Theorem psgnunilem5 17737

Description: Lemma for psgnuni 17742. 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 3878	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
2		psgnunilem2.id	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
3	2	difeq1d 3689	. . . . . . 7 ⊢ (𝜑 → ((𝐺 Σg 𝑊) ∖ I ) = (( I ↾ 𝐷) ∖ I ))
4	3	dmeqd 5248	. . . . . 6 ⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
5		resss 5342	. . . . . . . . 9 ⊢ ( I ↾ 𝐷) ⊆ I
6		ssdif0 3896	. . . . . . . . 9 ⊢ (( I ↾ 𝐷) ⊆ I ↔ (( I ↾ 𝐷) ∖ I ) = ∅)
7	5, 6	mpbi 219	. . . . . . . 8 ⊢ (( I ↾ 𝐷) ∖ I ) = ∅
8	7	dmeqi 5247	. . . . . . 7 ⊢ dom (( I ↾ 𝐷) ∖ I ) = dom ∅
9		dm0 5260	. . . . . . 7 ⊢ dom ∅ = ∅
10	8, 9	eqtri 2632	. . . . . 6 ⊢ dom (( I ↾ 𝐷) ∖ I ) = ∅
11	4, 10	syl6eq 2660	. . . . 5 ⊢ (𝜑 → dom ((𝐺 Σg 𝑊) ∖ I ) = ∅)
12	11	eleq2d 2673	. . . 4 ⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ) ↔ 𝐴 ∈ ∅))
13	1, 12	mtbiri 316	. . 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 17643	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp)
17		grpmnd 17252	. . . . . . . . 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 2610	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	19, 15, 20	symgtrf 17712	. . . . . . . . . . 11 ⊢ 𝑇 ⊆ (Base‘𝐺)
22		sswrd 13168	. . . . . . . . . . 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 3569	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺))
26		swrdcl 13271	. . . . . . . . 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 17200	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑊 substr ⟨0, 𝐼⟩) ∈ Word (Base‘𝐺)) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺))
29	18, 27, 28	syl2anc 691	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ (Base‘𝐺))
30	15, 20	symgbasf1o 17626	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)):𝐷–1-1-onto→𝐷)
33		wrdf 13165	. . . . . . . . . 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 6565	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(#‘𝑊)) = (0..^𝐿))
38	35, 37	eleqtrrd 2691	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^(#‘𝑊)))
39	34, 38	ffvelrnd 6268	. . . . . . . 8 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇)
40	21, 39	sseldi 3566	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺))
41	15, 20	symgbasf1o 17626	. . . . . . 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 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊‘𝐼):𝐷–1-1-onto→𝐷)
44	15, 20	symgsssg 17710	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺))
45		subgsubm 17439	. . . . . . . . . . . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺))
48		fzossfz 12357	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (0..^𝐿) ⊆ (0...𝐿)
49	48, 35	sseldi 3566	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → 𝐼 ∈ (0...𝐿))
50		elfzuz3 12210	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ≥‘𝐼))
51	49, 50	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝐿 ∈ (ℤ≥‘𝐼))
52	36, 51	eqeltrd 2688	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (#‘𝑊) ∈ (ℤ≥‘𝐼))
53		fzoss2 12365	. . . . . . . . . . . . . . . . . 18 ⊢ ((#‘𝑊) ∈ (ℤ≥‘𝐼) → (0..^𝐼) ⊆ (0..^(#‘𝑊)))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(#‘𝑊)))
55	54	sselda 3568	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(#‘𝑊)))
56	34	ffvelrnda 6267	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇)
57	21, 56	sseldi 3566	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(#‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺))
58	55, 57	syldan 486	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺))
59		psgnunilem2.al	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))
60		fveq2 6103	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠))
61	60	difeq1d 3689	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I ))
62	61	dmeqd 5248	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
63	62	eleq2d 2673	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
64	63	notbid 307	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
65	64	cbvralv 3147	. . . . . . . . . . . . . . . . 17 ⊢ (∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
66	59, 65	sylib 207	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
67	66	r19.21bi 2916	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
68		difeq1 3683	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I ))
69	68	dmeqd 5248	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
70	69	sseq1d 3595	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})))
71		disj2 3976	. . . . . . . . . . . . . . . . . 18 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))
72		disjsn 4192	. . . . . . . . . . . . . . . . . 18 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
