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Theorem psgnunilem5 19404

Description: Lemma for psgnuni 19409. 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.) (Proof shortened by AV, 12-Oct-2022.)

**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 Σ_g 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 4330	. . . 4 ⊢ ¬ 𝐴 ∈ ∅
2		psgnunilem2.id	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σ_g 𝑊) = ( I ↾ 𝐷))
3	2	difeq1d 4121	. . . . . . 7 ⊢ (𝜑 → ((𝐺 Σ_g 𝑊) ∖ I ) = (( I ↾ 𝐷) ∖ I ))
4	3	dmeqd 5905	. . . . . 6 ⊢ (𝜑 → dom ((𝐺 Σ_g 𝑊) ∖ I ) = dom (( I ↾ 𝐷) ∖ I ))
5		resss 6006	. . . . . . . . 9 ⊢ ( I ↾ 𝐷) ⊆ I
6		ssdif0 4363	. . . . . . . . 9 ⊢ (( I ↾ 𝐷) ⊆ I ↔ (( I ↾ 𝐷) ∖ I ) = ∅)
7	5, 6	mpbi 229	. . . . . . . 8 ⊢ (( I ↾ 𝐷) ∖ I ) = ∅
8	7	dmeqi 5904	. . . . . . 7 ⊢ dom (( I ↾ 𝐷) ∖ I ) = dom ∅
9		dm0 5920	. . . . . . 7 ⊢ dom ∅ = ∅
10	8, 9	eqtri 2759	. . . . . 6 ⊢ dom (( I ↾ 𝐷) ∖ I ) = ∅
11	4, 10	eqtrdi 2787	. . . . 5 ⊢ (𝜑 → dom ((𝐺 Σ_g 𝑊) ∖ I ) = ∅)
12	11	eleq2d 2818	. . . 4 ⊢ (𝜑 → (𝐴 ∈ dom ((𝐺 Σ_g 𝑊) ∖ I ) ↔ 𝐴 ∈ ∅))
13	1, 12	mtbiri 327	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ dom ((𝐺 Σ_g 𝑊) ∖ I ))
14		psgnunilem2.d	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
15		psgnunilem2.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
16	15	symggrp 19310	. . . . . . . . 9 ⊢ (𝐷 ∈ 𝑉 → 𝐺 ∈ Grp)
17		grpmnd 18863	. . . . . . . . 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 2731	. . . . . . . . . . . 12 ⊢ (Base‘𝐺) = (Base‘𝐺)
21	19, 15, 20	symgtrf 19379	. . . . . . . . . . 11 ⊢ 𝑇 ⊆ (Base‘𝐺)
22		sswrd 14477	. . . . . . . . . . 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 3983	. . . . . . . . 9 ⊢ (𝜑 → 𝑊 ∈ Word (Base‘𝐺))
26		pfxcl 14632	. . . . . . . . 9 ⊢ (𝑊 ∈ Word (Base‘𝐺) → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺))
27	25, 26	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺))
28	20	gsumwcl 18757	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺)) → (𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ (Base‘𝐺))
29	18, 27, 28	syl2anc 583	. . . . . . 7 ⊢ (𝜑 → (𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ (Base‘𝐺))
30	15, 20	symgbasf1o 19284	. . . . . . 7 ⊢ ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ (Base‘𝐺) → (𝐺 Σ_g (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷)
31	29, 30	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐺 Σ_g (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷)
32	31	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σ_g (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷)
33		wrdf 14474	. . . . . . . . . 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 7428	. . . . . . . . . 10 ⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
38	35, 37	eleqtrrd 2835	. . . . . . . . 9 ⊢ (𝜑 → 𝐼 ∈ (0..^(♯‘𝑊)))
39	34, 38	ffvelcdmd 7087	. . . . . . . 8 ⊢ (𝜑 → (𝑊‘𝐼) ∈ 𝑇)
40	21, 39	sselid 3980	. . . . . . 7 ⊢ (𝜑 → (𝑊‘𝐼) ∈ (Base‘𝐺))
41	15, 20	symgbasf1o 19284	. . . . . . 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 19377	. . . . . . . . . . 11 ⊢ (𝐷 ∈ 𝑉 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubGrp‘𝐺))
45		subgsubm 19065	. . . . . . . . . . 11 ⊢ ({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	. . . . . . . . . 10 ⊢ (𝜑 → {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺))
47		fzossfz 13656	. . . . . . . . . . . . . . . 16 ⊢ (0..^𝐿) ⊆ (0...𝐿)
48	47, 35	sselid 3980	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐼 ∈ (0...𝐿))
49	36	oveq2d 7428	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (0...(♯‘𝑊)) = (0...𝐿))
