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Theorem psgnunilem3 19514
Description: Lemma for psgnuni 19517. 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 (𝜑𝐷𝑉)
psgnunilem3.w1 (𝜑𝑊 ∈ 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.
StepHypRef Expression
1 psgnunilem3.l . . . 4 (𝜑 → (♯‘𝑊) = 𝐿)
2 psgnunilem3.w2 . . . 4 (𝜑 → (♯‘𝑊) ∈ ℕ)
31, 2eqeltrrd 2842 . . 3 (𝜑𝐿 ∈ ℕ)
43nnnn0d 12587 . 2 (𝜑𝐿 ∈ ℕ0)
5 psgnunilem3.w1 . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
6 wrdf 14557 . . . . . . 7 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
75, 6syl 17 . . . . . 6 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8 0nn0 12541 . . . . . . . . 9 0 ∈ ℕ0
98a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℕ0)
103nngt0d 12315 . . . . . . . 8 (𝜑 → 0 < 𝐿)
11 elfzo0 13740 . . . . . . . 8 (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0𝐿 ∈ ℕ ∧ 0 < 𝐿))
129, 3, 10, 11syl3anbrc 1344 . . . . . . 7 (𝜑 → 0 ∈ (0..^𝐿))
131oveq2d 7447 . . . . . . 7 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
1412, 13eleqtrrd 2844 . . . . . 6 (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
157, 14ffvelcdmd 7105 . . . . 5 (𝜑 → (𝑊‘0) ∈ 𝑇)
16 eqid 2737 . . . . . 6 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17 psgnunilem3.t . . . . . 6 𝑇 = ran (pmTrsp‘𝐷)
1816, 17pmtrfmvdn0 19480 . . . . 5 ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
1915, 18syl 17 . . . 4 (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20 n0 4353 . . . 4 (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
2119, 20sylib 218 . . 3 (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22 fzonel 13713 . . . . . . . 8 ¬ 𝐿 ∈ (0..^𝐿)
23 simpr1 1195 . . . . . . . 8 ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) → 𝐿 ∈ (0..^𝐿))
2422, 23mto 197 . . . . . . 7 ¬ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
2524a1i 11 . . . . . 6 (𝑤 ∈ Word 𝑇 → ¬ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
2625nrex 3074 . . . . 5 ¬ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
27 eleq1 2829 . . . . . . . . . 10 (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28 fveq2 6906 . . . . . . . . . . . . 13 (𝑎 = 0 → (𝑤𝑎) = (𝑤‘0))
2928difeq1d 4125 . . . . . . . . . . . 12 (𝑎 = 0 → ((𝑤𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
3029dmeqd 5916 . . . . . . . . . . 11 (𝑎 = 0 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
3130eleq2d 2827 . . . . . . . . . 10 (𝑎 = 0 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32 oveq2 7439 . . . . . . . . . . 11 (𝑎 = 0 → (0..^𝑎) = (0..^0))
3332raleqdv 3326 . . . . . . . . . 10 (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
3427, 31, 333anbi123d 1438 . . . . . . . . 9 (𝑎 = 0 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
3534anbi2d 630 . . . . . . . 8 (𝑎 = 0 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
3635rexbidv 3179 . . . . . . 7 (𝑎 = 0 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
3736imbi2d 340 . . . . . 6 (𝑎 = 0 → (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))) ↔ ((𝜑𝑒 ∈ 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 ((𝑤𝑐) ∖ I ))))))
38 eleq1 2829 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39 fveq2 6906 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑤𝑎) = (𝑤𝑏))
4039difeq1d 4125 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑤𝑎) ∖ I ) = ((𝑤𝑏) ∖ I ))
4140dmeqd 5916 . . . . . . . . . . . 12 (𝑎 = 𝑏 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝑏) ∖ I ))
4241eleq2d 2827 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝑏) ∖ I )))
43 oveq2 7439 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
4443raleqdv 3326 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
4538, 42, 443anbi123d 1438 . . . . . . . . . 10 (𝑎 = 𝑏 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
4645anbi2d 630 . . . . . . . . 9 (𝑎 = 𝑏 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
4746rexbidv 3179 . . . . . . . 8 (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
48 oveq2 7439 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
4948eqeq1d 2739 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50 fveqeq2 6915 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
5149, 50anbi12d 632 . . . . . . . . . 10 (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52 fveq1 6905 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑏) = (𝑥𝑏))
5352difeq1d 4125 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → ((𝑤𝑏) ∖ I ) = ((𝑥𝑏) ∖ I ))
5453dmeqd 5916 . . . . . . . . . . . 12 (𝑤 = 𝑥 → dom ((𝑤𝑏) ∖ I ) = dom ((𝑥𝑏) ∖ I ))
5554eleq2d 2827 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑏) ∖ I )))
56 fveq1 6905 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (𝑤𝑐) = (𝑥𝑐))
5756difeq1d 4125 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥 → ((𝑤𝑐) ∖ I ) = ((𝑥𝑐) ∖ I ))
5857dmeqd 5916 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑥𝑐) ∖ I ))
5958eleq2d 2827 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6059notbid 318 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6160ralbidv 3178 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
62 fveq2 6906 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑑 → (𝑥𝑐) = (𝑥𝑑))
