Theorem psgnunilem4 17917

Description: Lemma for psgnuni 17919. 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 13323	. . . . 5 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
4		hashcl 13147	. . . . 5 ⊢ (𝑊 ∈ Fin → (#‘𝑊) ∈ ℕ0)
5	1, 3, 4	3syl 18	. . . 4 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ0)
6		nn0uz 11722	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
7	5, 6	syl6eleq 2711	. . 3 ⊢ (𝜑 → (#‘𝑊) ∈ (ℤ≥‘0))
8		fveq2 6191	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
9		hash0 13158	. . . . . . . . 9 ⊢ (#‘∅) = 0
10	8, 9	syl6eq 2672	. . . . . . . 8 ⊢ (𝑤 = ∅ → (#‘𝑤) = 0)
11	10	oveq2d 6666	. . . . . . 7 ⊢ (𝑤 = ∅ → (-1↑(#‘𝑤)) = (-1↑0))
12		neg1cn 11124	. . . . . . . 8 ⊢ -1 ∈ ℂ
13		exp0 12864	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
15	11, 14	syl6eq 2672	. . . . . 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 1064	. . . . . . . . . . . . . 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 1116	. . . . . . . . . . . . 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 2623	. . . . . . . . . . . . 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 13323	. . . . . . . . . . . . . . 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 1065	. . . . . . . . . . . . . 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 13157	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((#‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
28	27	biimpar 502	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (#‘𝑤) ∈ ℕ)
29	25, 26, 28	syl2anc 693	. . . . . . . . . . . . 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 1117	. . . . . . . . . . . . 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 6191	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
32	31	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((#‘𝑥) = ((#‘𝑤) − 2) ↔ (#‘𝑦) = ((#‘𝑤) − 2)))
33		oveq2 6658	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
34	33	eqeq1d 2624	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
35	32, 34	anbi12d 747	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
36	35	cbvrexv 3172	. . . . . . . . . . . . . . . 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 17916	. . . . . . . . . . . 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 2918	. . . . . . . . . . 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 794	. . . . . . . . . . . . . . . . 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 805	. . . . . . . . . . . . . . . . 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 13323	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin)
49		hashcl 13147	. . . . . . . . . . . . . . . . . 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 1089	. . . . . . . . . . . . . . . . . . . 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 1062	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
54	52, 53, 28	syl2anc 693	. . . . . . . . . . . . . . . . . 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 804	. . . . . . . . . . . . . . . . . 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 11035	. . . . . . . . . . . . . . . . . . 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 11837	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ+
59		ltsubrp 11866	. . . . . . . . . . . . . . . . . . 19 ⊢ (((#‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((#‘𝑤) − 2) < (#‘𝑤))
60	57, 58, 59	sylancl 694	. . . . . . . . . . . . . . . . . 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 4675	. . . . . . . . . . . . . . . . 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 12508	. . . . . . . . . . . . . . . . 17 ⊢ ((#‘𝑥) ∈ (0..^(#‘𝑤)) ↔ ((#‘𝑥) ∈ ℕ0 ∧ (#‘𝑤) ∈ ℕ ∧ (#‘𝑥) < (#‘𝑤)))
63	50, 55, 61, 62	syl3anbrc 1246	. . . . . . . . . . . . . . . 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 6666	. . . . . . . . . . . . . . . . 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 11126	. . . . . . . . . . . . . . . . . . 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 11409	. . . . . . . . . . . . . . . . . . 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 11481	. . . . . . . . . . . . . . . . . 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 13019	. . . . . . . . . . . . . . . . 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 12959	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1↑2) = 1
76	75	oveq2i 6661	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1↑(#‘𝑤)) / (-1↑2)) = ((-1↑(#‘𝑤)) / 1)
77		m1expcl 12883	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ ℤ)
78	77	zcnd 11483	. . . . . . . . . . . . . . . . . . . 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 10793	. . . . . . . . . . . . . . . . . 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 2668	. . . . . . . . . . . . . . . . 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 2660	. . . . . . . . . . . . . . . 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 2624	. . . . . . . . . . . . . . 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 1845	. . . . . . . . . . 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 1902	. . . . . . . . . . 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 1264	. . . . . . . 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 2877	. . . 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 1080	. . 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 2689	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇))
98		oveq2 6658	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
99	98	eqeq1d 2624	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
100	97, 99	anbi12d 747	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
101		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
102	101	oveq2d 6666	. . . . 5 ⊢ (𝑤 = 𝑥 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑥)))
103	102	eqeq1d 2624	. . . 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 2689	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇))
106		oveq2 6658	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
107	106	eqeq1d 2624	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
108	105, 107	anbi12d 747	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
109		fveq2 6191	. . . . . 6 ⊢ (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
110	109	oveq2d 6666	. . . . 5 ⊢ (𝑤 = 𝑊 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑊)))
111	110	eqeq1d 2624	. . . 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 12781	. 2 ⊢ (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(#‘𝑊)) = 1))
114	1, 2, 113	mp2and 715	1 ⊢ (𝜑 → (-1↑(#‘𝑊)) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  ∅c0 3915   class class class wbr 4653   I cid 5023  ran crn 5115   ↾ cres 5116  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  ℂcc 9934  ℝcr 9935  0cc0 9936  1c1 9937   < clt 10074   − cmin 10266  -cneg 10267   / cdiv 10684  ℕcn 11020  2c2 11070  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ℝ⁺crp 11832  ...cfz 12326  ..^cfzo 12465  ↑cexp 12860  #chash 13117  Word cword 13291   Σ_g cgsu 16101  SymGrpcsymg 17797  pmTrspcpmtr 17861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-xor 1465  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-ot 4186  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  df-splice 13304  df-s2 13593  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-tset 15960  df-0g 16102  df-gsum 16103  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-grp 17425  df-minusg 17426  df-subg 17591  df-symg 17798  df-pmtr 17862

This theorem is referenced by:  psgnuni  17919