Theorem psgnunilem3 19104

Description: Lemma for psgnuni 19107. 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 2840	. . 3 ⊢ (𝜑 → 𝐿 ∈ ℕ)
4	3	nnnn0d 12293	. 2 ⊢ (𝜑 → 𝐿 ∈ ℕ0)
5		psgnunilem3.w1	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Word 𝑇)
6		wrdf 14222	. . . . . . 7 ⊢ (𝑊 ∈ Word 𝑇 → 𝑊:(0..^(♯‘𝑊))⟶𝑇)
7	5, 6	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑊:(0..^(♯‘𝑊))⟶𝑇)
8		0nn0 12248	. . . . . . . . 9 ⊢ 0 ∈ ℕ0
9	8	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 0 ∈ ℕ0)
10	3	nngt0d 12022	. . . . . . . 8 ⊢ (𝜑 → 0 < 𝐿)
11		elfzo0 13428	. . . . . . . 8 ⊢ (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0 ∧ 𝐿 ∈ ℕ ∧ 0 < 𝐿))
12	9, 3, 10, 11	syl3anbrc 1342	. . . . . . 7 ⊢ (𝜑 → 0 ∈ (0..^𝐿))
13	1	oveq2d 7291	. . . . . . 7 ⊢ (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
14	12, 13	eleqtrrd 2842	. . . . . 6 ⊢ (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
15	7, 14	ffvelrnd 6962	. . . . 5 ⊢ (𝜑 → (𝑊‘0) ∈ 𝑇)
16		eqid 2738	. . . . . 6 ⊢ (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17		psgnunilem3.t	. . . . . 6 ⊢ 𝑇 = ran (pmTrsp‘𝐷)
18	16, 17	pmtrfmvdn0 19070	. . . . 5 ⊢ ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
19	15, 18	syl 17	. . . 4 ⊢ (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20		n0 4280	. . . 4 ⊢ (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
21	19, 20	sylib 217	. . 3 ⊢ (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22		fzonel 13401	. . . . . . . 8 ⊢ ¬ 𝐿 ∈ (0..^𝐿)
23		simpr1 1193	. . . . . . . 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 196	. . . . . . 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 3197	. . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑎 = 0 → (𝑤‘𝑎) = (𝑤‘0))
29	28	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (𝑎 = 0 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
30	29	dmeqd 5814	. . . . . . . . . . 11 ⊢ (𝑎 = 0 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
31	30	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑎 = 0 → (0..^𝑎) = (0..^0))
33	32	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
34	27, 31, 33	3anbi123d 1435	. . . . . . . . 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 629	. . . . . . . 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 3226	. . . . . . 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 341	. . . . . 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 2826	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑎 = 𝑏 → (𝑤‘𝑎) = (𝑤‘𝑏))
40	39	difeq1d 4056	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝑏 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝑏) ∖ I ))
41	40	dmeqd 5814	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝑏) ∖ I ))
42	41	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝑏) ∖ I )))
43		oveq2 7283	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
44	43	raleqdv 3348	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
45	38, 42, 44	3anbi123d 1435	. . . . . . . . . 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 629	. . . . . . . . 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 3226	. . . . . . . 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 7283	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
49	48	eqeq1d 2740	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50		fveqeq2 6783	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
51	49, 50	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52		fveq1 6773	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑏) = (𝑥‘𝑏))
53	52	difeq1d 4056	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → ((𝑤‘𝑏) ∖ I ) = ((𝑥‘𝑏) ∖ I ))
54	53	dmeqd 5814	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑏) ∖ I ) = dom ((𝑥‘𝑏) ∖ I ))
55	54	eleq2d 2824	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑏) ∖ I )))
56		fveq1 6773	. . . . . . . . . . . . . . . . 17 ⊢ (𝑤 = 𝑥 → (𝑤‘𝑐) = (𝑥‘𝑐))
57	56	difeq1d 4056	. . . . . . . . . . . . . . . 16 ⊢ (𝑤 = 𝑥 → ((𝑤‘𝑐) ∖ I ) = ((𝑥‘𝑐) ∖ I ))
58	57	dmeqd 5814	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑥 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑥‘𝑐) ∖ I ))
59	58	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
60	59	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
61	60	ralbidv 3112	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I )))
62		fveq2 6774	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = 𝑑 → (𝑥‘𝑐) = (𝑥‘𝑑))
63	62	difeq1d 4056	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = 𝑑 → ((𝑥‘𝑐) ∖ I ) = ((𝑥‘𝑑) ∖ I ))
64	63	dmeqd 5814	. . . . . . . . . . . . . . 15 ⊢ (𝑐 = 𝑑 → dom ((𝑥‘𝑐) ∖ I ) = dom ((𝑥‘𝑑) ∖ I ))
65	64	eleq2d 2824	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
66	65	notbid 318	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
67	66	cbvralvw 3383	. . . . . . . . . . . 12 ⊢ (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I ))
68	61, 67	bitrdi 287	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥‘𝑑) ∖ I )))
69	55, 68	3anbi23d 1438	. . . . . . . . . 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 631	. . . . . . . . 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	cbvrexvw 3384	. . . . . . . 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	bitrdi 287	. . . . . . 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 341	. . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑎 = (𝑏 + 1) → (𝑤‘𝑎) = (𝑤‘(𝑏 + 1)))
76	75	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (𝑎 = (𝑏 + 1) → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
77	76	dmeqd 5814	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑏 + 1) → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
78	77	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
80	79	raleqdv 3348	. . . . . . . . . 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 1435	. . . . . . . . 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 629	. . . . . . . 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 3226	. . . . . . 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 341	. . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86		fveq2 6774	. . . . . . . . . . . . 13 ⊢ (𝑎 = 𝐿 → (𝑤‘𝑎) = (𝑤‘𝐿))
87	86	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝐿 → ((𝑤‘𝑎) ∖ I ) = ((𝑤‘𝐿) ∖ I ))
88	87	dmeqd 5814	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐿 → dom ((𝑤‘𝑎) ∖ I ) = dom ((𝑤‘𝐿) ∖ I ))
89	88	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤‘𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘𝐿) ∖ I )))
90		oveq2 7283	. . . . . . . . . . 11 ⊢ (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
91	90	raleqdv 3348	. . . . . . . . . 10 ⊢ (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I )))
92	85, 89, 91	3anbi123d 1435	. . . . . . . . 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 629	. . . . . . . 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 3226	. . . . . . 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 341	. . . . . 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 4443	. . . . . . . . . 10 ⊢ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )
103		fzo0 13411	. . . . . . . . . . 11 ⊢ (0..^0) = ∅
104	103	raleqi 3346	. . . . . . . . . 10 ⊢ (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I ))
105	102, 104	mpbir 230	. . . . . . . . 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 1127	. . . . . . 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 7283	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
109	108	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
110		fveqeq2 6783	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111	109, 110	anbi12d 631	. . . . . . . . 9 ⊢ (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112		fveq1 6773	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113	112	difeq1d 4056	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114	113	dmeqd 5814	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115	114	eleq2d 2824	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116		fveq1 6773	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑊 → (𝑤‘𝑐) = (𝑊‘𝑐))
117	116	difeq1d 4056	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑊 → ((𝑤‘𝑐) ∖ I ) = ((𝑊‘𝑐) ∖ I ))
118	117	dmeqd 5814	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑊 → dom ((𝑤‘𝑐) ∖ I ) = dom ((𝑊‘𝑐) ∖ I ))
119	118	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
120	119	notbid 318	. . . . . . . . . . 11 ⊢ (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
121	120	ralbidv 3112	. . . . . . . . . 10 ⊢ (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤‘𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊‘𝑐) ∖ I )))
122	115, 121	3anbi23d 1438	. . . . . . . . 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 631	. . . . . . . 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 3561	. . . . . . 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 834	. . . . . 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 723	. . . . . . . . . 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 768	. . . . . . . . . 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 764	. . . . . . . . . . 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 726	. . . . . . . . . 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 766	. . . . . . . . . . 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 726	. . . . . . . . . 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 1193	. . . . . . . . . . 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 726	. . . . . . . . . 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 1194	. . . . . . . . . . 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 726	. . . . . . . . . 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 1195	. . . . . . . . . . 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 726	. . . . . . . . . 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 6783	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142		oveq2 7283	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
143	142	eqeq1d 2740	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
144	141, 143	anbi12d 631	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145	144	cbvrexvw 3384	. . . . . . . . . . . 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 328	. . . . . . . . . . 11 ⊢ (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147	146	ad2antrr 723	. . . . . . . . . 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 19103	. . . . . . . . 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 3214	. . . . . . . 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 12415	. . . . 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 198	. . . 4 ⊢ (𝐿 ∈ ℕ0 → ¬ (𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154	153	con2i 139	. . 3 ⊢ ((𝜑 ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
155	21, 154	exlimddv 1938	. 2 ⊢ (𝜑 → ¬ 𝐿 ∈ ℕ0)
156	4, 155	pm2.65i 193	1 ⊢ ¬ 𝜑

