cyc3genpm - Mathbox for Thierry Arnoux

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Theorem cyc3genpm 32578

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 766	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑣 ∈ Word ran (pmTrsp‘𝐷))
2		lencl 14488	. . . . . . . 8 ⊢ (𝑣 ∈ Word ran (pmTrsp‘𝐷) → (♯‘𝑣) ∈ ℕ₀)
3	2	ad2antlr 724	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (♯‘𝑣) ∈ ℕ₀)
4	3	nn0zd 12589	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (♯‘𝑣) ∈ ℤ)
5		simpr 484	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝑄 = (𝑆 Σ_g 𝑣))
6	5	fveq2d 6896	. . . . . . 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 772	. . . . . . . 8 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 𝐷 ∈ Fin)
8		simpllr 773	. . . . . . . . 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 2842	. . . . . . . 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 2731	. . . . . . . . 9 ⊢ (Base‘𝑆) = (Base‘𝑆)
13		eqid 2731	. . . . . . . . 9 ⊢ (pmSgn‘𝐷) = (pmSgn‘𝐷)
14	11, 12, 13	psgnevpm 21362	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (pmEven‘𝐷)) → ((pmSgn‘𝐷)‘𝑄) = 1)
15	7, 10, 14	syl2anc 583	. . . . . . 7 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → ((pmSgn‘𝐷)‘𝑄) = 1)
16		eqid 2731	. . . . . . . . 9 ⊢ ran (pmTrsp‘𝐷) = ran (pmTrsp‘𝐷)
17	11, 16, 13	psgnvalii 19419	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) → ((pmSgn‘𝐷)‘(𝑆 Σ_g 𝑣)) = (-1↑(♯‘𝑣)))
18	7, 1, 17	syl2anc 583	. . . . . . 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 2780	. . . . . 6 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → (-1↑(♯‘𝑣)) = 1)
20		m1exp1 16324	. . . . . . 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 583	. . . . 5 ⊢ ((((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) ∧ 𝑣 ∈ Word ran (pmTrsp‘𝐷)) ∧ 𝑄 = (𝑆 Σ_g 𝑣)) → 2 ∥ (♯‘𝑣))
23		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g ∅))
24	23	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑥 = ∅ → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤)))
25	24	rexbidv 3177	. . . . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑥 = 𝑢 → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑢))
28	27	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑥 = 𝑢 → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)))
29	28	rexbidv 3177	. . . . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)))
32	31	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑥 = (𝑢 ++ ⟨“𝑖𝑗”⟩) → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g (𝑢 ++ ⟨“𝑖𝑗”⟩)) = (𝑆 Σ_g 𝑤)))
33	32	rexbidv 3177	. . . . . . . 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 7420	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑣))
36	35	eqeq1d 2733	. . . . . . . . 9 ⊢ (𝑥 = 𝑣 → ((𝑆 Σ_g 𝑥) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
37	36	rexbidv 3177	. . . . . . . 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 14494	. . . . . . . . 9 ⊢ ∅ ∈ Word 𝐶
40	39	a1i 11	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → ∅ ∈ Word 𝐶)
41		simpr 484	. . . . . . . . . 10 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → 𝑤 = ∅)
42	41	oveq2d 7428	. . . . . . . . 9 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → (𝑆 Σ_g 𝑤) = (𝑆 Σ_g ∅))
43	42	eqeq2d 2742	. . . . . . . 8 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 = ∅) → ((𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤) ↔ (𝑆 Σ_g ∅) = (𝑆 Σ_g ∅)))
44		eqidd 2732	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → (𝑆 Σ_g ∅) = (𝑆 Σ_g ∅))
45	40, 43, 44	rspcedvd 3615	. . . . . . 7 ⊢ (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g ∅) = (𝑆 Σ_g 𝑤))
46		ccatcl 14529	. . . . . . . . . . . . . 14 ⊢ ((𝑣 ∈ Word 𝐶 ∧ 𝑐 ∈ Word 𝐶) → (𝑣 ++ 𝑐) ∈ Word 𝐶)
47	46	ad5ant24 758	. . . . . . . . . . . . 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 7420	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = (𝑣 ++ 𝑐) → (𝑆 Σ_g 𝑤) = (𝑆 Σ_g (𝑣 ++ 𝑐)))
49	48	eqeq2d 2742	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 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 19310	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝑆 ∈ Grp)
55		grpmnd 18863	. . . . . . . . . . . . . . . . . 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 19379	. . . . . . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . . . . . 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 723	. . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . . . 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 2731	. . . . . . . . . . . . . . . . . 18 ⊢ (+_g‘𝑆) = (+_g‘𝑆)
65	12, 64	gsumws2 18760	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ Mnd ∧ 𝑖 ∈ (Base‘𝑆) ∧ 𝑗 ∈ (Base‘𝑆)) → (𝑆 Σ_g ⟨“𝑖𝑗”⟩) = (𝑖(+_g‘𝑆)𝑗))
66	56, 61, 63, 65	syl3anc 1370	. . . . . . . . . . . . . . . 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 484	. . . . . . . . . . . . . . . 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 2771	. . . . . . . . . . . . . . 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 7430	. . . . . . . . . . . . . 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 14477	. . . . . . . . . . . . . . . . 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 786	. . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . 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 14827	. . . . . . . . . . . . . . 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 18759	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 6057	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑀 “ (^◡♯ “ {3})) = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
80	77, 79	eqtri 2759	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝐶 = ((toCyc‘𝐷) “ (^◡♯ “ {3}))
81	80, 9	cyc3evpm 32576	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ 𝐴)
82	11, 12	evpmss 21359	. . . . . . . . . . . . . . . . . . 19 ⊢ (pmEven‘𝐷) ⊆ (Base‘𝑆)
83	9, 82	eqsstri 4017	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐴 ⊆ (Base‘𝑆)