50	48, 49	eleqtrrd 2835	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ (0...(♯‘𝑊)))
51		pfxmpt 14633	. . . . . . . . . . . . . 14 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝐼 ∈ (0...(♯‘𝑊))) → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
52	24, 50, 51	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑊 prefix 𝐼) = (𝑠 ∈ (0..^𝐼) ↦ (𝑊‘𝑠)))
53		difeq1 4115	. . . . . . . . . . . . . . . . 17 ⊢ (𝑗 = (𝑊‘𝑠) → (𝑗 ∖ I ) = ((𝑊‘𝑠) ∖ I ))
54	53	dmeqd 5905	. . . . . . . . . . . . . . . 16 ⊢ (𝑗 = (𝑊‘𝑠) → dom (𝑗 ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
55	54	sseq1d 4013	. . . . . . . . . . . . . . 15 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴})))
56		disj2 4457	. . . . . . . . . . . . . . . 16 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}))
57		disjsn 4715	. . . . . . . . . . . . . . . 16 ⊢ ((dom ((𝑊‘𝑠) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
58	56, 57	bitr3i 277	. . . . . . . . . . . . . . 15 ⊢ (dom ((𝑊‘𝑠) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
59	55, 58	bitrdi 287	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝑊‘𝑠) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
60		elfzuz3 13503	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝐼 ∈ (0...𝐿) → 𝐿 ∈ (ℤ_≥‘𝐼))
61	48, 60	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝐿 ∈ (ℤ_≥‘𝐼))
62	36, 61	eqeltrd 2832	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (♯‘𝑊) ∈ (ℤ_≥‘𝐼))
63		fzoss2 13665	. . . . . . . . . . . . . . . . 17 ⊢ ((♯‘𝑊) ∈ (ℤ_≥‘𝐼) → (0..^𝐼) ⊆ (0..^(♯‘𝑊)))
64	62, 63	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (0..^𝐼) ⊆ (0..^(♯‘𝑊)))
65	64	sselda 3982	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → 𝑠 ∈ (0..^(♯‘𝑊)))
66	34	ffvelcdmda 7086	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ 𝑇)
67	21, 66	sselid 3980	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^(♯‘𝑊))) → (𝑊‘𝑠) ∈ (Base‘𝐺))
68	65, 67	syldan 590	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ (Base‘𝐺))
69		psgnunilem2.al	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ))
70		fveq2 6891	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑘 = 𝑠 → (𝑊‘𝑘) = (𝑊‘𝑠))
71	70	difeq1d 4121	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑘 = 𝑠 → ((𝑊‘𝑘) ∖ I ) = ((𝑊‘𝑠) ∖ I ))
72	71	dmeqd 5905	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑘 = 𝑠 → dom ((𝑊‘𝑘) ∖ I ) = dom ((𝑊‘𝑠) ∖ I ))
73	72	eleq2d 2818	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑘 = 𝑠 → (𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
74	73	notbid 318	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑠 → (¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I )))
75	74	cbvralvw 3233	. . . . . . . . . . . . . . . 16 ⊢ (∀𝑘 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑘) ∖ I ) ↔ ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
76	69, 75	sylib 217	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ∀𝑠 ∈ (0..^𝐼) ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
77	76	r19.21bi 3247	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → ¬ 𝐴 ∈ dom ((𝑊‘𝑠) ∖ I ))
78	59, 68, 77	elrabd 3685	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑠 ∈ (0..^𝐼)) → (𝑊‘𝑠) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
79	52, 78	fmpt3d 7117	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
80	79	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
81		iswrdi 14473	. . . . . . . . . . 11 ⊢ ((𝑊 prefix 𝐼):(0..^𝐼)⟶{𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
