Theorem psgnunilem4 19530

Description: Lemma for psgnuni 19532. 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 14567	. . . . 5 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
4		hashcl 14392	. . . . 5 ⊢ (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ0)
5	1, 3, 4	3syl 18	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ0)
6		nn0uz 12918	. . . 4 ⊢ ℕ0 = (ℤ≥‘0)
7	5, 6	eleqtrdi 2849	. . 3 ⊢ (𝜑 → (♯‘𝑊) ∈ (ℤ≥‘0))
8		fveq2 6907	. . . . . . . . 9 ⊢ (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
9		hash0 14403	. . . . . . . . 9 ⊢ (♯‘∅) = 0
10	8, 9	eqtrdi 2791	. . . . . . . 8 ⊢ (𝑤 = ∅ → (♯‘𝑤) = 0)
11	10	oveq2d 7447	. . . . . . 7 ⊢ (𝑤 = ∅ → (-1↑(♯‘𝑤)) = (-1↑0))
12		neg1cn 12378	. . . . . . . 8 ⊢ -1 ∈ ℂ
13		exp0 14103	. . . . . . . 8 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
14	12, 13	ax-mp 5	. . . . . . 7 ⊢ (-1↑0) = 1
15	11, 14	eqtrdi 2791	. . . . . 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 1190	. . . . . . . . . . . . . 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 1227	. . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 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 14567	. . . . . . . . . . . . . . 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 1191	. . . . . . . . . . . . . 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 14402	. . . . . . . . . . . . . . 15 ⊢ (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
28	27	biimpar 477	. . . . . . . . . . . . . 14 ⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (♯‘𝑤) ∈ ℕ)
29	25, 26, 28	syl2anc 584	. . . . . . . . . . . . 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 1228	. . . . . . . . . . . . 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 6916	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2)))
32		oveq2 7439	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
33	32	eqeq1d 2737	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
34	31, 33	anbi12d 632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
35	34	cbvrexvw 3236	. . . . . . . . . . . . . . . 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 320	. . . . . . . . . . . . . . 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 216	. . . . . . . . . . . . . 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 481	. . . . . . . . . . . . 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 19529	. . . . . . . . . . . 12 ⊢ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
40		iman 401	. . . . . . . . . . . 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 231	. . . . . . . . . . 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 3069	. . . . . . . . . . 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 218	. . . . . . . . . 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 511	. . . . . . . . . . . . . . . 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 14567	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin)
48		hashcl 14392	. . . . . . . . . . . . . . . . . 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 1200	. . . . . . . . . . . . . . . . . . . 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 1136	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
53	51, 52, 28	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
54	53	adantr 480	. . . . . . . . . . . . . . . . 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 12279	. . . . . . . . . . . . . . . . . . 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 13037	. . . . . . . . . . . . . . . . . . 19 ⊢ 2 ∈ ℝ+
58		ltsubrp 13069	. . . . . . . . . . . . . . . . . . 19 ⊢ (((♯‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) < (♯‘𝑤))
59	56, 57, 58	sylancl 586	. . . . . . . . . . . . . . . . . 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 5170	. . . . . . . . . . . . . . . . 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 13737	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧ (♯‘𝑤) ∈ ℕ ∧ (♯‘𝑥) < (♯‘𝑤)))
62	49, 54, 60, 61	syl3anbrc 1342	. . . . . . . . . . . . . . . 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 7447	. . . . . . . . . . . . . . . . 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 12380	. . . . . . . . . . . . . . . . . . 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 12647	. . . . . . . . . . . . . . . . . . 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 12638	. . . . . . . . . . . . . . . . . 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 14194	. . . . . . . . . . . . . . . . 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 14232	. . . . . . . . . . . . . . . . . . 19 ⊢ (-1↑2) = 1
75	74	oveq2i 7442	. . . . . . . . . . . . . . . . . 18 ⊢ ((-1↑(♯‘𝑤)) / (-1↑2)) = ((-1↑(♯‘𝑤)) / 1)
76		m1expcl 14124	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ)
77	76	zcnd 12721	. . . . . . . . . . . . . . . . . . . 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 12033	. . . . . . . . . . . . . . . . . 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	eqtrid 2787	. . . . . . . . . . . . . . . . 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 2779	. . . . . . . . . . . . . . . 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 2737	. . . . . . . . . . . . . . 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 239	. . . . . . . . . . . . . 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 412	. . . . . . . . . . . . 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 1914	. . . . . . . . . . 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 1940	. . . . . . . . . . 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	imbitrdi 251	. . . . . . . . . 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 1118	. . . . . . . 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 410	. . . . 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 3023	. . . 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 1130	. . 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 2827	. . . . 5 ⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇))
97		oveq2 7439	. . . . . 6 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
98	97	eqeq1d 2737	. . . . 5 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
99	96, 98	anbi12d 632	. . . 4 ⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
100		fveq2 6907	. . . . . 6 ⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
101	100	oveq2d 7447	. . . . 5 ⊢ (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑥)))
102	101	eqeq1d 2737	. . . 4 ⊢ (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑥)) = 1))
103	99, 102	imbi12d 344	. . 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 2827	. . . . 5 ⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇))
105		oveq2 7439	. . . . . 6 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
106	105	eqeq1d 2737	. . . . 5 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
107	104, 106	anbi12d 632	. . . 4 ⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
108		fveq2 6907	. . . . . 6 ⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
109	108	oveq2d 7447	. . . . 5 ⊢ (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊)))
110	109	eqeq1d 2737	. . . 4 ⊢ (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑊)) = 1))
111	107, 110	imbi12d 344	. . 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 14020	. 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 395   ∧ w3a 1086  ∀wal 1535   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∃wrex 3068  ∅c0 4339   class class class wbr 5148   I cid 5582  ran crn 5690   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431  Fincfn 8984  ℂcc 11151  ℝcr 11152  0cc0 11153  1c1 11154   < clt 11293   − cmin 11490  -cneg 11491   / cdiv 11918  ℕcn 12264  2c2 12319  ℕ₀cn0 12524  ℤcz 12611  ℤ_≥cuz 12876  ℝ⁺crp 13032  ...cfz 13544  ..^cfzo 13691  ↑cexp 14099  ♯chash 14366  Word cword 14549   Σ_g cgsu 17487  SymGrpcsymg 19401  pmTrspcpmtr 19474

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-xnn0 12598  df-z 12612  df-uz 12877  df-rp 13033  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-word 14550  df-lsw 14598  df-concat 14606  df-s1 14631  df-substr 14676  df-pfx 14706  df-splice 14785  df-s2 14884  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-tset 17317  df-0g 17488  df-gsum 17489  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-efmnd 18895  df-grp 18967  df-minusg 18968  df-subg 19154  df-symg 19402  df-pmtr 19475

This theorem is referenced by:  psgnuni  19532