84	81, 83	sstrdi 3995	. . . . . . . . . . . . . . . . 17 ⊢ (𝐷 ∈ Fin → 𝐶 ⊆ (Base‘𝑆))
85		sswrd 14477	. . . . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . . . 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 18759	. . . . . . . . . . . . . . 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 1370	. . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . 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 3615	. . . . . . . . . . . 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 785	. . . . . . . . . . . . . . 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 783	. . . . . . . . . . . . . . 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 773	. . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . 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 781	. . . . . . . . . . . . . . . 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 495	. . . . . . . . . . . . . . 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 770	. . . . . . . . . . . . . . 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 733	. . . . . . . . . . . . . . 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 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”⟩))) → 𝑒 ≠ 𝑓)
105		simprl 768	. . . . . . . . . . . . . . 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 32577	. . . . . . . . . . . . . 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 785	. . . . . . . . . . . . . . 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 787	. . . . . . . . . . . . . . 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 32549	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ Fin ∧ 𝑗 ∈ ran (pmTrsp‘𝐷)) → ∃𝑔 ∈ 𝐷 ∃ℎ ∈ 𝐷 (𝑔 ≠ ℎ ∧ 𝑗 = (𝑀‘⟨“𝑔ℎ”⟩)))
110	107, 108, 109	syl2anc 583	. . . . . . . . . . . . . 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 3212	. . . . . . . . . . . . 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 32549	. . . . . . . . . . . . . 14 ⊢ ((𝐷 ∈ Fin ∧ 𝑖 ∈ ran (pmTrsp‘𝐷)) → ∃𝑒 ∈ 𝐷 ∃𝑓 ∈ 𝐷 (𝑒 ≠ 𝑓 ∧ 𝑖 = (𝑀‘⟨“𝑒𝑓”⟩)))
113	52, 59, 112	syl2anc 583	. . . . . . . . . . . . 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 3212	. . . . . . . . . . . 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 3161	. . . . . . . . . . 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 747	. . . . . . . . . 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 484	. . . . . . . . . . . 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 766	. . . . . . . . . . . 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 7420	. . . . . . . . . . . . 13 ⊢ (𝑣 = 𝑤 → (𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤))
121	120	eqeq2d 2742	. . . . . . . . . . . 12 ⊢ (𝑣 = 𝑤 → ((𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑣) ↔ (𝑆 Σ_g 𝑢) = (𝑆 Σ_g 𝑤)))
122	121	cbvrexvw 3234	. . . . . . . . . . 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 3161	. . . . . . . . 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 412	. . . . . . . 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 1345	. . . . . . 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 32381	. . . . . 6 ⊢ ((𝑣 ∈ Word ran (pmTrsp‘𝐷) ∧ 2 ∥ (♯‘𝑣)) → (𝐷 ∈ Fin → ∃𝑤 ∈ Word 𝐶(𝑆 Σ_g 𝑣) = (𝑆 Σ_g 𝑤)))
128	127	imp 406	. . . . 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 835	. . . 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 2733	. . . . 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 3177	. . . 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 3979	. . . 4 ⊢ (𝑄 ∈ 𝐴 → 𝑄 ∈ (Base‘𝑆))
134	11, 12, 16	psgnfitr 19427	. . . . 5 ⊢ (𝐷 ∈ Fin → (𝑄 ∈ (Base‘𝑆) ↔ ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣)))
135	134	biimpa 476	. . . 4 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ (Base‘𝑆)) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣))
136	133, 135	sylan2 592	. . 3 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑣 ∈ Word ran (pmTrsp‘𝐷)𝑄 = (𝑆 Σ_g 𝑣))
137	132, 136	r19.29a 3161	. 2 ⊢ ((𝐷 ∈ Fin ∧ 𝑄 ∈ 𝐴) → ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤))
138		simpr 484	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 = (𝑆 Σ_g 𝑤))
139	11	altgnsg 32575	. . . . . . . . 9 ⊢ (𝐷 ∈ Fin → (pmEven‘𝐷) ∈ (NrmSGrp‘𝑆))
140	9, 139	eqeltrid 2836	. . . . . . . 8 ⊢ (𝐷 ∈ Fin → 𝐴 ∈ (NrmSGrp‘𝑆))
141		nsgsubg 19075	. . . . . . . 8 ⊢ (𝐴 ∈ (NrmSGrp‘𝑆) → 𝐴 ∈ (SubGrp‘𝑆))
142		subgsubm 19065	. . . . . . . 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 14477	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐴 → Word 𝐶 ⊆ Word 𝐴)
146	81, 145	syl 17	. . . . . . 7 ⊢ (𝐷 ∈ Fin → Word 𝐶 ⊆ Word 𝐴)
147	146	sselda 3983	. . . . . 6 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → 𝑤 ∈ Word 𝐴)
148		gsumwsubmcl 18755	. . . . . 6 ⊢ ((𝐴 ∈ (SubMnd‘𝑆) ∧ 𝑤 ∈ Word 𝐴) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
149	144, 147, 148	syl2anc 583	. . . . 5 ⊢ ((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
150	149	adantr 480	. . . 4 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → (𝑆 Σ_g 𝑤) ∈ 𝐴)
151	138, 150	eqeltrd 2832	. . 3 ⊢ (((𝐷 ∈ Fin ∧ 𝑤 ∈ Word 𝐶) ∧ 𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 ∈ 𝐴)
152	151	r19.29an 3157	. 2 ⊢ ((𝐷 ∈ Fin ∧ ∃𝑤 ∈ Word 𝐶𝑄 = (𝑆 Σ_g 𝑤)) → 𝑄 ∈ 𝐴)
153	137, 152	impbida 798	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 395 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∃wrex 3069 ⊆ wss 3949 ∅c0 4323 {csn 4629 class class class wbr 5149 ^◡ccnv 5676 ran crn 5678 “ cima 5680 ‘cfv 6544 (class class class)co 7412 Fincfn 8942 1c1 11114 -cneg 11450 2c2 12272 3c3 12273 ℕ₀cn0 12477 ℤcz 12563 ↑cexp 14032 ♯chash 14295 Word cword 14469 ++ cconcat 14525 ⟨“cs2 14797 ∥ cdvds 16202 Basecbs 17149 +_gcplusg 17202 Σ_g cgsu 17391 Mndcmnd 18660 SubMndcsubmnd 18705 Grpcgrp 18856 SubGrpcsubg 19037 NrmSGrpcnsg 19038 SymGrpcsymg 19276 pmTrspcpmtr 19351 pmSgncpsgn 19399 pmEvencevpm 19400 toCycctocyc 32532