82	80, 81	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
83		gsumwsubmcl 18755	. . . . . . . . . 10 ⊢ (({𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ∈ (SubMnd‘𝐺) ∧ (𝑊 prefix 𝐼) ∈ Word {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})}) → (𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
84	46, 82, 83	syl2an2r 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})})
85		difeq1 4115	. . . . . . . . . . . . . 14 ⊢ (𝑗 = (𝐺 Σ_g (𝑊 prefix 𝐼)) → (𝑗 ∖ I ) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
86	85	dmeqd 5905	. . . . . . . . . . . . 13 ⊢ (𝑗 = (𝐺 Σ_g (𝑊 prefix 𝐼)) → dom (𝑗 ∖ I ) = dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
87	86	sseq1d 4013	. . . . . . . . . . . 12 ⊢ (𝑗 = (𝐺 Σ_g (𝑊 prefix 𝐼)) → (dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴}) ↔ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})))
88	87	elrab 3683	. . . . . . . . . . 11 ⊢ ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} ↔ ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴})))
89	88	simprbi 496	. . . . . . . . . 10 ⊢ ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))
90		disj2 4457	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}))
91		disjsn 4715	. . . . . . . . . . 11 ⊢ ((dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∩ {𝐴}) = ∅ ↔ ¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
92	90, 91	bitr3i 277	. . . . . . . . . 10 ⊢ (dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊆ (V ∖ {𝐴}) ↔ ¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
93	89, 92	sylib 217	. . . . . . . . 9 ⊢ ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ {𝑗 ∈ (Base‘𝐺) ∣ dom (𝑗 ∖ I ) ⊆ (V ∖ {𝐴})} → ¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
94	84, 93	syl 17	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ))
95		psgnunilem2.a	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
96	95	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))
97	94, 96	jca 511	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
98	97	olcd 871	. . . . . 6 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))))
99		excxor 1514	. . . . . 6 ⊢ ((𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ↔ ((𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∧ ¬ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )) ∨ (¬ 𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ∧ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))))
100	98, 99	sylibr 233	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I )))
101		f1omvdco3 19359	. . . . 5 ⊢ (((𝐺 Σ_g (𝑊 prefix 𝐼)):𝐷–1-1-onto→𝐷 ∧ (𝑊‘𝐼):𝐷–1-1-onto→𝐷 ∧ (𝐴 ∈ dom ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∖ I ) ⊻ 𝐴 ∈ dom ((𝑊‘𝐼) ∖ I ))) → 𝐴 ∈ dom (((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I ))
102	32, 43, 100, 101	syl3anc 1370	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom (((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I ))
103		elfzo0 13678	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) ↔ (𝐼 ∈ ℕ₀ ∧ 𝐿 ∈ ℕ ∧ 𝐼 < 𝐿))
104	103	simp2bi 1145	. . . . . . . . . . . . . 14 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐿 ∈ ℕ)
105	35, 104	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℕ)
106	36, 105	eqeltrd 2832	. . . . . . . . . . . 12 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
107		wrdfin 14487	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin)
108		hashnncl 14331	. . . . . . . . . . . . 13 ⊢ (𝑊 ∈ Fin → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