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884  ∅c0 4256   class class class wbr 5074   I cid 5488  dom cdm 5589  ran crn 5590   ↾ cres 5591  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  0cc0 10871  1c1 10872   + caddc 10874   < clt 11009   − cmin 11205  ℕcn 11973  2c2 12028  ℕ₀cn0 12233  ..^cfzo 13382  ♯chash 14044  Word cword 14217   Σ_g cgsu 17151  SymGrpcsymg 18974  pmTrspcpmtr 19049

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1507  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-ot 4570  df-uni 4840  df-int 4880  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-1st 7831  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-fin 8737  df-card 9697  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-4 12038  df-5 12039  df-6 12040  df-7 12041  df-8 12042  df-9 12043  df-n0 12234  df-xnn0 12306  df-z 12320  df-uz 12583  df-fz 13240  df-fzo 13383  df-seq 13722  df-hash 14045  df-word 14218  df-lsw 14266  df-concat 14274  df-s1 14301  df-substr 14354  df-pfx 14384  df-splice 14463  df-s2 14561  df-struct 16848  df-sets 16865  df-slot 16883  df-ndx 16895  df-base 16913  df-ress 16942  df-plusg 16975  df-tset 16981  df-0g 17152  df-gsum 17153  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-submnd 18431  df-efmnd 18508  df-grp 18580  df-minusg 18581  df-subg 18752  df-symg 18975  df-pmtr 19050

This theorem is referenced by:  psgnunilem4  19105