Theorem psgnunilem4 18889

Description: Lemma for psgnuni 18891. 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 14087	. . . . 5 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
4		hashcl 13923	. . . . 5 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ0)
5	1, 3, 4	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0)
6		nn0uz 12476	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
7	5, 6	eleqtrdi 2848	. . 3 ⊢ (𝜑 → (♯‘𝑊) ∈ (ℤ≥‘0))
8		fveq2 6717	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
9		hash0 13934	. . . . . . . . 9 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2794	. . . . . . . 8 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0)
11	10	oveq2d 7229	. . . . . . 7 ⊢ (𝑤 = ∅ → (-1↑(♯‘𝑤)) = (-1↑0))
12		neg1cn 11944	. . . . . . . 8 ⊢ -1 ∈ ℂ
13		exp0 13639	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
15	11, 14	eqtrdi 2794	. . . . . 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 1193	. . . . . . . . . . . . . 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 1230	. . . . . . . . . . . . 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 2738	. . . . . . . . . . . . 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 14087	. . . . . . . . . . . . . . 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 1194	. . . . . . . . . . . . . 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 13933	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
28	27	biimpar 481	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (♯‘𝑤) ∈ ℕ)
29	25, 26, 28	syl2anc 587	. . . . . . . . . . . . 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 1231	. . . . . . . . . . . . 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		fveqeq2 6726	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2)))
32		oveq2 7221	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
33	32	eqeq1d 2739	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
34	31, 33	anbi12d 634	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
35	34	cbvrexvw 3359	. . . . . . . . . . . . . . . 16 ⊢ (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
36	35	notbii 323	. . . . . . . . . . . . . . 15 ⊢ (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
37	36	biimpi 219	. . . . . . . . . . . . . 14 ⊢ (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
38	37	adantl 485	. . . . . . . . . . . . 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)))
39	17, 18, 21, 22, 23, 29, 30, 38	psgnunilem3 18888	. . . . . . . . . . . 12 ⊢ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
40		iman 405	. . . . . . . . . . . 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))))
41	39, 40	mpbir 234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
42		df-rex 3067	. . . . . . . . . . 11 ⊢ (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
43	41, 42	sylib 221	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
44		simprl 771	. . . . . . . . . . . . . . . . 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)
45		simprrr 782	. . . . . . . . . . . . . . . . 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))
46	44, 45	jca 515	. . . . . . . . . . . . . . . 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)))
47		wrdfin 14087	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin)
48		hashcl 13923	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0)
49	44, 47, 48	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)
50		simp3l 1203	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
51	50, 24	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
52		simp2 1139	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
53	51, 52, 28	syl2anc 587	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
54	53	adantr 484	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℕ)
55		simprrl 781	. . . . . . . . . . . . . . . . . 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))
56	54	nnred 11845	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℝ)
57		2rp 12591	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ+
58		ltsubrp 12622	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) < (♯‘𝑤))
59	56, 57, 58	sylancl 589	. . . . . . . . . . . . . . . . . 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))
60	55, 59	eqbrtrd 5075	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) < (♯‘𝑤))
61		elfzo0 13283	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧ (♯‘𝑤) ∈ ℕ ∧ (♯‘𝑥) < (♯‘𝑤)))
62	49, 54, 60, 61	syl3anbrc 1345	. . . . . . . . . . . . . . . 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)))
63		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)))
64	63	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)))
65	46, 62, 64	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))
66	55	oveq2d 7229	. . . . . . . . . . . . . . . . 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)))
67	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)
68		neg1ne0 11946	. . . . . . . . . . . . . . . . . . 19 ⊢ -1 ≠ 0
69	68	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)
70		2z 12209	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℤ
71	70	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)
72	54	nnzd 12281	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℤ)
73	67, 69, 71, 72	expsubd 13727	. . . . . . . . . . . . . . . . 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)))
74		neg1sqe1 13765	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1↑2) = 1
75	74	oveq2i 7224	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1↑(♯‘𝑤)) / (-1↑2)) = ((-1↑(♯‘𝑤)) / 1)
76		m1expcl 13658	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ)
77	76	zcnd 12283	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℂ)
78	72, 77	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)
79	78	div1d 11600	. . . . . . . . . . . . . . . . . 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)))
80	75, 79	syl5eq 2790	. . . . . . . . . . . . . . . . 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)))
81	66, 73, 80	3eqtrd 2781	. . . . . . . . . . . . . . . 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)))
82	81	eqeq1d 2739	. . . . . . . . . . . . . . 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))
83	65, 82	sylibd 242	. . . . . . . . . . . . . 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))
84	83	ex 416	. . . . . . . . . . . . 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)))
85	84	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)))
86	85	alimdv 1924	. . . . . . . . . . 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)))
87		19.23v 1950	. . . . . . . . . . 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))
88	86, 87	syl6ib 254	. . . . . . . . . 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)))
89	43, 88	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))
90	89	3exp 1121	. . . . . . . 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))))
91	90	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))))
92	91	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))))
93	92	impd 414	. . . . 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)))
94	16, 93	pm2.61ine 3025	. . . 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))
95	94	3adant2 1133	. . 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))
96		eleq1 2825	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇))
97		oveq2 7221	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
98	97	eqeq1d 2739	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
99	96, 98	anbi12d 634	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
100		fveq2 6717	. . . . . 6 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
101	100	oveq2d 7229	. . . . 5 ⊢ (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑥)))
102	101	eqeq1d 2739	. . . 4 ⊢ (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑥)) = 1))
103	99, 102	imbi12d 348	. . 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)))
104		eleq1 2825	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇))
105		oveq2 7221	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
106	105	eqeq1d 2739	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
107	104, 106	anbi12d 634	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
108		fveq2 6717	. . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
109	108	oveq2d 7229	. . . . 5 ⊢ (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊)))
110	109	eqeq1d 2739	. . . 4 ⊢ (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑊)) = 1))
111	107, 110	imbi12d 348	. . 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)))
112	1, 7, 95, 103, 111, 100, 108	uzindi 13555	. 2 ⊢ (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑊)) = 1))
113	1, 2, 112	mp2and 699	1 ⊢ (𝜑 → (-1↑(♯‘𝑊)) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2110   ≠ wne 2940  ∃wrex 3062  ∅c0 4237   class class class wbr 5053   I cid 5454  ran crn 5552   ↾ cres 5553  ‘cfv 6380  (class class class)co 7213  Fincfn 8626  ℂcc 10727  ℝcr 10728  0cc0 10729  1c1 10730   < clt 10867   − cmin 11062  -cneg 11063   / cdiv 11489  ℕcn 11830  2c2 11885  ℕ₀cn0 12090  ℤcz 12176  ℤ_≥cuz 12438  ℝ⁺crp 12586  ...cfz 13095  ..^cfzo 13238  ↑cexp 13635  ♯chash 13896  Word cword 14069   Σ_g cgsu 16945  SymGrpcsymg 18759  pmTrspcpmtr 18833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-se 5510  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-isom 6389  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-div 11490  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-rp 12587  df-fz 13096  df-fzo 13239  df-seq 13575  df-exp 13636  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-splice 14315  df-s2 14413  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-tset 16821  df-0g 16946  df-gsum 16947  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-efmnd 18296  df-grp 18368  df-minusg 18369  df-subg 18540  df-symg 18760  df-pmtr 18834

This theorem is referenced by:  psgnuni  18891