Theorem psgnunilem5 17914

Description: Lemma for psgnuni 17919. It is impossible to shift a transposition off the end because if the active transposition is at the right end, it is the only transposition moving 𝐴 in contradiction to this being a representation of the identity. (Contributed by Stefan O'Rear, 25-Aug-2015.) (Revised by Mario Carneiro, 28-Feb-2016.)

Hypotheses

Ref	Expression
psgnunilem2.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnunilem2.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem2.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
psgnunilem2.w	⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
psgnunilem2.id	⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
psgnunilem2.l	⊢ (𝜑 → (#‘𝑊) = 𝐿)
psgnunilem2.ix	⊢ (𝜑 → 𝐼 ∈ (0..^𝐿))
psgnunilem2.a	⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
psgnunilem2.al	⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))

Assertion

Ref	Expression
psgnunilem5	⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Distinct variable groups:   𝐴,𝑘   𝑘,𝐺   𝑘,𝐼   𝑘,𝑊

Allowed substitution hints:   𝜑(𝑘)   𝐷(𝑘)   𝑇(𝑘)   𝐿(𝑘)   𝑉(𝑘)

Proof of Theorem psgnunilem5

Dummy variables 𝑗 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3919	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
2		psgnunilem2.id	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
3	2	difeq1d 3727	. . . . . . 7 ⊢ (𝜑 → ((𝐺 Σg 𝑊) ∖ I ) = (( I ↾ 𝐷) ∖ I ))
4	3	dmeqd 5326	. . . . . 6 ⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
5		resss 5422	. . . . . . . . 9 ⊢ ( I ↾ 𝐷) ⊆ I
6		ssdif0 3942	. . . . . . . . 9 ⊢ (( I ↾ 𝐷) ⊆ I ↔ (( I ↾ 𝐷) ∖ I ) = ∅)
7	5, 6	mpbi 220	. . . . . . . 8 ⊢ (( I ↾ 𝐷) ∖ I ) = ∅
8	7	dmeqi 5325	. . . . . . 7 ⊢ dom (( I ↾ 𝐷) ∖ I ) = dom ∅
9		dm0 5339	. . . . . . 7 ⊢ dom ∅ = ∅
10	8, 9	eqtri 2644	. . . . . 6 ⊢ dom (( I ↾ 𝐷) ∖ I ) = ∅
11	4, 10	syl6eq 2672	. . . . 5 ⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = ∅)
12	11	eleq2d 2687	. . . 4 ⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ) ↔ 𝐴 ∈ ∅))
13	1, 12	mtbiri 317	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ))
14		psgnunilem2.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
15		psgnunilem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
16	15	symggrp 17820	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp)
17		grpmnd 17429	. . . . . . . . 9 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
18	14, 16, 17	3syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ Mnd)
19		psgnunilem2.t	. . . . . . . . . . . 12 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
20		eqid 2622	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	19, 15, 20	symgtrf 17889	. . . . . . . . . . 11 ⊢ 𝑇 ⊆ (Base‘𝐺)
22		sswrd 13313	. . . . . . . . . . 11 ⊢ (𝑇 ⊆ (Base‘𝐺) → Word 𝑇 ⊆ Word (Base‘𝐺))
23	21, 22	mp1i 13	. . . . . . . . . 10 ⊢ (𝜑 → Word 𝑇 ⊆ Word (Base‘𝐺))
24		psgnunilem2.w	. . . . . . . . . 10 ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
25	23, 24	sseldd 3604	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺))
26		swrdcl 13419	. . . . . . . . 9 ⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 substr ⟨0, 𝐼⟩) ∈ Word (Base‘𝐺))
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩) ∈ Word (Base‘𝐺))
28	20	gsumwcl 17377	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑊 substr ⟨0, 𝐼⟩) ∈ Word (Base‘𝐺)) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺))
29	18, 27, 28	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺))
30	15, 20	symgbasf1o 17803	. . . . . . 7 ⊢ ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)):𝐷–1-1-onto→𝐷)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)):𝐷–1-1-onto→𝐷)
32	31	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)):𝐷–1-1-onto→𝐷)
33		wrdf 13310	. . . . . . . . . 10 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(#‘𝑊))⟶𝑇)
34	24, 33	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑊:(0..^(#‘𝑊))⟶𝑇)
35		psgnunilem2.ix	. . . . . . . . . 10 ⊢ (𝜑 → 𝐼 ∈ (0..^𝐿))
36		psgnunilem2.l	. . . . . . . . . . 11 ⊢ (𝜑 → (#‘𝑊) = 𝐿)
37	36	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(#‘𝑊)) = (0..^𝐿))
38	35, 37	eleqtrrd 2704	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^(#‘𝑊)))
39	34, 38	ffvelrnd 6360	. . . . . . . 8 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇)
40	21, 39	sseldi 3601	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺))
41	15, 20	symgbasf1o 17803	. . . . . . 7 ⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷)
42	40, 41	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑊‘𝐼):𝐷–1-1-onto→𝐷)
43	42	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷)
44	15, 20	symgsssg 17887	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺))
45		subgsubm 17616	. . . . . . . . . . . 12 ⊢ ({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺) → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺))
46	14, 44, 45	3syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺))
47	46	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺))
48		fzossfz 12488	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0..^𝐿) ⊆ (0...𝐿)
49	48, 35	sseldi 3601	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ∈ (0...𝐿))
50		elfzuz3 12339	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐼))
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 ∈ (ℤ≥‘𝐼))
52	36, 51	eqeltrd 2701	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝑊) ∈ (ℤ≥‘𝐼))
53		fzoss2 12496	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑊) ∈ (ℤ≥‘𝐼) → (0..^𝐼) ⊆ (0..^(#‘𝑊)))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(#‘𝑊)))
55	54	sselda 3603	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(#‘𝑊)))
56	34	ffvelrnda 6359	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇)
