Theorem psgnunilem4 18377

Description: Lemma for psgnuni 18379. 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 13683	. . . . 5 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
4		hashcl 13525	. . . . 5 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ0)
5	1, 3, 4	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0)
6		nn0uz 12087	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
7	5, 6	syl6eleq 2870	. . 3 ⊢ (𝜑 → (♯‘𝑊) ∈ (ℤ≥‘0))
8		fveq2 6493	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
9		hash0 13536	. . . . . . . . 9 ⊢ (♯‘∅) = 0
10	8, 9	syl6eq 2824	. . . . . . . 8 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0)
11	10	oveq2d 6986	. . . . . . 7 ⊢ (𝑤 = ∅ → (-1↑(♯‘𝑤)) = (-1↑0))
12		neg1cn 11554	. . . . . . . 8 ⊢ -1 ∈ ℂ
13		exp0 13241	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
15	11, 14	syl6eq 2824	. . . . . 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 1171	. . . . . . . . . . . . . 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 1208	. . . . . . . . . . . . 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 2773	. . . . . . . . . . . . 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 13683	. . . . . . . . . . . . . . 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 1172	. . . . . . . . . . . . . 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 13535	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
28	27	biimpar 470	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (♯‘𝑤) ∈ ℕ)
29	25, 26, 28	syl2anc 576	. . . . . . . . . . . . 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 1209	. . . . . . . . . . . . 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 6502	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2)))
32		oveq2 6978	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
33	32	eqeq1d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
34	31, 33	anbi12d 621	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
35	34	cbvrexv 3378	. . . . . . . . . . . . . . . 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 312	. . . . . . . . . . . . . . 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 208	. . . . . . . . . . . . . 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 474	. . . . . . . . . . . . 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 18376	. . . . . . . . . . . 12 ⊢ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
40		iman 393	. . . . . . . . . . . 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 223	. . . . . . . . . . 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 3088	. . . . . . . . . . 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 210	. . . . . . . . . 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 758	. . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . 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 504	. . . . . . . . . . . . . . . 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 13683	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin)
48		hashcl 13525	. . . . . . . . . . . . . . . . . 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 1181	. . . . . . . . . . . . . . . . . . . 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 1117	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
53	51, 52, 28	syl2anc 576	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
54	53	adantr 473	. . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . 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 11448	. . . . . . . . . . . . . . . . . . 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 12202	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ+
58		ltsubrp 12235	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) < (♯‘𝑤))
59	56, 57, 58	sylancl 577	. . . . . . . . . . . . . . . . . 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 4945	. . . . . . . . . . . . . . . . 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 12886	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧ (♯‘𝑤) ∈ ℕ ∧ (♯‘𝑥) < (♯‘𝑤)))
62	49, 54, 60, 61	syl3anbrc 1323	. . . . . . . . . . . . . . . 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 6986	. . . . . . . . . . . . . . . . 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 11556	. . . . . . . . . . . . . . . . . . 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 11820	. . . . . . . . . . . . . . . . . . 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 11892	. . . . . . . . . . . . . . . . . 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 13329	. . . . . . . . . . . . . . . . 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 13367	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1↑2) = 1
75	74	oveq2i 6981	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1↑(♯‘𝑤)) / (-1↑2)) = ((-1↑(♯‘𝑤)) / 1)
76		m1expcl 13260	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ)
77	76	zcnd 11894	. . . . . . . . . . . . . . . . . . . 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 11201	. . . . . . . . . . . . . . . . . 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 2820	. . . . . . . . . . . . . . . . 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 2812	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 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 231	. . . . . . . . . . . . . 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 405	. . . . . . . . . . . . 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 1875	. . . . . . . . . . 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 1901	. . . . . . . . . . 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 243	. . . . . . . . . 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 1099	. . . . . . . 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 402	. . . . 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 3045	. . . 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 1111	. . 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 2847	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇))
97		oveq2 6978	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
98	97	eqeq1d 2774	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
99	96, 98	anbi12d 621	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
100		fveq2 6493	. . . . . 6 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
101	100	oveq2d 6986	. . . . 5 ⊢ (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑥)))
102	101	eqeq1d 2774	. . . 4 ⊢ (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑥)) = 1))
103	99, 102	imbi12d 337	. . 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 2847	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇))
105		oveq2 6978	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
106	105	eqeq1d 2774	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
107	104, 106	anbi12d 621	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
108		fveq2 6493	. . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
109	108	oveq2d 6986	. . . . 5 ⊢ (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊)))
110	109	eqeq1d 2774	. . . 4 ⊢ (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑊)) = 1))
111	107, 110	imbi12d 337	. . 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 13158	. 2 ⊢ (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑊)) = 1))
113	1, 2, 112	mp2and 686	1 ⊢ (𝜑 → (-1↑(♯‘𝑊)) = 1)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 387   ∧ w3a 1068  ∀wal 1505   = wceq 1507  ∃wex 1742   ∈ wcel 2048   ≠ wne 2961  ∃wrex 3083  ∅c0 4173   class class class wbr 4923   I cid 5304  ran crn 5401   ↾ cres 5402  ‘cfv 6182  (class class class)co 6970  Fincfn 8298  ℂcc 10325  ℝcr 10326  0cc0 10327  1c1 10328   < clt 10466   − cmin 10662  -cneg 10663   / cdiv 11090  ℕcn 11431  2c2 11488  ℕ₀cn0 11700  ℤcz 11786  ℤ_≥cuz 12051  ℝ⁺crp 12197  ...cfz 12701  ..^cfzo 12842  ↑cexp 13237  ♯chash 13498  Word cword 13662   Σ_g cgsu 16560  SymGrpcsymg 18256  pmTrspcpmtr 18320