73	71, 72	bitr3i 265	. . . . . . . . . . . . . . . . 17 ⊢ (dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
74	70, 73	syl6bb 275	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
75	74	elrab 3331	. . . . . . . . . . . . . . 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 695	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
77		eqid 2610	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠))
78	76, 77	fmptd 6292	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
79	36	oveq2d 6565	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (0...(#‘𝑊)) = (0...𝐿))
80	49, 79	eleqtrrd 2691	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐼 ∈ (0...(#‘𝑊)))
81		swrd0val 13273	. . . . . . . . . . . . . . . 16 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(#‘𝑊))) → (𝑊 substr ⟨0, 𝐼⟩) = (𝑊 ↾ (0..^𝐼)))
82	24, 80, 81	syl2anc 691	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩) = (𝑊 ↾ (0..^𝐼)))
83	34	feqmptd 6159	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑊 = (𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)))
84	83	reseq1d 5316	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑊 ↾ (0..^𝐼)) = ((𝑠 ∈ (0..^(#‘𝑊)) ↦ (𝑊‘𝑠)) ↾ (0..^𝐼)))
85		resmpt 5369	. . . . . . . . . . . . . . . 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 2648	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
88	87	feq1d 5943	. . . . . . . . . . . . 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 246	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
90	89	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 substr ⟨0, 𝐼⟩):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
91		iswrdi 13164	. . . . . . . . . . 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 17198	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
95		difeq1 3683	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) → (𝑗 ∖ I ) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
96	95	dmeqd 5248	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
97	96	sseq1d 3595	. . . . . . . . . . . 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 3331	. . . . . . . . . . 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 479	. . . . . . . . . 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 3976	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴}))
101		disjsn 4192	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
102	100, 101	bitr3i 265	. . . . . . . . . 10 ⊢ (dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ))
103	99, 102	sylib 207	. . . . . . . . 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 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
107	104, 106	jca 553	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
108	107	olcd 407	. . . . . 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 1461	. . . . . 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 223	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
111		f1omvdco3 17692	. . . . 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 1318	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
113	24	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ∈ Word 𝑇)
114		elfzo0 12376	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿))
115	114	simp2bi 1070	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ)
116	35, 115	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℕ)
117	36, 116	eqeltrd 2688	. . . . . . . . . . . 12 ⊢ (𝜑 → (#‘𝑊) ∈ ℕ)
118		wrdfin 13178	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
119		hashnncl 13018	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
120	24, 118, 119	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
121	117, 120	mpbid 221	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ≠ ∅)
122	121	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅)
123		swrdccatwrd 13320	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = 𝑊)
124	123	eqcomd 2616	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩))
125	113, 122, 124	syl2anc 691	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩))
126	36	oveq1d 6564	. . . . . . . . . . . 12 ⊢ (𝜑 → ((#‘𝑊) − 1) = (𝐿 − 1))
127	126	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((#‘𝑊) − 1) = (𝐿 − 1))
128	116	nncnd 10913	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
129		1cnd 9935	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
130		elfzoelz 12339	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ)
131	35, 130	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
132	131	zcnd 11359	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ ℂ)
133	128, 129, 132	subadd2d 10290	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿))