57	21, 56	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺))
58	55, 57	syldan 487	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺))
59		psgnunilem2.al	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))
60		fveq2 6191	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠))
61	60	difeq1d 3727	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I ))
62	61	dmeqd 5326	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
63	62	eleq2d 2687	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
64	63	notbid 308	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
65	64	cbvralv 3171	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
66	59, 65	sylib 208	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
67	66	r19.21bi 2932	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
68		difeq1 3721	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I ))
69	68	dmeqd 5326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
70	69	sseq1d 3632	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})))
71		disj2 4024	. . . . . . . . . . . . . . . . . 18 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))
72		disjsn 4246	. . . . . . . . . . . . . . . . . 18 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
73	71, 72	bitr3i 266	. . . . . . . . . . . . . . . . 17 ⊢ (dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
74	70, 73	syl6bb 276	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
75	74	elrab 3363	. . . . . . . . . . . . . . 15 ⊢ ((𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝑊‘𝑠) ∈ (Base‘𝐺) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
76	58, 67, 75	sylanbrc 698	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
77		eqid 2622	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))
78	76, 77	fmptd 6385	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
79	36	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...(#‘𝑊)) = (0...𝐿))
80	49, 79	eleqtrrd 2704	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0...(#‘𝑊)))
81		swrd0val 13421	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨0, 𝐼⟩) = (𝑊 ↾ (0..^𝐼)))
82	24, 80, 81	syl2anc 693	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩) = (𝑊 ↾ (0..^𝐼)))
83	34	feqmptd 6249	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 = (𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)))
84	83	reseq1d 5395	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐼)) = ((𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼)))
85		resmpt 5449	. . . . . . . . . . . . . . . 16 ⊢ ((0..^𝐼) ⊆ (0..^(#‘𝑊)) → ((𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
86	52, 53, 85	3syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
87	82, 84, 86	3eqtrd 2660	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
88	87	feq1d 6030	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}))
89	78, 88	mpbird 247	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
90	89	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
91		iswrdi 13309	. . . . . . . . . . 11 ⊢ ((𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → (𝑊 substr ⟨0, 𝐼⟩) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
92	90, 91	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 substr ⟨0, 𝐼⟩) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
93		gsumwsubmcl 17375	. . . . . . . . . 10 ⊢ (({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺) ∧ (𝑊 substr ⟨0, 𝐼⟩) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
94	47, 92, 93	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
95		difeq1 3721	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) → (𝑗 ∖ I ) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
96	95	dmeqd 5326	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
97	96	sseq1d 3632	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴})))
98	97	elrab 3363	. . . . . . . . . . 11 ⊢ ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺) ∧ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴})))
99	98	simprbi 480	. . . . . . . . . 10 ⊢ ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴}))
100		disj2 4024	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴}))
101		disjsn 4246	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
102	100, 101	bitr3i 266	. . . . . . . . . 10 ⊢ (dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
103	99, 102	sylib 208	. . . . . . . . 9 ⊢ ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
104	94, 103	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
105		psgnunilem2.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
106	105	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
107	104, 106	jca 554	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
108	107	olcd 408	. . . . . 6 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))))
109		excxor 1469	. . . . . 6 ⊢ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ↔ ((𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))))
110	108, 109	sylibr 224	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
111		f1omvdco3 17869	. . . . 5 ⊢ (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)):𝐷–1-1-onto→𝐷 ∧ (𝑊‘𝐼):𝐷–1-1-onto→𝐷 ∧ (𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
112	32, 43, 110, 111	syl3anc 1326	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
113	24	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ∈ Word 𝑇)
114		elfzo0 12508	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿))
115	114	simp2bi 1077	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ)
116	35, 115	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℕ)
117	36, 116	eqeltrd 2701	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ)
118		wrdfin 13323	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
119		hashnncl 13157	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