6362difeq1d 4125 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑑 → ((𝑥𝑐) ∖ I ) = ((𝑥𝑑) ∖ I ))
6463dmeqd 5916 . . . . . . . . . . . . . . 15 (𝑐 = 𝑑 → dom ((𝑥𝑐) ∖ I ) = dom ((𝑥𝑑) ∖ I ))
6564eleq2d 2827 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6665notbid 318 . . . . . . . . . . . . 13 (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6766cbvralvw 3237 . . . . . . . . . . . 12 (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
6861, 67bitrdi 287 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6955, 683anbi23d 1441 . . . . . . . . . 10 (𝑤 = 𝑥 → ((𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))
7051, 69anbi12d 632 . . . . . . . . 9 (𝑤 = 𝑥 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))))
7170cbvrexvw 3238 . . . . . . . 8 (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))
7247, 71bitrdi 287 . . . . . . 7 (𝑎 = 𝑏 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))))
7372imbi2d 340 . . . . . 6 (𝑎 = 𝑏 → (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))) ↔ ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))))
74 eleq1 2829 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75 fveq2 6906 . . . . . . . . . . . . 13 (𝑎 = (𝑏 + 1) → (𝑤𝑎) = (𝑤‘(𝑏 + 1)))
7675difeq1d 4125 . . . . . . . . . . . 12 (𝑎 = (𝑏 + 1) → ((𝑤𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
7776dmeqd 5916 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
7877eleq2d 2827 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79 oveq2 7439 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
8079raleqdv 3326 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
8174, 78, 803anbi123d 1438 . . . . . . . . 9 (𝑎 = (𝑏 + 1) → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
8281anbi2d 630 . . . . . . . 8 (𝑎 = (𝑏 + 1) → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ ((𝑏 + 1) ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I ) ∧ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
8382rexbidv 3179 . . . . . . 7 (𝑎 = (𝑏 + 1) → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ 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 ((𝑤𝑐) ∖ I )))))
8483imbi2d 340 . . . . . 6 (𝑎 = (𝑏 + 1) → (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))) ↔ ((𝜑𝑒 ∈ 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 ((𝑤𝑐) ∖ I ))))))
85 eleq1 2829 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86 fveq2 6906 . . . . . . . . . . . . 13 (𝑎 = 𝐿 → (𝑤𝑎) = (𝑤𝐿))
8786difeq1d 4125 . . . . . . . . . . . 12 (𝑎 = 𝐿 → ((𝑤𝑎) ∖ I ) = ((𝑤𝐿) ∖ I ))
8887dmeqd 5916 . . . . . . . . . . 11 (𝑎 = 𝐿 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝐿) ∖ I ))
8988eleq2d 2827 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝐿) ∖ I )))
90 oveq2 7439 . . . . . . . . . . 11 (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
9190raleqdv 3326 . . . . . . . . . 10 (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
9285, 89, 913anbi123d 1438 . . . . . . . . 9 (𝑎 = 𝐿 → ((𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
9392anbi2d 630 . . . . . . . 8 (𝑎 = 𝐿 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
9493rexbidv 3179 . . . . . . 7 (𝑎 = 𝐿 → (∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
9594imbi2d 340 . . . . . 6 (𝑎 = 𝐿 → (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝑎 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))) ↔ ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))))
965adantr 480 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇)
97 psgnunilem3.w3 . . . . . . . . 9 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
9897, 1jca 511 . . . . . . . 8 (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
9998adantr 480 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
10012adantr 480 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈ (0..^𝐿))
101 simpr 484 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
102 ral0 4513 . . . . . . . . . 10 𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
103 fzo0 13723 . . . . . . . . . . 11 (0..^0) = ∅
104103raleqi 3324 . . . . . . . . . 10 (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
105102, 104mpbir 231 . . . . . . . . 9 𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
106105a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
107100, 101, 1063jca 1129 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
108 oveq2 7439 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
109108eqeq1d 2739 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
110 fveqeq2 6915 . . . . . . . . . 10 (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111109, 110anbi12d 632 . . . . . . . . 9 (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112 fveq1 6905 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113112difeq1d 4125 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114113dmeqd 5916 . . . . . . . . . . 11 (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115114eleq2d 2827 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116 fveq1 6905 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑤𝑐) = (𝑊𝑐))
117116difeq1d 4125 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((𝑤𝑐) ∖ I ) = ((𝑊𝑐) ∖ I ))
118117dmeqd 5916 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑊𝑐) ∖ I ))