109	24, 107, 108	3syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) ∈ ℕ ↔ 𝑊 ≠ ∅))
110	106, 109	mpbid 231	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑊 ≠ ∅)
111	110	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 ≠ ∅)
112		pfxlswccat 14668	. . . . . . . . . . 11 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩) = 𝑊)
113	112	eqcomd 2737	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Word 𝑇 ∧ 𝑊 ≠ ∅) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩))
114	24, 111, 113	syl2an2r 682	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩))
115	36	oveq1d 7427	. . . . . . . . . . . 12 ⊢ (𝜑 → ((♯‘𝑊) − 1) = (𝐿 − 1))
116	115	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = (𝐿 − 1))
117	105	nncnd 12233	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐿 ∈ ℂ)
118		1cnd 11214	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
119		elfzoelz 13637	. . . . . . . . . . . . . . 15 ⊢ (𝐼 ∈ (0..^𝐿) → 𝐼 ∈ ℤ)
120	35, 119	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐼 ∈ ℤ)
121	120	zcnd 12672	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐼 ∈ ℂ)
122	117, 118, 121	subadd2d 11595	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿 − 1) = 𝐼 ↔ (𝐼 + 1) = 𝐿))
123	122	biimpar 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐿 − 1) = 𝐼)
124	116, 123	eqtrd 2771	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((♯‘𝑊) − 1) = 𝐼)
125		oveq2 7420	. . . . . . . . . . . 12 ⊢ (((♯‘𝑊) − 1) = 𝐼 → (𝑊 prefix ((♯‘𝑊) − 1)) = (𝑊 prefix 𝐼))
126	125	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (𝑊 prefix ((♯‘𝑊) − 1)) = (𝑊 prefix 𝐼))
127		lsw 14519	. . . . . . . . . . . . . 14 ⊢ (𝑊 ∈ Word 𝑇 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
128	24, 127	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (lastS‘𝑊) = (𝑊‘((♯‘𝑊) − 1)))
129		fveq2 6891	. . . . . . . . . . . . 13 ⊢ (((♯‘𝑊) − 1) = 𝐼 → (𝑊‘((♯‘𝑊) − 1)) = (𝑊‘𝐼))
130	128, 129	sylan9eq 2791	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → (lastS‘𝑊) = (𝑊‘𝐼))
131	130	s1eqd 14556	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → ⟨“(lastS‘𝑊)”⟩ = ⟨“(𝑊‘𝐼)”⟩)
132	126, 131	oveq12d 7430	. . . . . . . . . 10 ⊢ ((𝜑 ∧ ((♯‘𝑊) − 1) = 𝐼) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩) = ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩))
133	124, 132	syldan 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝑊 prefix ((♯‘𝑊) − 1)) ++ ⟨“(lastS‘𝑊)”⟩) = ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩))
134	114, 133	eqtrd 2771	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝑊 = ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩))
135	134	oveq2d 7428	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σ_g 𝑊) = (𝐺 Σ_g ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩)))
136	40	s1cld 14558	. . . . . . . . 9 ⊢ (𝜑 → ⟨“(𝑊‘𝐼)”⟩ ∈ Word (Base‘𝐺))
137		eqid 2731	. . . . . . . . . 10 ⊢ (+_g‘𝐺) = (+_g‘𝐺)
138	20, 137	gsumccat 18759	. . . . . . . . 9 ⊢ ((𝐺 ∈ Mnd ∧ (𝑊 prefix 𝐼) ∈ Word (Base‘𝐺) ∧ ⟨“(𝑊‘𝐼)”⟩ ∈ Word (Base‘𝐺)) → (𝐺 Σ_g ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)))
139	18, 27, 136, 138	syl3anc 1370	. . . . . . . 8 ⊢ (𝜑 → (𝐺 Σ_g ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)))
140	139	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σ_g ((𝑊 prefix 𝐼) ++ ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)))
141	20	gsumws1 18756	. . . . . . . . . . 11 ⊢ ((𝑊‘𝐼) ∈ (Base‘𝐺) → (𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩) = (𝑊‘𝐼))
142	40, 141	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩) = (𝑊‘𝐼))