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-reg 9590 ax-ac2 10461 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 ax-mulf 11193

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-ot 4638 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7859 df-1st 7978 df-2nd 7979 df-tpos 8214 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-map 8825 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-sup 9440 df-inf 9441 df-card 9937 df-ac 10114 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-dec 12683 df-uz 12828 df-rp 12980 df-fz 13490 df-fzo 13633 df-fl 13762 df-mod 13840 df-seq 13972 df-exp 14033 df-hash 14296 df-word 14470 df-lsw 14518 df-concat 14526 df-s1 14551 df-substr 14596 df-pfx 14626 df-splice 14705 df-reverse 14714 df-csh 14744 df-s2 14804 df-s3 14805 df-dvds 16203 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-0g 17392 df-gsum 17393 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-mhm 18706 df-submnd 18707 df-efmnd 18787 df-grp 18859 df-minusg 18860 df-sbg 18861 df-subg 19040 df-nsg 19041 df-ghm 19129 df-gim 19174 df-oppg 19252 df-symg 19277 df-pmtr 19352 df-psgn 19401 df-evpm 19402 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-cring 20131 df-oppr 20226 df-dvdsr 20249 df-unit 20250 df-invr 20280 df-dvr 20293 df-drng 20503 df-cnfld 21146 df-tocyc 32533

This theorem is referenced by: (None)

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