119118eleq2d 2827 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
120119notbid 318 . . . . . . . . . . 11 (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
121120ralbidv 3178 . . . . . . . . . 10 (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
122115, 1213anbi23d 1441 . . . . . . . . 9 (𝑤 = 𝑊 → ((0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )) ↔ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))))
123111, 122anbi12d 632 . . . . . . . 8 (𝑤 = 𝑊 → ((((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))) ↔ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))))
124123rspcev 3622 . . . . . . 7 ((𝑊 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ))))
12596, 99, 107, 124syl12anc 837 . . . . . 6 ((𝜑𝑒 ∈ 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 ((𝑤𝑐) ∖ I ))))
126 psgnunilem3.g . . . . . . . . . 10 𝐺 = (SymGrp‘𝐷)
127 psgnunilem3.d . . . . . . . . . . 11 (𝜑𝐷𝑉)
128127ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → 𝐷𝑉)
129 simprl 771 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → 𝑥 ∈ Word 𝑇)
130 simpll 767 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
131130ad2antll 729 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
132 simplr 769 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → (♯‘𝑥) = 𝐿)
133132ad2antll 729 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → (♯‘𝑥) = 𝐿)
134 simpr1 1195 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → 𝑏 ∈ (0..^𝐿))
135134ad2antll 729 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → 𝑏 ∈ (0..^𝐿))
136 simpr2 1196 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → 𝑒 ∈ dom ((𝑥𝑏) ∖ I ))
137136ad2antll 729 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → 𝑒 ∈ dom ((𝑥𝑏) ∖ I ))
138 simpr3 1197 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
139138ad2antll 729 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
140 psgnunilem3.in . . . . . . . . . . . 12 (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
141 fveqeq2 6915 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142 oveq2 7439 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
143142eqeq1d 2739 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
144141, 143anbi12d 632 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145144cbvrexvw 3238 . . . . . . . . . . . 12 (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
146140, 145sylnib 328 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147146ad2antrr 726 . . . . . . . . . 10 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))))) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
148126, 17, 128, 129, 131, 133, 135, 137, 139, 147psgnunilem2 19513 . . . . . . . . 9 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) ∧ (𝑥 ∈ Word 𝑇 ∧ (((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ 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 ((𝑤𝑐) ∖ I ))))
149148rexlimdvaa 3156 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ 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 ((𝑤𝑐) ∖ I )))))
150149a2i 14 . . . . . . 7 (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))) → ((𝜑𝑒 ∈ 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 ((𝑤𝑐) ∖ I )))))
151150a1i 11 . . . . . 6 (𝑏 ∈ ℕ0 → (((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑥 ∈ Word 𝑇(((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))) → ((𝜑𝑒 ∈ 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 ((𝑤𝑐) ∖ I ))))))
15237, 73, 84, 95, 125, 151nn0ind 12713 . . . . 5 (𝐿 ∈ ℕ0 → ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∃𝑤 ∈ Word 𝑇(((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ∧ (𝐿 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑤𝐿) ∖ I ) ∧ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))))
15326, 152mtoi 199 . . . 4 (𝐿 ∈ ℕ0 → ¬ (𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154153con2i 139 . . 3 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
15521, 154exlimddv 1935 . 2 (𝜑 → ¬ 𝐿 ∈ ℕ0)
1564, 155pm2.65i 194 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  wne 2940  wral 3061  wrex 3070  cdif 3948  c0 4333   class class class wbr 5143   I cid 5577  dom cdm 5685  ran crn 5686  cres 5687  wf 6557  cfv 6561  (class class class)co 7431  0cc0 11155  1c1 11156   + caddc 11158   < clt 11295  cmin 11492  cn 12266  2c2 12321  0cn0 12526  ..^cfzo 13694  chash 14369  Word cword 14552   Σg cgsu 17485  SymGrpcsymg 19386  pmTrspcpmtr 19459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-xor 1512  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-card 9979  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-xnn0 12600  df-z 12614  df-uz 12879  df-fz 13548  df-fzo 13695  df-seq 14043  df-hash 14370  df-word 14553  df-lsw 14601  df-concat 14609  df-s1 14634  df-substr 14679  df-pfx 14709  df-splice 14788  df-s2 14887  df-struct 17184  df-sets 17201  df-slot 17219  df-ndx 17231  df-base 17248  df-ress 17275  df-plusg 17310  df-tset 17316  df-0g 17486  df-gsum 17487  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-submnd 18797  df-efmnd 18882  df-grp 18954  df-minusg 18955  df-subg 19141  df-symg 19387  df-pmtr 19460
This theorem is referenced by:  psgnunilem4  19515
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