𝐷)
12		eqid 2730	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2730	. . . . . . . . 9 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
14	11, 12, 13	psgnevpm 21361	. . . . . . . 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 Σ_g 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16		eqid 2730	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
17	11, 16, 13	psgnvalii 19418	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σ_g 𝑣)) = (-1↑(♯‘𝑣)))
18	7, 1, 17	syl2anc 582	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → ((pmSgn‘𝐷)‘(𝑆 Σ_g 𝑣)) = (-1↑(♯‘𝑣)))
19	6, 15, 18	3eqtr3rd 2779	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20		m1exp1 16323	. . . . . . 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 Σ_g 𝑣)) → 2 ∥ (♯‘𝑣))
23		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g ∅))
24	23	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤)))
25	24	rexbidv 3176	. . . . . . . 8 ⊢ (𝑥 = ∅ → (∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤)))
26	25	imbi2d 339	. . . . . . 7 ⊢ (𝑥 = ∅ → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤))))
27		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑢))
28	27	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)))
29	28	rexbidv 3176	. . . . . . . 8 ⊢ (𝑥 = 𝑢 → (∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)))
30	29	imbi2d 339	. . . . . . 7 ⊢ (𝑥 = 𝑢 → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))))
31		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)))
32	31	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤)))
33	32	rexbidv 3176	. . . . . . . 8 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤)))
34	33	imbi2d 339	. . . . . . 7 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤))))
35		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑣))
36	35	eqeq1d 2732	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
37	36	rexbidv 3176	. . . . . . . 8 ⊢ (𝑥 = 𝑣 → (∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
38	37	imbi2d 339	. . . . . . 7 ⊢ (𝑥 = 𝑣 → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤)) ↔ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤))))
39		wrd0 14493	. . . . . . . . 9 ⊢ ∅ ∈ Word 𝐶
40	39	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41		simpr 483	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
42	41	oveq2d 7427	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σ_g 𝑤) = (𝑆 Σ_g ∅))
43	42	eqeq2d 2741	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g ∅) = (𝑆 Σ_g ∅)))
44		eqidd 2731	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆 Σ_g ∅) = (𝑆 Σ_g ∅))
45	40, 43, 44	rspcedvd 3613	. . . . . . 7 ⊢ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤))
46		ccatcl 14528	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word 𝐶 ∧ 𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
47	46	ad5ant24 757	. . . . . . . . . . . . 13 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
48		oveq2 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σ_g 𝑤) = (𝑆 Σ_g (𝑣 ++ 𝑐)))
49	48	eqeq2d 2741	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑣 ++ 𝑐) → ((𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g (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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) ∧ 𝑤 = (𝑣 ++ 𝑐)) → ((𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g (𝑣 ++ 𝑐))))
51		simpllr 772	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣))
52		simpllr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → 𝐷 ∈ Fin)
53	52	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝐷 ∈ Fin)
54	11	symggrp 19309	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55		grpmnd 18862	. . . . . . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑆 ∈ Mnd)
57	16, 11, 12	symgtrf 19378	. . . . . . . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → ran (pmTrsp‘𝐷) ⊆ (Base‘𝑆))
59		simp-5r 782	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → 𝑖 ∈ ran (pmTrsp‘𝐷))
60	59	ad2antrr 722	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑖 ∈ ran (pmTrsp‘𝐷))
61	58, 60	sseldd 3982	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑖 ∈ (Base‘𝑆))
62		simp-6r 784	. . . . . . . . . . . . . . . . . 18 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑗 ∈ ran (pmTrsp‘𝐷))
63	58, 62	sseldd 3982	. . . . . . . . . . . . . . . . 17 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑗 ∈ (Base‘𝑆))
64		eqid 2730	. . . . . . . . . . . . . . . . . 18 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
65	12, 64	gsumws2 18759	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σ_g ⟨“𝑖𝑗”⟩) = (𝑖(+_g‘𝑆)𝑗))
66	56, 61, 63, 65	syl3anc 1369	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g ⟨“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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐))
68	66, 67	eqtrd 2770	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g ⟨“𝑖𝑗”⟩) = (𝑆 Σ_g 𝑐))
