Theorem psgnunilem4 17857

Description: Lemma for psgnuni 17859. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Hypotheses

Ref	Expression
psgnunilem4.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnunilem4.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem4.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
psgnunilem4.w1	⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
psgnunilem4.w2	⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))

Assertion

Ref	Expression
psgnunilem4	⊢ (𝜑 → (-1↑(#‘𝑊)) = 1)

Proof of Theorem psgnunilem4

Dummy variables 𝑥 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnunilem4.w1	. 2 ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
2		psgnunilem4.w2	. 2 ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
3		wrdfin 13278	. . . . 5 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
4		hashcl 13103	. . . . 5 ⊢ (𝑊 ∈ Fin → (#‘𝑊) ∈ ℕ0)
5	1, 3, 4	3syl 18	. . . 4 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ0)
6		nn0uz 11682	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
7	5, 6	syl6eleq 2708	. . 3 ⊢ (𝜑 → (#‘𝑊) ∈ (ℤ≥‘0))
8		fveq2 6158	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
9		hash0 13114	. . . . . . . . 9 ⊢ (#‘∅) = 0
10	8, 9	syl6eq 2671	. . . . . . . 8 ⊢ (𝑤 = ∅ → (#‘𝑤) = 0)
11	10	oveq2d 6631	. . . . . . 7 ⊢ (𝑤 = ∅ → (-1↑(#‘𝑤)) = (-1↑0))
12		neg1cn 11084	. . . . . . . 8 ⊢ -1 ∈ ℂ
13		exp0 12820	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
15	11, 14	syl6eq 2671	. . . . . 6 ⊢ (𝑤 = ∅ → (-1↑(#‘𝑤)) = 1)
16	15	2a1d 26	. . . . 5 ⊢ (𝑤 = ∅ → ((𝜑 ∧ ∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1)))
17		psgnunilem4.g	. . . . . . . . . . . . 13 ⊢ 𝐺 = (SymGrp‘𝐷)
18		psgnunilem4.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
19		simpl1 1062	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝜑)
20		psgnunilem4.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
21	19, 20	syl 17	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝐷 ∈ 𝑉)
22		simpl3l 1114	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
23		eqidd 2622	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (#‘𝑤) = (#‘𝑤))
24		wrdfin 13278	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Word 𝑇 → 𝑤 ∈ Fin)
25	22, 24	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
26		simpl2 1063	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
27		hashnncl 13113	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((#‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
28	27	biimpar 502	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (#‘𝑤) ∈ ℕ)
29	25, 26, 28	syl2anc 692	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (#‘𝑤) ∈ ℕ)
30		simpl3r 1115	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (𝐺 Σg 𝑤) = ( I ↾ 𝐷))
31		fveq2 6158	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
32	31	eqeq1d 2623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = ((#‘𝑤) − 2) ↔ (#‘𝑦) = ((#‘𝑤) − 2)))
33		oveq2 6623	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
34	33	eqeq1d 2623	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
35	32, 34	anbi12d 746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
36	35	cbvrexv 3164	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
37	36	notbii 310	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
38	37	biimpi 206	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
39	38	adantl 482	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
40	17, 18, 21, 22, 23, 29, 30, 39	psgnunilem3 17856	. . . . . . . . . . . 12 ⊢ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
41		iman 440	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) ↔ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
42	40, 41	mpbir 221	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
43		df-rex 2914	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
44	42, 43	sylib 208	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
45		simprl 793	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 𝑥 ∈ Word 𝑇)
46		simprrr 804	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
47	45, 46	jca 554	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
48		wrdfin 13278	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin)
49		hashcl 13103	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0)
50	45, 48, 49	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) ∈ ℕ0)
51		simp3l 1087	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
52	51, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
53		simp2 1060	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
54	52, 53, 28	syl2anc 692	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (#‘𝑤) ∈ ℕ)
55	54	adantr 481	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℕ)
56		simprrl 803	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) = ((#‘𝑤) − 2))
57	55	nnred 10995	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℝ)
58		2rp 11797	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ+
59		ltsubrp 11826	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((#‘𝑤) − 2) < (#‘𝑤))
60	57, 58, 59	sylancl 693	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((#‘𝑤) − 2) < (#‘𝑤))
61	56, 60	eqbrtrd 4645	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) < (#‘𝑤))
62		elfzo0 12465	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑥) ∈ (0..^(#‘𝑤)) ↔ ((#‘𝑥) ∈ ℕ0 ∧ (#‘𝑤) ∈ ℕ ∧ (#‘𝑥) < (#‘𝑤)))
63	50, 55, 61, 62	syl3anbrc 1244	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) ∈ (0..^(#‘𝑤)))
64		id 22	. . . . . . . . . . . . . . . . 17 ⊢ (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → ((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)))
65	64	com13 88	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ((#‘𝑥) ∈ (0..^(#‘𝑤)) → (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑥)) = 1)))
