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Theorem psgnunilem3 19200
Description: Lemma for psgnuni 19203. 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 2838 . . 3 (𝜑𝐿 ∈ ℕ)
43nnnn0d 12394 . 2 (𝜑𝐿 ∈ ℕ0)
5 psgnunilem3.w1 . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
6 wrdf 14322 . . . . . . 7 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
75, 6syl 17 . . . . . 6 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8 0nn0 12349 . . . . . . . . 9 0 ∈ ℕ0
98a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℕ0)
103nngt0d 12123 . . . . . . . 8 (𝜑 → 0 < 𝐿)
11 elfzo0 13529 . . . . . . . 8 (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0𝐿 ∈ ℕ ∧ 0 < 𝐿))
129, 3, 10, 11syl3anbrc 1342 . . . . . . 7 (𝜑 → 0 ∈ (0..^𝐿))
131oveq2d 7353 . . . . . . 7 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
1412, 13eleqtrrd 2840 . . . . . 6 (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
157, 14ffvelcdmd 7018 . . . . 5 (𝜑 → (𝑊‘0) ∈ 𝑇)
16 eqid 2736 . . . . . 6 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17 psgnunilem3.t . . . . . 6 𝑇 = ran (pmTrsp‘𝐷)
1816, 17pmtrfmvdn0 19166 . . . . 5 ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
1915, 18syl 17 . . . 4 (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20 n0 4293 . . . 4 (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
2119, 20sylib 217 . . 3 (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22 fzonel 13502 . . . . . . . 8 ¬ 𝐿 ∈ (0..^𝐿)
23 simpr1 1193 . . . . . . . 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 196 . . . . . . 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 2824 . . . . . . . . . 10 (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28 fveq2 6825 . . . . . . . . . . . . 13 (𝑎 = 0 → (𝑤𝑎) = (𝑤‘0))
2928difeq1d 4068 . . . . . . . . . . . 12 (𝑎 = 0 → ((𝑤𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
3029dmeqd 5847 . . . . . . . . . . 11 (𝑎 = 0 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
3130eleq2d 2822 . . . . . . . . . 10 (𝑎 = 0 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32 oveq2 7345 . . . . . . . . . . 11 (𝑎 = 0 → (0..^𝑎) = (0..^0))
3332raleqdv 3309 . . . . . . . . . 10 (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
3427, 31, 333anbi123d 1435 . . . . . . . . 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 629 . . . . . . . 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 3171 . . . . . . 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 2824 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39 fveq2 6825 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑤𝑎) = (𝑤𝑏))
4039difeq1d 4068 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑤𝑎) ∖ I ) = ((𝑤𝑏) ∖ I ))
4140dmeqd 5847 . . . . . . . . . . . 12 (𝑎 = 𝑏 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝑏) ∖ I ))
4241eleq2d 2822 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝑏) ∖ I )))
43 oveq2 7345 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
4443raleqdv 3309 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
4538, 42, 443anbi123d 1435 . . . . . . . . . 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 629 . . . . . . . . 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 3171 . . . . . . . 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 7345 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
4948eqeq1d 2738 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50 fveqeq2 6834 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
5149, 50anbi12d 631 . . . . . . . . . 10 (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52 fveq1 6824 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑏) = (𝑥𝑏))
5352difeq1d 4068 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → ((𝑤𝑏) ∖ I ) = ((𝑥𝑏) ∖ I ))
5453dmeqd 5847 . . . . . . . . . . . 12 (𝑤 = 𝑥 → dom ((𝑤𝑏) ∖ I ) = dom ((𝑥𝑏) ∖ I ))
5554eleq2d 2822 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑏) ∖ I )))
56 fveq1 6824 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (𝑤𝑐) = (𝑥𝑐))
5756difeq1d 4068 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥 → ((𝑤𝑐) ∖ I ) = ((𝑥𝑐) ∖ I ))
5857dmeqd 5847 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑥𝑐) ∖ I ))
5958eleq2d 2822 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6059notbid 317 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6160ralbidv 3170 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
62 fveq2 6825 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑑 → (𝑥𝑐) = (𝑥𝑑))
6362difeq1d 4068 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑑 → ((𝑥𝑐) ∖ I ) = ((𝑥𝑑) ∖ I ))
6463dmeqd 5847 . . . . . . . . . . . . . . 15 (𝑐 = 𝑑 → dom ((𝑥𝑐) ∖ I ) = dom ((𝑥𝑑) ∖ I ))
6564eleq2d 2822 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6665notbid 317 . . . . . . . . . . . . 13 (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6766cbvralvw 3221 . . . . . . . . . . . 12 (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
6861, 67bitrdi 286 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6955, 683anbi23d 1438 . . . . . . . . . 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 631 . . . . . . . . 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 3222 . . . . . . . 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 286 . . . . . . 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 2824 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75 fveq2 6825 . . . . . . . . . . . . 13 (𝑎 = (𝑏 + 1) → (𝑤𝑎) = (𝑤‘(𝑏 + 1)))
7675difeq1d 4068 . . . . . . . . . . . 12 (𝑎 = (𝑏 + 1) → ((𝑤𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
7776dmeqd 5847 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
7877eleq2d 2822 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79 oveq2 7345 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
8079raleqdv 3309 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
8174, 78, 803anbi123d 1435 . . . . . . . . 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 629 . . . . . . . 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 3171 . . . . . . 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 2824 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86 fveq2 6825 . . . . . . . . . . . . 13 (𝑎 = 𝐿 → (𝑤𝑎) = (𝑤𝐿))
8786difeq1d 4068 . . . . . . . . . . . 12 (𝑎 = 𝐿 → ((𝑤𝑎) ∖ I ) = ((𝑤𝐿) ∖ I ))
8887dmeqd 5847 . . . . . . . . . . 11 (𝑎 = 𝐿 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝐿) ∖ I ))
