Theorem psgnunilem3 18355

Description: Lemma for psgnuni 18358. Any nonempty representation of the identity can be incrementally transformed into a representation two shorter. (Contributed by Stefan O'Rear, 25-Aug-2015.)

Hypotheses

Ref	Expression
psgnunilem3.g	⊢ 𝐺 = (SymGrp‘𝐷)
psgnunilem3.t	⊢ 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem3.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
psgnunilem3.w1	⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
psgnunilem3.l	⊢ (𝜑 → (♯‘𝑊) = 𝐿)
psgnunilem3.w2	⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
psgnunilem3.w3	⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
psgnunilem3.in	⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))

Assertion

Ref	Expression
psgnunilem3	⊢ ¬ 𝜑

Distinct variable groups:   𝑥,𝐷   𝑥,𝐺   𝑥,𝐿   𝑥,𝑇   𝑥,𝑊   𝜑,𝑥

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem psgnunilem3

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑒 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		psgnunilem3.l	. . . 4 ⊢ (𝜑 → (♯‘𝑊) = 𝐿)
2		psgnunilem3.w2	. . . 4 ⊢ (𝜑 → (♯‘𝑊) ∈ ℕ)
3	1, 2	eqeltrrd 2884	. . 3 ⊢ (𝜑 → 𝐿 ∈ ℕ)
4	3	nnnn0d 11803	. 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
5		psgnunilem3.w1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
6		wrdf 13712	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇)
8		0nn0 11760	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℕ0)
10	3	nngt0d 11534	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐿)
11		elfzo0 12928	. . . . . . . 8 ⊢ (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 0 < 𝐿))
12	9, 3, 10, 11	syl3anbrc 1336	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝐿))
13	1	oveq2d 7032	. . . . . . 7 ⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
14	12, 13	eleqtrrd 2886	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
15	7, 14	ffvelrnd 6717	. . . . 5 ⊢ (𝜑 → (𝑊‘0) ∈ 𝑇)
16		eqid 2795	. . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17		psgnunilem3.t	. . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
18	16, 17	pmtrfmvdn0 18321	. . . . 5 ⊢ ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
19	15, 18	syl 17	. . . 4 ⊢ (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20		n0 4230	. . . 4 ⊢ (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
21	19, 20	sylib 219	. . 3 ⊢ (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22		fzonel 12901	. . . . . . . 8 ⊢ ¬ 𝐿 ∈ (0..^𝐿)
23		simpr1 1187	. . . . . . . 8 ⊢ ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) → 𝐿 ∈ (0..^𝐿))
24	22, 23	mto 198	. . . . . . 7 ⊢ ¬ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
25	24	a1i 11	. . . . . 6 ⊢ (𝑤 ∈ Word 𝑇 → ¬ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
26	25	nrex 3232	. . . . 5 ⊢ ¬ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
27		eleq1 2870	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28		fveq2 6538	. . . . . . . . . . . . 13 ⊢ (𝑎 = 0 → (𝑤‘𝑎) = (𝑤‘0))
29	28	difeq1d 4019	. . . . . . . . . . . 12 ⊢ (𝑎 = 0 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
30	29	dmeqd 5660	. . . . . . . . . . 11 ⊢ (𝑎 = 0 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
31	30	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑎 = 0 → (0..^𝑎) = (0..^0))
33	32	raleqdv 3375	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
34	27, 31, 33	3anbi123d 1428	. . . . . . . . 9 ⊢ (𝑎 = 0 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
35	34	anbi2d 628	. . . . . . . 8 ⊢ (𝑎 = 0 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
36	35	rexbidv 3260	. . . . . . 7 ⊢ (𝑎 = 0 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
37	36	imbi2d 342	. . . . . 6 ⊢ (𝑎 = 0 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))))
38		eleq1 2870	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39		fveq2 6538	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑤‘𝑎) = (𝑤‘𝑏))
40	39	difeq1d 4019	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝑏) ∖ I ))
41	40	dmeqd 5660	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝑏) ∖ I ))
42	41	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I )))
43		oveq2 7024	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
44	43	raleqdv 3375	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
45	38, 42, 44	3anbi123d 1428	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
46	45	anbi2d 628	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
47	46	rexbidv 3260	. . . . . . . 8 ⊢ (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
48		oveq2 7024	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
49	48	eqeq1d 2797	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50		fveqeq2 6547	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
51	49, 50	anbi12d 630	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52		fveq1 6537	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑏) = (𝑥‘𝑏))
53	52	difeq1d 4019	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → ((𝑤‘𝑏) ∖ I ) = ((𝑥‘𝑏) ∖ I ))
54	53	dmeqd 5660	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑏) ∖ I ) = dom ((𝑥‘𝑏) ∖ I ))
55	54	eleq2d 2868	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I )))
56		fveq1 6537	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑐) = (𝑥‘𝑐))
57	56	difeq1d 4019	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → ((𝑤‘𝑐) ∖ I ) = ((𝑥‘𝑐) ∖ I ))
58	57	dmeqd 5660	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑥‘𝑐) ∖ I ))
59	58	eleq2d 2868	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
60	59	notbid 319	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
61	60	ralbidv 3164	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
62		fveq2 6538	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → (𝑥‘𝑐) = (𝑥‘𝑑))
63	62	difeq1d 4019	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → ((𝑥‘𝑐) ∖ I ) = ((𝑥‘𝑑) ∖ I ))
64	63	dmeqd 5660	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → dom ((𝑥‘𝑐) ∖ I ) = dom ((𝑥‘𝑑) ∖ I ))
65	64	eleq2d 2868	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
66	65	notbid 319	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
67	66	cbvralv 3403	. . . . . . . . . . . 12 ⊢ (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))
68	61, 67	syl6bb 288	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
69	55, 68	3anbi23d 1431	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → ((𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))
70	51, 69	anbi12d 630	. . . . . . . . 9 ⊢ (𝑤 = 𝑥 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))))
71	70	cbvrexv 3404	. . . . . . . 8 ⊢ (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))
72	47, 71	syl6bb 288	. . . . . . 7 ⊢ (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))))
73	72	imbi2d 342	. . . . . 6 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))))
74		eleq1 2870	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75		fveq2 6538	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 + 1) → (𝑤‘𝑎) = (𝑤‘(𝑏 + 1)))
76	75	difeq1d 4019	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑏 + 1) → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
77	76	dmeqd 5660	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑏 + 1) → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