134	133	biimpar 501	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼)
135	127, 134	eqtrd 2644	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((#‘𝑊) − 1) = 𝐼)
136		opeq2 4341	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) − 1) = 𝐼 → ⟨0, ((#‘𝑊) − 1)⟩ = ⟨0, 𝐼⟩)
137	136	oveq2d 6565	. . . . . . . . . . . 12 ⊢ (((#‘𝑊) − 1) = 𝐼 → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑊 substr ⟨0, 𝐼⟩))
138	137	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → (𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) = (𝑊 substr ⟨0, 𝐼⟩))
139		lsw 13204	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑇 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
140	24, 139	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → ( lastS ‘𝑊) = (𝑊‘((#‘𝑊) − 1)))
141		fveq2 6103	. . . . . . . . . . . . 13 ⊢ (((#‘𝑊) − 1) = 𝐼 → (𝑊‘((#‘𝑊) − 1)) = (𝑊‘𝐼))
142	140, 141	sylan9eq 2664	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ( lastS ‘𝑊) = (𝑊‘𝐼))
143	142	s1eqd 13234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ⟨“( lastS ‘𝑊)”⟩ = ⟨“(𝑊‘𝐼)”⟩)
144	138, 143	oveq12d 6567	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((#‘𝑊) − 1) = 𝐼) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
145	135, 144	syldan 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 substr ⟨0, ((#‘𝑊) − 1)⟩) ++ ⟨“( lastS ‘𝑊)”⟩) = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
146	125, 145	eqtrd 2644	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩))
147	146	oveq2d 6565	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = (𝐺 Σg ((𝑊 substr ⟨0, 𝐼⟩) ++ ⟨“(𝑊‘𝐼)”⟩)))
148	40	s1cld 13236	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑊‘𝐼)”⟩ ∈ Word (Base‘𝐺))
149		eqid 2610	. . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺)
150	20, 149	gsumccat 17201	. . . . . . . . 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 1318	. . . . . . . 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 480	. . . . . . 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 17199	. . . . . . . . . . 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 6565	. . . . . . . . 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 17633	. . . . . . . . . 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 691	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩))(+g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
158	155, 157	eqtrd 2644	. . . . . . . 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 480	. . . . . . 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 2648	. . . . . 6 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σg 𝑊) = ((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)))
161	160	difeq1d 3689	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σg 𝑊) ∖ I ) = (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
162	161	dmeqd 5248	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σg 𝑊) ∖ I ) = dom (((𝐺 Σg (𝑊 substr ⟨0, 𝐼⟩)) ∘ (𝑊‘𝐼)) ∖ I ))
163	112, 162	eleqtrrd 2691	. . 3 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σg 𝑊) ∖ I ))
164	13, 163	mtand 689	. 2 ⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿)
165		fzostep1 12446	. . . 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 391	. 2 ⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿))
168	164, 167	mt3d 139	1 ⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ⊻ wxo 1456   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  {crab 2900  Vcvv 3173   ∖ cdif 3537   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583   ↦ cmpt 4643   I cid 4948  dom cdm 5038  ran crn 5039   ↾ cres 5040   ∘ ccom 5042  ⟶wf 5800  –1-1-onto→wf1o 5803  ‘cfv 5804  (class class class)co 6549  Fincfn 7841  0cc0 9815  1c1 9816   + caddc 9818   < clt 9953   − cmin 10145  ℕcn 10897  ℕ₀cn0 11169  ℤcz 11254  ℤ_≥cuz 11563  ...cfz 12197  ..^cfzo 12334  #chash 12979  Word cword 13146   lastS clsw 13147   ++ cconcat 13148  ⟨“cs1 13149   substr csubstr 13150  Basecbs 15695  +_gcplusg 15768   Σ_g cgsu 15924  Mndcmnd 17117  SubMndcsubmnd 17157  Grpcgrp 17245  SubGrpcsubg 17411  SymGrpcsymg 17620  pmTrspcpmtr 17684

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-xor 1457  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-seq 12664  df-hash 12980  df-word 13154  df-lsw 13155  df-concat 13156  df-s1 13157  df-substr 13158  df-struct 15697  df-ndx 15698  df-slot 15699  df-base 15700  df-sets 15701  df-ress 15702  df-plusg 15781  df-tset 15787  df-0g 15925  df-gsum 15926  df-mgm 17065  df-sgrp 17107  df-mnd 17118  df-submnd 17159  df-grp 17248  df-minusg 17249  df-subg 17414  df-symg 17621  df-pmtr 17685

This theorem is referenced by:  psgnunilem2  17738