8988eleq2d 2822 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝐿) ∖ I )))
90 oveq2 7345 . . . . . . . . . . 11 (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
9190raleqdv 3309 . . . . . . . . . 10 (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
9285, 89, 913anbi123d 1435 . . . . . . . . 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 629 . . . . . . . 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 3171 . . . . . . 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 481 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇)
97 psgnunilem3.w3 . . . . . . . . 9 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
9897, 1jca 512 . . . . . . . 8 (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
9998adantr 481 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
10012adantr 481 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈ (0..^𝐿))
101 simpr 485 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
102 ral0 4457 . . . . . . . . . 10 𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
103 fzo0 13512 . . . . . . . . . . 11 (0..^0) = ∅
104103raleqi 3307 . . . . . . . . . 10 (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
105102, 104mpbir 230 . . . . . . . . 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 1127 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
108 oveq2 7345 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
109108eqeq1d 2738 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
110 fveqeq2 6834 . . . . . . . . . 10 (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111109, 110anbi12d 631 . . . . . . . . 9 (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112 fveq1 6824 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113112difeq1d 4068 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114113dmeqd 5847 . . . . . . . . . . 11 (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115114eleq2d 2822 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116 fveq1 6824 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑤𝑐) = (𝑊𝑐))
117116difeq1d 4068 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((𝑤𝑐) ∖ I ) = ((𝑊𝑐) ∖ I ))
118117dmeqd 5847 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑊𝑐) ∖ I ))
119118eleq2d 2822 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
120119notbid 317 . . . . . . . . . . 11 (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
121120ralbidv 3170 . . . . . . . . . 10 (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
122115, 1213anbi23d 1438 . . . . . . . . 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 631 . . . . . . . 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 3570 . . . . . . 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 834 . . . . . 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 723 . . . . . . . . . 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 768 . . . . . . . . . 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 764 . . . . . . . . . . 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 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 ))))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
132 simplr 766 . . . . . . . . . . 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 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 ))))) → (♯‘𝑥) = 𝐿)
134 simpr1 1193 . . . . . . . . . . 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 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 ))))) → 𝑏 ∈ (0..^𝐿))
136 simpr2 1194 . . . . . . . . . . 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 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 ))))) → 𝑒 ∈ dom ((𝑥𝑏) ∖ I ))
138 simpr3 1195 . . . . . . . . . . 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 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 ))))) → ∀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 6834 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142 oveq2 7345 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
143142eqeq1d 2738 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
144141, 143anbi12d 631 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145144cbvrexvw 3222 . . . . . . . . . . . 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 327 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147146ad2antrr 723 . . . . . . . . . 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 19199 . . . . . . . . 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 3149 . . . . . . . 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 12516 . . . . 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 198 . . . 4 (𝐿 ∈ ℕ0 → ¬ (𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154153con2i 139 . . 3 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
15521, 154exlimddv 1937 . 2 (𝜑 → ¬ 𝐿 ∈ ℕ0)
1564, 155pm2.65i 193 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1086   = wceq 1540  wex 1780  wcel 2105  wne 2940  wral 3061  wrex 3070  cdif 3895  c0 4269   class class class wbr 5092   I cid 5517  dom cdm 5620  ran crn 5621  cres 5622  wf 6475  cfv 6479  (class class class)co 7337  0cc0 10972  1c1 10973   + caddc 10975   < clt 11110  cmin 11306  cn 12074  2c2 12129  0cn0 12334  ..^cfzo 13483  chash 14145  Word cword 14317   Σg cgsu 17248  SymGrpcsymg 19070  pmTrspcpmtr 19145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-xor 1509  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-ot 4582  df-uni 4853  df-int 4895  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-2o 8368  df-er 8569  df-map 8688  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-card 9796  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-nn 12075  df-2 12137  df-3 12138  df-4 12139  df-5 12140  df-6 12141  df-7 12142  df-8 12143  df-9 12144  df-n0 12335  df-xnn0 12407  df-z 12421  df-uz 12684  df-fz 13341  df-fzo 13484  df-seq 13823  df-hash 14146  df-word 14318  df-lsw 14366  df-concat 14374  df-s1 14400  df-substr 14452  df-pfx 14482  df-splice 14561  df-s2 14660  df-struct 16945  df-sets 16962  df-slot 16980  df-ndx 16992  df-base 17010  df-ress 17039  df-plusg 17072  df-tset 17078  df-0g 17249  df-gsum 17250  df-mgm 18423  df-sgrp 18472  df-mnd 18483  df-submnd 18528  df-efmnd 18604  df-grp 18676  df-minusg 18677  df-subg 18848  df-symg 19071  df-pmtr 19146
This theorem is referenced by:  psgnunilem4  19201
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