78	77	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
80	79	raleqdv 3375	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
81	74, 78, 80	3anbi123d 1428	. . . . . . . . 9 ⊢ (𝑎 = (𝑏 + 1) → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
82	81	anbi2d 628	. . . . . . . 8 ⊢ (𝑎 = (𝑏 + 1) → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
83	82	rexbidv 3260	. . . . . . 7 ⊢ (𝑎 = (𝑏 + 1) → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
84	83	imbi2d 342	. . . . . 6 ⊢ (𝑎 = (𝑏 + 1) → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))))
85		eleq1 2870	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86		fveq2 6538	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐿 → (𝑤‘𝑎) = (𝑤‘𝐿))
87	86	difeq1d 4019	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐿 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝐿) ∖ I ))
88	87	dmeqd 5660	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐿 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝐿) ∖ I ))
89	88	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I )))
90		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
91	90	raleqdv 3375	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
92	85, 89, 91	3anbi123d 1428	. . . . . . . . 9 ⊢ (𝑎 = 𝐿 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
93	92	anbi2d 628	. . . . . . . 8 ⊢ (𝑎 = 𝐿 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
94	93	rexbidv 3260	. . . . . . 7 ⊢ (𝑎 = 𝐿 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
95	94	imbi2d 342	. . . . . 6 ⊢ (𝑎 = 𝐿 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))) ↔ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))))
96	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇)
97		psgnunilem3.w3	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
98	97, 1	jca 512	. . . . . . . 8 ⊢ (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
99	98	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
100	12	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈ (0..^𝐿))
101		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
102		ral0 4370	. . . . . . . . . 10 ⊢ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )
103		fzo0 12911	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
104	103	raleqi 3373	. . . . . . . . . 10 ⊢ (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))
105	102, 104	mpbir 232	. . . . . . . . 9 ⊢ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )
106	105	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))
107	100, 101, 106	3jca 1121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
108		oveq2 7024	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
109	108	eqeq1d 2797	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
110		fveqeq2 6547	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111	109, 110	anbi12d 630	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112		fveq1 6537	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113	112	difeq1d 4019	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114	113	dmeqd 5660	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115	114	eleq2d 2868	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116		fveq1 6537	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑐) = (𝑊‘𝑐))
117	116	difeq1d 4019	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑐) ∖ I ) = ((𝑊‘𝑐) ∖ I ))
118	117	dmeqd 5660	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑊‘𝑐) ∖ I ))
119	118	eleq2d 2868	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
120	119	notbid 319	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
121	120	ralbidv 3164	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
122	115, 121	3anbi23d 1431	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → ((0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))))
123	111, 122	anbi12d 630	. . . . . . . 8 ⊢ (𝑤 = 𝑊 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))))
124	123	rspcev 3559	. . . . . . 7 ⊢ ((𝑊 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
125	96, 99, 107, 124	syl12anc 833	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
126		psgnunilem3.g	. . . . . . . . . 10 ⊢ 𝐺 = (SymGrp‘𝐷)
127		psgnunilem3.d	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
128	127	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝐷 ∈ 𝑉)
129		simprl 767	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑥 ∈ Word 𝑇)
130		simpll 763	. . . . . . . . . . 11 ⊢ ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
131	130	ad2antll 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
132		simplr 765	. . . . . . . . . . 11 ⊢ ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → (♯‘𝑥) = 𝐿)
133	132	ad2antll 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → (♯‘𝑥) = 𝐿)
134		simpr1 1187	. . . . . . . . . . 11 ⊢ ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → 𝑏 ∈ (0..^𝐿))
135	134	ad2antll 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑏 ∈ (0..^𝐿))
136		simpr2 1188	. . . . . . . . . . 11 ⊢ ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ))
137	136	ad2antll 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ))
138		simpr3 1189	. . . . . . . . . . 11 ⊢ ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))
139	138	ad2antll 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))
140		psgnunilem3.in	. . . . . . . . . . . 12 ⊢ (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
141		fveqeq2 6547	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142		oveq2 7024	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
143	142	eqeq1d 2797	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
144	141, 143	anbi12d 630	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145	144	cbvrexv 3404	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
146	140, 145	sylnib 329	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147	146	ad2antrr 722	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
148	126, 17, 128, 129, 131, 133, 135, 137, 139, 147	psgnunilem2 18354	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))
149	148	rexlimdvaa 3248	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
150	149	a2i 14	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) → ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
151	150	a1i 11	. . . . . 6 ⊢ (𝑏 ∈ ℕ0 → (((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))) → ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ))))))
152	37, 73, 84, 95, 125, 151	nn0ind 11926	. . . . 5 ⊢ (𝐿 ∈ ℕ0 → ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))))
153	26, 152	mtoi 200	. . . 4 ⊢ (𝐿 ∈ ℕ0 → ¬ (𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154	153	con2i 141	. . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
155	21, 154	exlimddv 1913	. 2 ⊢ (𝜑 → ¬ 𝐿 ∈ ℕ0)
156	4, 155	pm2.65i 195	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1080   = wceq 1522  ∃wex 1761   ∈ wcel 2081   ≠ wne 2984  ∀wral 3105  ∃wrex 3106   ∖ cdif 3856  ∅c0 4211   class class class wbr 4962   I cid 5347  dom cdm 5443  ran crn 5444   ↾ cres 5445  ⟶wf 6221  ‘cfv 6225  (class class class)co 7016  0cc0 10383  1c1 10384   + caddc 10386   < clt 10521   − cmin 10717  ℕcn 11486  2c2 11540  ℕ₀cn0 11745  ..^cfzo 12883  ♯chash 13540  Word cword 13707   Σ_g cgsu 16543  SymGrpcsymg 18236  pmTrspcpmtr 18300