69	51, 68	oveq12d 7429	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → ((𝑆 Σ_g 𝑢)(+_g‘𝑆)(𝑆 Σ_g ⟨“𝑖𝑗”⟩)) = ((𝑆 Σ_g 𝑣)(+_g‘𝑆)(𝑆 Σ_g 𝑐)))
70		sswrd 14476	. . . . . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → Word ran (pmTrsp‘𝐷) ⊆ Word (Base‘𝑆))
72		simp-7l 785	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑢 ∈ Word ran (pmTrsp‘𝐷))
73	71, 72	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑢 ∈ Word (Base‘𝑆))
74	61, 63	s2cld 14826	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆))
75	12, 64	gsumccat 18758	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Mnd ∧ 𝑢 ∈ Word (Base‘𝑆) ∧ ⟨“𝑖𝑗”⟩ ∈ Word (Base‘𝑆)) → (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σ_g 𝑢)(+_g‘𝑆)(𝑆 Σ_g ⟨“𝑖𝑗”⟩)))
76	56, 73, 74, 75	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = ((𝑆 Σ_g 𝑢)(+_g‘𝑆)(𝑆 Σ_g ⟨“𝑖𝑗”⟩)))
77		cyc3genpm.t	. . . . . . . . . . . . . . . . . . . 20 ⊢ 𝐶 = (𝑀 “ (^◡♯ “ {3}))
78		cyc3genpm.m	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑀 = (toCyc‘𝐷)
79	78	imaeq1i 6055	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 “ (^◡♯ “ {3})) = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
80	77, 79	eqtri 2758	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
81	80, 9	cyc3evpm 32579	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
82	11, 12	evpmss 21358	. . . . . . . . . . . . . . . . . . 19 ⊢ (pmEven‘𝐷) ⊆ (Base‘𝑆)
83	9, 82	eqsstri 4015	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 ⊆ (Base‘𝑆)
84	81, 83	sstrdi 3993	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85		sswrd 14476	. . . . . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → Word 𝐶 ⊆ Word (Base‘𝑆))
87		simp-4r 780	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑣 ∈ Word 𝐶)
88	86, 87	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑣 ∈ Word (Base‘𝑆))
89		simplr 765	. . . . . . . . . . . . . . . 16 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑐 ∈ Word 𝐶)
90	86, 89	sseldd 3982	. . . . . . . . . . . . . . 15 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → 𝑐 ∈ Word (Base‘𝑆))
91	12, 64	gsumccat 18758	. . . . . . . . . . . . . . 15 ⊢ ((𝑆 ∈ Mnd ∧ 𝑣 ∈ Word (Base‘𝑆) ∧ 𝑐 ∈ Word (Base‘𝑆)) → (𝑆 Σ_g (𝑣 ++ 𝑐)) = ((𝑆 Σ_g 𝑣)(+_g‘𝑆)(𝑆 Σ_g 𝑐)))
92	56, 88, 90, 91	syl3anc 1369	. . . . . . . . . . . . . 14 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g (𝑣 ++ 𝑐)) = ((𝑆 Σ_g 𝑣)(+_g‘𝑆)(𝑆 Σ_g 𝑐)))
93	69, 76, 92	3eqtr4d 2780	. . . . . . . . . . . . 13 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g (𝑣 ++ 𝑐)))
94	47, 50, 93	rspcedvd 3613	. . . . . . . . . . . 12 ⊢ ((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑐 ∈ Word 𝐶) ∧ (𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤))
95		cyc3genpm.n	. . . . . . . . . . . . . . 15 ⊢ 𝑁 = (♯‘𝐷)
96		simp-6r 784	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑒 ∈ 𝐷)
97		simp-5r 782	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑓 ∈ 𝐷)
98		simpllr 772	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑔 ∈ 𝐷)
99		simplr 765	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → ℎ ∈ 𝐷)
100		simp-4r 780	. . . . . . . . . . . . . . . 16 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))
102		simprr 769	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 732	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) ∧ 𝑔 ∈ 𝐷) ∧ ℎ ∈ 𝐷) ∧ (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩))) → 𝑒 ≠ 𝑓)
105		simprl 767	. . . . . . . . . . . . . . 15 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 32580	. . . . . . . . . . . . . 14 ⊢ ((((((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 Σ_g 𝑐))
107		simp-6r 784	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝐷 ∈ Fin)
108		simp-7r 786	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → 𝑗 ∈ ran (pmTrsp‘𝐷))
109	16, 78	trsp2cyc 32552	. . . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
111	106, 110	r19.29vva 3211	. . . . . . . . . . . . 13 ⊢ (((((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) ∧ 𝑒 ∈ 𝐷) ∧ 𝑓 ∈ 𝐷) ∧ (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩))) → ∃𝑐 ∈ Word 𝐶(𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐))
112	16, 78	trsp2cyc 32552	. . . . . . . . . . . . . 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
114	111, 113	r19.29vva 3211	. . . . . . . . . . . 12 ⊢ ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → ∃𝑐 ∈ Word 𝐶(𝑖(+_g‘𝑆)𝑗) = (𝑆 Σ_g 𝑐))
115	94, 114	r19.29a 3160	. . . . . . . . . . 11 ⊢ ((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤))
116	115	adantl3r 746	. . . . . . . . . 10 ⊢ (((((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) ∧ 𝑣 ∈ Word 𝐶) ∧ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤))
117		simpr 483	. . . . . . . . . . . 12 ⊢ (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) → 𝐷 ∈ Fin)
118		simplr 765	. . . . . . . . . . . 12 ⊢ (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))
120		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤))