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10383  ax-resscn 10384  ax-1cn 10385  ax-icn 10386  ax-addcl 10387  ax-addrcl 10388  ax-mulcl 10389  ax-mulrcl 10390  ax-mulcom 10391  ax-addass 10392  ax-mulass 10393  ax-distr 10394  ax-i2m1 10395  ax-1ne0 10396  ax-1rid 10397  ax-rnegex 10398  ax-rrecex 10399  ax-cnre 10400  ax-pre-lttri 10401  ax-pre-lttrn 10402  ax-pre-ltadd 10403  ax-pre-mulgt0 10404

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-xor 1489  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-pss 3841  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-ot 4444  df-uni 4707  df-int 4744  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5305  df-eprel 5310  df-po 5319  df-so 5320  df-fr 5359  df-se 5360  df-we 5361  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-isom 6191  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-om 7391  df-1st 7494  df-2nd 7495  df-wrecs 7743  df-recs 7805  df-rdg 7843  df-1o 7897  df-2o 7898  df-oadd 7901  df-er 8081  df-map 8200  df-en 8299  df-dom 8300  df-sdom 8301  df-fin 8302  df-card 9154  df-pnf 10468  df-mnf 10469  df-xr 10470  df-ltxr 10471  df-le 10472  df-sub 10664  df-neg 10665  df-div 11091  df-nn 11432  df-2 11496  df-3 11497  df-4 11498  df-5 11499  df-6 11500  df-7 11501  df-8 11502  df-9 11503  df-n0 11701  df-xnn0 11773  df-z 11787  df-uz 12052  df-rp 12198  df-fz 12702  df-fzo 12843  df-seq 13178  df-exp 13238  df-hash 13499  df-word 13663  df-lsw 13716  df-concat 13724  df-s1 13749  df-substr 13794  df-pfx 13843  df-splice 13950  df-s2 14062  df-struct 16331  df-ndx 16332  df-slot 16333  df-base 16335  df-sets 16336  df-ress 16337  df-plusg 16424  df-tset 16430  df-0g 16561  df-gsum 16562  df-mgm 17700  df-sgrp 17742  df-mnd 17753  df-submnd 17794  df-grp 17884  df-minusg 17885  df-subg 18050  df-symg 18257  df-pmtr 18321

This theorem is referenced by:  psgnuni  18379