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319  ax-cnex 10439  ax-resscn 10440  ax-1cn 10441  ax-icn 10442  ax-addcl 10443  ax-addrcl 10444  ax-mulcl 10445  ax-mulrcl 10446  ax-mulcom 10447  ax-addass 10448  ax-mulass 10449  ax-distr 10450  ax-i2m1 10451  ax-1ne0 10452  ax-1rid 10453  ax-rnegex 10454  ax-rrecex 10455  ax-cnre 10456  ax-pre-lttri 10457  ax-pre-lttrn 10458  ax-pre-ltadd 10459  ax-pre-mulgt0 10460

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-xor 1497  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-nel 3091  df-ral 3110  df-rex 3111  df-reu 3112  df-rmo 3113  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-pss 3876  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-tp 4477  df-op 4479  df-ot 4481  df-uni 4746  df-int 4783  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-tr 5064  df-id 5348  df-eprel 5353  df-po 5362  df-so 5363  df-fr 5402  df-we 5404  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-pred 6023  df-ord 6069  df-on 6070  df-lim 6071  df-suc 6072  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-riota 6977  df-ov 7019  df-oprab 7020  df-mpo 7021  df-om 7437  df-1st 7545  df-2nd 7546  df-wrecs 7798  df-recs 7860  df-rdg 7898  df-1o 7953  df-2o 7954  df-oadd 7957  df-er 8139  df-map 8258  df-en 8358  df-dom 8359  df-sdom 8360  df-fin 8361  df-card 9214  df-pnf 10523  df-mnf 10524  df-xr 10525  df-ltxr 10526  df-le 10527  df-sub 10719  df-neg 10720  df-nn 11487  df-2 11548  df-3 11549  df-4 11550  df-5 11551  df-6 11552  df-7 11553  df-8 11554  df-9 11555  df-n0 11746  df-xnn0 11816  df-z 11830  df-uz 12094  df-fz 12743  df-fzo 12884  df-seq 13220  df-hash 13541  df-word 13708  df-lsw 13761  df-concat 13769  df-s1 13794  df-substr 13839  df-pfx 13869  df-splice 13948  df-s2 14046  df-struct 16314  df-ndx 16315  df-slot 16316  df-base 16318  df-sets 16319  df-ress 16320  df-plusg 16407  df-tset 16413  df-0g 16544  df-gsum 16545  df-mgm 17681  df-sgrp 17723  df-mnd 17734  df-submnd 17775  df-grp 17864  df-minusg 17865  df-subg 18030  df-symg 18237  df-pmtr 18301

This theorem is referenced by:  psgnunilem4  18356