121	120	eqeq2d 2741	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣) ↔ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)))
122	121	cbvrexvw 3233	. . . . . . . . . . 11 ⊢ (∃𝑣 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑣 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣))
124	116, 123	r19.29a 3160	. . . . . . . . 9 ⊢ (((((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) ∧ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 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 Σ_g 𝑢) = (𝑆 Σ_g 𝑤))) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤)))
126	125	ex3 1344	. . . . . . 7 ⊢ ((𝑢 ∈ Word ran (pmTrsp‘𝐷) ∧ 𝑖 ∈ ran (pmTrsp‘𝐷) ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ((𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤))))
127	26, 30, 34, 38, 45, 126	wrdt2ind 32384	. . . . . 6 ⊢ ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
128	127	imp 405	. . . . 5 ⊢ (((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) ∧ 𝐷 ∈ Fin) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤))
129	1, 22, 7, 128	syl21anc 834	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤))
130	5	eqeq1d 2732	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (𝑄 = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
131	130	rexbidv 3176	. . . 4 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤) ↔ ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
132	129, 131	mpbird 256	. . 3 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤))
133	83	sseli 3977	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝑆))
134	11, 12, 16	psgnfitr 19426	. . . . 5 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣)))
135	134	biimpa 475	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣))
136	133, 135	sylan2 591	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣))
137	132, 136	r19.29a 3160	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤))
138		simpr 483	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 = (𝑆 Σ_g 𝑤))
139	11	altgnsg 32578	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
140	9, 139	eqeltrid 2835	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141		nsgsubg 19074	. . . . . . . 8 ⊢ (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142		subgsubm 19064	. . . . . . . 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 14476	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → Word 𝐶 ⊆ Word 𝐴)
146	81, 145	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147	146	sselda 3981	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148		gsumwsubmcl 18754	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
149	144, 147, 148	syl2anc 582	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
150	149	adantr 479	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
151	138, 150	eqeltrd 2831	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 ∈ 𝐴)
152	151	r19.29an 3156	. 2 ⊢ ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 ∈ 𝐴)
153	137, 152	impbida 797	1 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ 𝐴 ↔ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1539 ∈ wcel 2104 ≠ wne 2938 ∃wrex 3068 ⊆ wss 3947 ∅c0 4321 {csn 4627 class class class wbr 5147 ^◡ccnv 5674 ran crn 5676 “ cima 5678 ‘cfv 6542 (class class class)co 7411 Fincfn 8941 1c1 11113 -cneg 11449 2c2 12271 3c3 12272 ℕ₀cn0 12476 ℤcz 12562 ↑cexp 14031 ♯chash 14294 Word cword 14468 ++ cconcat 14524 ⟨“cs2 14796 ∥ cdvds 16201 Basecbs 17148 +_gcplusg 17201 Σ_g cgsu 17390 Mndcmnd 18659 SubMndcsubmnd 18704 Grpcgrp 18855 SubGrpcsubg 19036 NrmSGrpcnsg 19037 SymGrpcsymg 19275 pmTrspcpmtr 19350 pmSgncpsgn 19398 pmEvencevpm 19399 toCycctocyc 32535

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-reg 9589 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-xor 1508 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-card 9936 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-xnn0 12549 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12979 df-fz 13489 df-fzo 13632 df-fl 13761 df-mod 13839 df-seq 13971 df-exp 14032 df-hash 14295 df-word 14469 df-lsw 14517 df-concat 14525 df-s1 14550 df-substr 14595 df-pfx 14625 df-splice 14704 df-reverse 14713 df-csh 14743 df-s2 14803 df-s3 14804 df-dvds 16202 df-struct 17084 df-sets 17101 df-slot 17119 df-ndx 17131 df-base 17149 df-ress 17178 df-plusg 17214 df-mulr 17215 df-starv 17216 df-tset 17220 df-ple 17221 df-ds 17223 df-unif 17224 df-0g 17391 df-gsum 17392 df-mre 17534 df-mrc 17535 df-acs 17537 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18705 df-submnd 18706 df-efmnd 18786 df-grp 18858 df-minusg 18859 df-sbg 18860 df-subg 19039 df-nsg 19040 df-ghm 19128 df-gim 19173 df-oppg 19251 df-symg 19276 df-pmtr 19351 df-psgn 19400 df-evpm 19401 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-cring 20130 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20502 df-cnfld 21145 df-tocyc 32536

This theorem is referenced by: (None)

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