120	24, 118, 119	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
121	117, 120	mpbid 222	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ≠ ∅)
122	121	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅)
123		swrdccatwrd 13468	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)
124	123	eqcomd 2628	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩))
125	113, 122, 124	syl2anc 693	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩))
126	36	oveq1d 6665	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) − 1) = (𝐿 − 1))
127	126	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((#‘𝑊) − 1) = (𝐿 − 1))
128	116	nncnd 11036	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
129		1cnd 10056	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
130		elfzoelz 12470	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ)
131	35, 130	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
132	131	zcnd 11483	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ ℂ)
133	128, 129, 132	subadd2d 10411	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿))
134	133	biimpar 502	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼)
135	127, 134	eqtrd 2656	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((#‘𝑊) − 1) = 𝐼)
136		opeq2 4403	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) − 1) = 𝐼 → ⟨0, ((#‘𝑊) − 1)⟩ = ⟨0, 𝐼⟩)
137	136	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) − 1) = 𝐼 → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑊 substr ⟨0, 𝐼⟩))
138	137	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑊 substr ⟨0, 𝐼⟩))
139		lsw 13351	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑇 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
140	24, 139	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
141		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) − 1) = 𝐼 → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘𝐼))
142	140, 141	sylan9eq 2676	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ( lastS ‘𝑊) = (𝑊‘𝐼))
143	142	s1eqd 13381	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ⟨“( lastS ‘𝑊)”⟩ = ⟨“(𝑊‘𝐼)”⟩)
144	138, 143	oveq12d 6668	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
145	135, 144	syldan 487	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
146	125, 145	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
147	146	oveq2d 6666	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = (𝐺 Σg ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩)))
148	40	s1cld 13383	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑊‘𝐼)”⟩ ∈ Word (Base‘𝐺))
149		eqid 2622	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
150	20, 149	gsumccat 17378	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑊 substr ⟨0, 𝐼⟩) ∈ Word (Base‘𝐺) ∧ ⟨“(𝑊‘𝐼)”⟩ ∈ Word (Base‘𝐺)) → (𝐺 Σg ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)))
151	18, 27, 148, 150	syl3anc 1326	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σg ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)))
152	151	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)))
153	20	gsumws1 17376	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝐺 Σg ⟨“(𝑊‘𝐼)”⟩) = (𝑊‘𝐼))
154	40, 153	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 Σg ⟨“(𝑊‘𝐼)”⟩) = (𝑊‘𝐼))
155	154	oveq2d 6666	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝑊‘𝐼)))
156	15, 20, 149	symgov 17810	. . . . . . . . . 10 ⊢ (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺) ∧ (𝑊‘𝐼) ∈ (Base‘𝐺)) → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
157	29, 40, 156	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
158	155, 157	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
159	158	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝐺 Σg ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
160	147, 152, 159	3eqtrd 2660	. . . . . 6 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
161	160	difeq1d 3727	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg 𝑊) ∖ I ) = (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
162	161	dmeqd 5326	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
163	112, 162	eleqtrrd 2704	. . 3 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ))
164	13, 163	mtand 691	. 2 ⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿)
165		fzostep1 12584	. . . 4 ⊢ (𝐼 ∈ (0..^𝐿) → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿))
166	35, 165	syl 17	. . 3 ⊢ (𝜑 → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿))
167	166	ord 392	. 2 ⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿))
168	164, 167	mt3d 140	1 ⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ⊻ wxo 1464   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∀wral 2912  {crab 2916  Vcvv 3200   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   I cid 5023  dom cdm 5114  ran crn 5115   ↾ cres 5116   ∘ ccom 5118  ⟶wf 5884  –1-1-onto→wf1o 5887  ‘cfv 5888  (class class class)co 6650  Fincfn 7955  0cc0 9936  1c1 9937   + caddc 9939   < clt 10074   − cmin 10266  ℕcn 11020  ℕ₀cn0 11292  ℤcz 11377  ℤ_≥cuz 11687  ...cfz 12326  ..^cfzo 12465  #chash 13117  Word cword 13291   lastS clsw 13292   ++ cconcat 13293  ⟨“cs1 13294   substr csubstr 13295  Basecbs 15857  +_gcplusg 15941   Σ_g cgsu 16101  Mndcmnd 17294  SubMndcsubmnd 17334  Grpcgrp 17422  SubGrpcsubg 17588  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-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-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-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-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-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-word 13299  df-lsw 13300  df-concat 13301  df-s1 13302  df-substr 13303  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:  psgnunilem2  17915