66	47, 63, 65	sylc 65	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑥)) = 1))
67	56	oveq2d 6631	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(#‘𝑥)) = (-1↑((#‘𝑤) − 2)))
68	12	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ∈ ℂ)
69		neg1ne0 11086	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
70	69	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ≠ 0)
71		2z 11369	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
72	71	a1i 11	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 2 ∈ ℤ)
73	55	nnzd 11441	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℤ)
74	68, 70, 72, 73	expsubd 12975	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑((#‘𝑤) − 2)) = ((-1↑(#‘𝑤)) / (-1↑2)))
75		neg1sqe1 12915	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1↑2) = 1
76	75	oveq2i 6626	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1↑(#‘𝑤)) / (-1↑2)) = ((-1↑(#‘𝑤)) / 1)
77		m1expcl 12839	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ ℤ)
78	77	zcnd 11443	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ ℂ)
79	73, 78	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(#‘𝑤)) ∈ ℂ)
80	79	div1d 10753	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(#‘𝑤)) / 1) = (-1↑(#‘𝑤)))
81	76, 80	syl5eq 2667	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(#‘𝑤)) / (-1↑2)) = (-1↑(#‘𝑤)))
82	67, 74, 81	3eqtrd 2659	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(#‘𝑥)) = (-1↑(#‘𝑤)))
83	82	eqeq1d 2623	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(#‘𝑥)) = 1 ↔ (-1↑(#‘𝑤)) = 1))
84	66, 83	sylibd 229	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑤)) = 1))
85	84	ex 450	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ((𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑤)) = 1)))
86	85	com23 86	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → ((𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(#‘𝑤)) = 1)))
87	86	alimdv 1842	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → ∀𝑥((𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(#‘𝑤)) = 1)))
88		19.23v 1899	. . . . . . . . . . 11 ⊢ (∀𝑥((𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(#‘𝑤)) = 1) ↔ (∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(#‘𝑤)) = 1))
89	87, 88	syl6ib 241	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(#‘𝑤)) = 1)))
90	44, 89	mpid 44	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑤)) = 1))
91	90	3exp 1261	. . . . . . . 8 ⊢ (𝜑 → (𝑤 ≠ ∅ → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → (-1↑(#‘𝑤)) = 1))))
92	91	com34 91	. . . . . . 7 ⊢ (𝜑 → (𝑤 ≠ ∅ → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1))))
93	92	com12 32	. . . . . 6 ⊢ (𝑤 ≠ ∅ → (𝜑 → (∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1))))
94	93	impd 447	. . . . 5 ⊢ (𝑤 ≠ ∅ → ((𝜑 ∧ ∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1)))
95	16, 94	pm2.61ine 2873	. . . 4 ⊢ ((𝜑 ∧ ∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1))
96	95	3adant2 1078	. . 3 ⊢ ((𝜑 ∧ (#‘𝑤) ∈ (0...(#‘𝑊)) ∧ ∀𝑥((#‘𝑥) ∈ (0..^(#‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1))
97		eleq1 2686	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇))
98		oveq2 6623	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
99	98	eqeq1d 2623	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
100	97, 99	anbi12d 746	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
101		fveq2 6158	. . . . . 6 ⊢ (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
102	101	oveq2d 6631	. . . . 5 ⊢ (𝑤 = 𝑥 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑥)))
103	102	eqeq1d 2623	. . . 4 ⊢ (𝑤 = 𝑥 → ((-1↑(#‘𝑤)) = 1 ↔ (-1↑(#‘𝑥)) = 1))
104	100, 103	imbi12d 334	. . 3 ⊢ (𝑤 = 𝑥 → (((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1) ↔ ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(#‘𝑥)) = 1)))
105		eleq1 2686	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇))
106		oveq2 6623	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
107	106	eqeq1d 2623	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
108	105, 107	anbi12d 746	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
109		fveq2 6158	. . . . . 6 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
110	109	oveq2d 6631	. . . . 5 ⊢ (𝑤 = 𝑊 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑊)))
111	110	eqeq1d 2623	. . . 4 ⊢ (𝑤 = 𝑊 → ((-1↑(#‘𝑤)) = 1 ↔ (-1↑(#‘𝑊)) = 1))
112	108, 111	imbi12d 334	. . 3 ⊢ (𝑤 = 𝑊 → (((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(#‘𝑤)) = 1) ↔ ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(#‘𝑊)) = 1)))
113	1, 7, 96, 104, 112, 101, 109	uzindi 12737	. 2 ⊢ (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(#‘𝑊)) = 1))
114	1, 2, 113	mp2and 714	1 ⊢ (𝜑 → (-1↑(#‘𝑊)) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1036  ∀wal 1478   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790  ∃wrex 2909  ∅c0 3897   class class class wbr 4623   I cid 4994  ran crn 5085   ↾ cres 5086  ‘cfv 5857  (class class class)co 6615  Fincfn 7915  ℂcc 9894  ℝcr 9895  0cc0 9896  1c1 9897   < clt 10034   − cmin 10226  -cneg 10227   / cdiv 10644  ℕcn 10980  2c2 11030  ℕ₀cn0 11252  ℤcz 11337  ℤ_≥cuz 11647  ℝ⁺crp 11792  ...cfz 12284  ..^cfzo 12422  ↑cexp 12816  #chash 13073  Word cword 13246   Σ_g cgsu 16041  SymGrpcsymg 17737  pmTrspcpmtr 17801

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-seq 12758  df-exp 12817  df-hash 13074  df-word 13254  df-lsw 13255  df-concat 13256  df-s1 13257  df-substr 13258  df-splice 13259  df-s2 13546  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-tset 15900  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-grp 17365  df-minusg 17366  df-subg 17531  df-symg 17738  df-pmtr 17802

This theorem is referenced by:  psgnuni  17859