143	142	oveq2d 7428	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝑊‘𝐼)))
144	15, 20, 137	symgov 19293	. . . . . . . . . 10 ⊢ (((𝐺 Σ_g (𝑊 prefix 𝐼)) ∈ (Base‘𝐺) ∧ (𝑊‘𝐼) ∈ (Base‘𝐺)) → ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)))
145	29, 40, 144	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝑊‘𝐼)) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)))
146	143, 145	eqtrd 2771	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)))
147	146	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σ_g (𝑊 prefix 𝐼))(+_g‘𝐺)(𝐺 Σ_g ⟨“(𝑊‘𝐼)”⟩)) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)))
148	135, 140, 147	3eqtrd 2775	. . . . . 6 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → (𝐺 Σ_g 𝑊) = ((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)))
149	148	difeq1d 4121	. . . . 5 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → ((𝐺 Σ_g 𝑊) ∖ I ) = (((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I ))
150	149	dmeqd 5905	. . . 4 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → dom ((𝐺 Σ_g 𝑊) ∖ I ) = dom (((𝐺 Σ_g (𝑊 prefix 𝐼)) ∘ (𝑊‘𝐼)) ∖ I ))
151	102, 150	eleqtrrd 2835	. . 3 ⊢ ((𝜑 ∧ (𝐼 + 1) = 𝐿) → 𝐴 ∈ dom ((𝐺 Σ_g 𝑊) ∖ I ))
152	13, 151	mtand 813	. 2 ⊢ (𝜑 → ¬ (𝐼 + 1) = 𝐿)
153		fzostep1 13753	. . . 4 ⊢ (𝐼 ∈ (0..^𝐿) → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿))
154	35, 153	syl 17	. . 3 ⊢ (𝜑 → ((𝐼 + 1) ∈ (0..^𝐿) ∨ (𝐼 + 1) = 𝐿))
155	154	ord 861	. 2 ⊢ (𝜑 → (¬ (𝐼 + 1) ∈ (0..^𝐿) → (𝐼 + 1) = 𝐿))
156	152, 155	mt3d 148	1 ⊢ (𝜑 → (𝐼 + 1) ∈ (0..^𝐿))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 844 ⊻ wxo 1508 = wceq 1540 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 {crab 3431 Vcvv 3473 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 {csn 4628 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 dom cdm 5676 ran crn 5677 ↾ cres 5678 ∘ ccom 5680 ⟶wf 6539 –1-1-onto→wf1o 6542 ‘cfv 6543 (class class class)co 7412 Fincfn 8943 0cc0 11114 1c1 11115 + caddc 11117 < clt 11253 − cmin 11449 ℕcn 12217 ℕ₀cn0 12477 ℤcz 12563 ℤ_≥cuz 12827 ...cfz 13489 ..^cfzo 13632 ♯chash 14295 Word cword 14469 lastSclsw 14517 ++ cconcat 14525 ⟨“cs1 14550 prefix cpfx 14625 Basecbs 17149 +_gcplusg 17202 Σ_g cgsu 17391 Mndcmnd 18660 SubMndcsubmnd 18705 Grpcgrp 18856 SubGrpcsubg 19037 SymGrpcsymg 19276 pmTrspcpmtr 19351

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11170 ax-resscn 11171 ax-1cn 11172 ax-icn 11173 ax-addcl 11174 ax-addrcl 11175 ax-mulcl 11176 ax-mulrcl 11177 ax-mulcom 11178 ax-addass 11179 ax-mulass 11180 ax-distr 11181 ax-i2m1 11182 ax-1ne0 11183 ax-1rid 11184 ax-rnegex 11185 ax-rrecex 11186 ax-cnre 11187 ax-pre-lttri 11188 ax-pre-lttrn 11189 ax-pre-ltadd 11190 ax-pre-mulgt0 11191

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-xor 1509 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7860 df-1st 7979 df-2nd 7980 df-frecs 8270 df-wrecs 8301 df-recs 8375 df-rdg 8414 df-1o 8470 df-2o 8471 df-er 8707 df-map 8826 df-en 8944 df-dom 8945 df-sdom 8946 df-fin 8947 df-card 9938 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-xnn0 12550 df-z 12564 df-uz 12828 df-fz 13490 df-fzo 13633 df-seq 13972 df-hash 14296 df-word 14470 df-lsw 14518 df-concat 14526 df-s1 14551 df-substr 14596 df-pfx 14626 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-tset 17221 df-0g 17392 df-gsum 17393 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-efmnd 18787 df-grp 18859 df-minusg 18860 df-subg 19040 df-symg 19277 df-pmtr 19352

This theorem is referenced by: psgnunilem2 19405

W3C validator