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Theorem psgnunilem3 18888
Description: Lemma for psgnuni 18891. 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 2839 . . 3 (𝜑𝐿 ∈ ℕ)
43nnnn0d 12150 . 2 (𝜑𝐿 ∈ ℕ0)
5 psgnunilem3.w1 . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
6 wrdf 14074 . . . . . . 7 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
75, 6syl 17 . . . . . 6 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8 0nn0 12105 . . . . . . . . 9 0 ∈ ℕ0
98a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℕ0)
103nngt0d 11879 . . . . . . . 8 (𝜑 → 0 < 𝐿)
11 elfzo0 13283 . . . . . . . 8 (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0𝐿 ∈ ℕ ∧ 0 < 𝐿))
129, 3, 10, 11syl3anbrc 1345 . . . . . . 7 (𝜑 → 0 ∈ (0..^𝐿))
131oveq2d 7229 . . . . . . 7 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
1412, 13eleqtrrd 2841 . . . . . 6 (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
157, 14ffvelrnd 6905 . . . . 5 (𝜑 → (𝑊‘0) ∈ 𝑇)
16 eqid 2737 . . . . . 6 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17 psgnunilem3.t . . . . . 6 𝑇 = ran (pmTrsp‘𝐷)
1816, 17pmtrfmvdn0 18854 . . . . 5 ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
1915, 18syl 17 . . . 4 (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20 n0 4261 . . . 4 (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
2119, 20sylib 221 . . 3 (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22 fzonel 13256 . . . . . . . 8 ¬ 𝐿 ∈ (0..^𝐿)
23 simpr1 1196 . . . . . . . 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 200 . . . . . . 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 3188 . . . . 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 2825 . . . . . . . . . 10 (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28 fveq2 6717 . . . . . . . . . . . . 13 (𝑎 = 0 → (𝑤𝑎) = (𝑤‘0))
2928difeq1d 4036 . . . . . . . . . . . 12 (𝑎 = 0 → ((𝑤𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
3029dmeqd 5774 . . . . . . . . . . 11 (𝑎 = 0 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
3130eleq2d 2823 . . . . . . . . . 10 (𝑎 = 0 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32 oveq2 7221 . . . . . . . . . . 11 (𝑎 = 0 → (0..^𝑎) = (0..^0))
3332raleqdv 3325 . . . . . . . . . 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 632 . . . . . . . 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 3216 . . . . . . 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 344 . . . . . 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 2825 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39 fveq2 6717 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑤𝑎) = (𝑤𝑏))
4039difeq1d 4036 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑤𝑎) ∖ I ) = ((𝑤𝑏) ∖ I ))
4140dmeqd 5774 . . . . . . . . . . . 12 (𝑎 = 𝑏 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝑏) ∖ I ))
4241eleq2d 2823 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝑏) ∖ I )))
43 oveq2 7221 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
4443raleqdv 3325 . . . . . . . . . . 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 632 . . . . . . . . 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 3216 . . . . . . . 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 7221 . . . . . . . . . . . 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 6726 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
5149, 50anbi12d 634 . . . . . . . . . 10 (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52 fveq1 6716 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑏) = (𝑥𝑏))
5352difeq1d 4036 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → ((𝑤𝑏) ∖ I ) = ((𝑥𝑏) ∖ I ))
5453dmeqd 5774 . . . . . . . . . . . 12 (𝑤 = 𝑥 → dom ((𝑤𝑏) ∖ I ) = dom ((𝑥𝑏) ∖ I ))
5554eleq2d 2823 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑏) ∖ I )))
56 fveq1 6716 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (𝑤𝑐) = (𝑥𝑐))
5756difeq1d 4036 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥 → ((𝑤𝑐) ∖ I ) = ((𝑥𝑐) ∖ I ))
5857dmeqd 5774 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑥𝑐) ∖ I ))
5958eleq2d 2823 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6059notbid 321 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6160ralbidv 3118 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
62 fveq2 6717 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑑 → (𝑥𝑐) = (𝑥𝑑))
6362difeq1d 4036 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑑 → ((𝑥𝑐) ∖ I ) = ((𝑥𝑑) ∖ I ))
6463dmeqd 5774 . . . . . . . . . . . . . . 15 (𝑐 = 𝑑 → dom ((𝑥𝑐) ∖ I ) = dom ((𝑥𝑑) ∖ I ))
6564eleq2d 2823 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6665notbid 321 . . . . . . . . . . . . 13 (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6766cbvralvw 3358 . . . . . . . . . . . 12 (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
6861, 67bitrdi 290 . . . . . . . . . . 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 634 . . . . . . . . 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 3359 . . . . . . . 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 290 . . . . . . 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 344 . . . . . 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 2825 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75 fveq2 6717 . . . . . . . . . . . . 13 (𝑎 = (𝑏 + 1) → (𝑤𝑎) = (𝑤‘(𝑏 + 1)))
7675difeq1d 4036 . . . . . . . . . . . 12 (𝑎 = (𝑏 + 1) → ((𝑤𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
7776dmeqd 5774 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
7877eleq2d 2823 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79 oveq2 7221 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
8079raleqdv 3325 . . . . . . . . . 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 632 . . . . . . . 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 3216 . . . . . . 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 344 . . . . . 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 2825 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86 fveq2 6717 . . . . . . . . . . . . 13 (𝑎 = 𝐿 → (𝑤𝑎) = (𝑤𝐿))
8786difeq1d 4036 . . . . . . . . . . . 12 (𝑎 = 𝐿 → ((𝑤𝑎) ∖ I ) = ((𝑤𝐿) ∖ I ))
8887dmeqd 5774 . . . . . . . . . . 11 (𝑎 = 𝐿 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝐿) ∖ I ))
8988eleq2d 2823 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝐿) ∖ I )))
90 oveq2 7221 . . . . . . . . . . 11 (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
9190raleqdv 3325 . . . . . . . . . 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 632 . . . . . . . 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 3216 . . . . . . 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 344 . . . . . 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 484 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇)
97 psgnunilem3.w3 . . . . . . . . 9 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
9897, 1jca 515 . . . . . . . 8 (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
9998adantr 484 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
10012adantr 484 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈ (0..^𝐿))
101 simpr 488 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
102 ral0 4424 . . . . . . . . . 10 𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
103 fzo0 13266 . . . . . . . . . . 11 (0..^0) = ∅
104103raleqi 3323 . . . . . . . . . 10 (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
105102, 104mpbir 234 . . . . . . . . 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 1130 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
108 oveq2 7221 . . . . . . . . . . 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 6726 . . . . . . . . . 10 (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111109, 110anbi12d 634 . . . . . . . . 9 (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112 fveq1 6716 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113112difeq1d 4036 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114113dmeqd 5774 . . . . . . . . . . 11 (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115114eleq2d 2823 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116 fveq1 6716 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑤𝑐) = (𝑊𝑐))
117116difeq1d 4036 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((𝑤𝑐) ∖ I ) = ((𝑊𝑐) ∖ I ))
118117dmeqd 5774 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑊𝑐) ∖ I ))
119118eleq2d 2823 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
120119notbid 321 . . . . . . . . . . 11 (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
121120ralbidv 3118 . . . . . . . . . 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 634 . . . . . . . 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 3537 . . . . . . 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 1196 . . . . . . . . . . 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 1197 . . . . . . . . . . 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 1198 . . . . . . . . . . 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 6726 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142 oveq2 7221 . . . . . . . . . . . . . . 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 634 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145144cbvrexvw 3359 . . . . . . . . . . . 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 331 . . . . . . . . . . 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 18887 . . . . . . . . 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 3204 . . . . . . . 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 12272 . . . . 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 202 . . . 4 (𝐿 ∈ ℕ0 → ¬ (𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154153con2i 141 . . 3 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
15521, 154exlimddv 1943 . 2 (𝜑 → ¬ 𝐿 ∈ ℕ0)
1564, 155pm2.65i 197 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 399  w3a 1089   = wceq 1543  wex 1787  wcel 2110  wne 2940  wral 3061  wrex 3062  cdif 3863  c0 4237   class class class wbr 5053   I cid 5454  dom cdm 5551  ran crn 5552  cres 5553  wf 6376  cfv 6380  (class class class)co 7213  0cc0 10729  1c1 10730   + caddc 10732   < clt 10867  cmin 11062  cn 11830  2c2 11885  0cn0 12090  ..^cfzo 13238  chash 13896  Word cword 14069   Σg cgsu 16945  SymGrpcsymg 18759  pmTrspcpmtr 18833
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523  ax-cnex 10785  ax-resscn 10786  ax-1cn 10787  ax-icn 10788  ax-addcl 10789  ax-addrcl 10790  ax-mulcl 10791  ax-mulrcl 10792  ax-mulcom 10793  ax-addass 10794  ax-mulass 10795  ax-distr 10796  ax-i2m1 10797  ax-1ne0 10798  ax-1rid 10799  ax-rnegex 10800  ax-rrecex 10801  ax-cnre 10802  ax-pre-lttri 10803  ax-pre-lttrn 10804  ax-pre-ltadd 10805  ax-pre-mulgt0 10806
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-xor 1508  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rmo 3069  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-pss 3885  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-tp 4546  df-op 4548  df-ot 4550  df-uni 4820  df-int 4860  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-tr 5162  df-id 5455  df-eprel 5460  df-po 5468  df-so 5469  df-fr 5509  df-we 5511  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-pred 6160  df-ord 6216  df-on 6217  df-lim 6218  df-suc 6219  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-riota 7170  df-ov 7216  df-oprab 7217  df-mpo 7218  df-om 7645  df-1st 7761  df-2nd 7762  df-wrecs 8047  df-recs 8108  df-rdg 8146  df-1o 8202  df-2o 8203  df-er 8391  df-map 8510  df-en 8627  df-dom 8628  df-sdom 8629  df-fin 8630  df-card 9555  df-pnf 10869  df-mnf 10870  df-xr 10871  df-ltxr 10872  df-le 10873  df-sub 11064  df-neg 11065  df-nn 11831  df-2 11893  df-3 11894  df-4 11895  df-5 11896  df-6 11897  df-7 11898  df-8 11899  df-9 11900  df-n0 12091  df-xnn0 12163  df-z 12177  df-uz 12439  df-fz 13096  df-fzo 13239  df-seq 13575  df-hash 13897  df-word 14070  df-lsw 14118  df-concat 14126  df-s1 14153  df-substr 14206  df-pfx 14236  df-splice 14315  df-s2 14413  df-struct 16700  df-sets 16717  df-slot 16735  df-ndx 16745  df-base 16761  df-ress 16785  df-plusg 16815  df-tset 16821  df-0g 16946  df-gsum 16947  df-mgm 18114  df-sgrp 18163  df-mnd 18174  df-submnd 18219  df-efmnd 18296  df-grp 18368  df-minusg 18369  df-subg 18540  df-symg 18760  df-pmtr 18834
This theorem is referenced by:  psgnunilem4  18889
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