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Theorem psgnunilem3 18553
Description: Lemma for psgnuni 18556. 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 2911 . . 3 (𝜑𝐿 ∈ ℕ)
43nnnn0d 11943 . 2 (𝜑𝐿 ∈ ℕ0)
5 psgnunilem3.w1 . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
6 wrdf 13854 . . . . . . 7 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
75, 6syl 17 . . . . . 6 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8 0nn0 11900 . . . . . . . . 9 0 ∈ ℕ0
98a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℕ0)
103nngt0d 11674 . . . . . . . 8 (𝜑 → 0 < 𝐿)
11 elfzo0 13066 . . . . . . . 8 (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0𝐿 ∈ ℕ ∧ 0 < 𝐿))
129, 3, 10, 11syl3anbrc 1335 . . . . . . 7 (𝜑 → 0 ∈ (0..^𝐿))
131oveq2d 7161 . . . . . . 7 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
1412, 13eleqtrrd 2913 . . . . . 6 (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
157, 14ffvelrnd 6844 . . . . 5 (𝜑 → (𝑊‘0) ∈ 𝑇)
16 eqid 2818 . . . . . 6 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17 psgnunilem3.t . . . . . 6 𝑇 = ran (pmTrsp‘𝐷)
1816, 17pmtrfmvdn0 18519 . . . . 5 ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
1915, 18syl 17 . . . 4 (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20 n0 4307 . . . 4 (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
2119, 20sylib 219 . . 3 (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22 fzonel 13039 . . . . . . . 8 ¬ 𝐿 ∈ (0..^𝐿)
23 simpr1 1186 . . . . . . . 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 198 . . . . . . 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 3266 . . . . 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 2897 . . . . . . . . . 10 (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28 fveq2 6663 . . . . . . . . . . . . 13 (𝑎 = 0 → (𝑤𝑎) = (𝑤‘0))
2928difeq1d 4095 . . . . . . . . . . . 12 (𝑎 = 0 → ((𝑤𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
3029dmeqd 5767 . . . . . . . . . . 11 (𝑎 = 0 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
3130eleq2d 2895 . . . . . . . . . 10 (𝑎 = 0 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32 oveq2 7153 . . . . . . . . . . 11 (𝑎 = 0 → (0..^𝑎) = (0..^0))
3332raleqdv 3413 . . . . . . . . . 10 (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
3427, 31, 333anbi123d 1427 . . . . . . . . 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 628 . . . . . . . 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 3294 . . . . . . 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 342 . . . . . 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 2897 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39 fveq2 6663 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑤𝑎) = (𝑤𝑏))
4039difeq1d 4095 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑤𝑎) ∖ I ) = ((𝑤𝑏) ∖ I ))
4140dmeqd 5767 . . . . . . . . . . . 12 (𝑎 = 𝑏 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝑏) ∖ I ))
4241eleq2d 2895 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝑏) ∖ I )))
43 oveq2 7153 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
4443raleqdv 3413 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
4538, 42, 443anbi123d 1427 . . . . . . . . . 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 628 . . . . . . . . 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 3294 . . . . . . . 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 7153 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
4948eqeq1d 2820 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50 fveqeq2 6672 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
5149, 50anbi12d 630 . . . . . . . . . 10 (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
52 fveq1 6662 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑏) = (𝑥𝑏))
5352difeq1d 4095 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → ((𝑤𝑏) ∖ I ) = ((𝑥𝑏) ∖ I ))
5453dmeqd 5767 . . . . . . . . . . . 12 (𝑤 = 𝑥 → dom ((𝑤𝑏) ∖ I ) = dom ((𝑥𝑏) ∖ I ))
5554eleq2d 2895 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑏) ∖ I )))
56 fveq1 6662 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (𝑤𝑐) = (𝑥𝑐))
5756difeq1d 4095 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥 → ((𝑤𝑐) ∖ I ) = ((𝑥𝑐) ∖ I ))
5857dmeqd 5767 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑥𝑐) ∖ I ))
5958eleq2d 2895 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6059notbid 319 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6160ralbidv 3194 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
62 fveq2 6663 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑑 → (𝑥𝑐) = (𝑥𝑑))
6362difeq1d 4095 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑑 → ((𝑥𝑐) ∖ I ) = ((𝑥𝑑) ∖ I ))
6463dmeqd 5767 . . . . . . . . . . . . . . 15 (𝑐 = 𝑑 → dom ((𝑥𝑐) ∖ I ) = dom ((𝑥𝑑) ∖ I ))
6564eleq2d 2895 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6665notbid 319 . . . . . . . . . . . . 13 (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6766cbvralvw 3447 . . . . . . . . . . . 12 (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
6861, 67syl6bb 288 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6955, 683anbi23d 1430 . . . . . . . . . 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 630 . . . . . . . . 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 3448 . . . . . . . 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, 71syl6bb 288 . . . . . . 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 342 . . . . . 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 2897 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
75 fveq2 6663 . . . . . . . . . . . . 13 (𝑎 = (𝑏 + 1) → (𝑤𝑎) = (𝑤‘(𝑏 + 1)))
7675difeq1d 4095 . . . . . . . . . . . 12 (𝑎 = (𝑏 + 1) → ((𝑤𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
7776dmeqd 5767 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
7877eleq2d 2895 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
79 oveq2 7153 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
8079raleqdv 3413 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
8174, 78, 803anbi123d 1427 . . . . . . . . 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 628 . . . . . . . 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 3294 . . . . . . 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 342 . . . . . 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 2897 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
86 fveq2 6663 . . . . . . . . . . . . 13 (𝑎 = 𝐿 → (𝑤𝑎) = (𝑤𝐿))
8786difeq1d 4095 . . . . . . . . . . . 12 (𝑎 = 𝐿 → ((𝑤𝑎) ∖ I ) = ((𝑤𝐿) ∖ I ))
8887dmeqd 5767 . . . . . . . . . . 11 (𝑎 = 𝐿 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝐿) ∖ I ))
8988eleq2d 2895 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝐿) ∖ I )))
90 oveq2 7153 . . . . . . . . . . 11 (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
9190raleqdv 3413 . . . . . . . . . 10 (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
9285, 89, 913anbi123d 1427 . . . . . . . . 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 628 . . . . . . . 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 3294 . . . . . . 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 342 . . . . . 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 4452 . . . . . . . . . 10 𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
103 fzo0 13049 . . . . . . . . . . 11 (0..^0) = ∅
104103raleqi 3411 . . . . . . . . . 10 (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
105102, 104mpbir 232 . . . . . . . . 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 1120 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
108 oveq2 7153 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
109108eqeq1d 2820 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
110 fveqeq2 6672 . . . . . . . . . 10 (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
111109, 110anbi12d 630 . . . . . . . . 9 (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
112 fveq1 6662 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
113112difeq1d 4095 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
114113dmeqd 5767 . . . . . . . . . . 11 (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
115114eleq2d 2895 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
116 fveq1 6662 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑤𝑐) = (𝑊𝑐))
117116difeq1d 4095 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((𝑤𝑐) ∖ I ) = ((𝑊𝑐) ∖ I ))
118117dmeqd 5767 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑊𝑐) ∖ I ))
119118eleq2d 2895 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
120119notbid 319 . . . . . . . . . . 11 (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
121120ralbidv 3194 . . . . . . . . . 10 (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
122115, 1213anbi23d 1430 . . . . . . . . 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 630 . . . . . . . 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 3620 . . . . . . 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 832 . . . . . 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 722 . . . . . . . . . 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 767 . . . . . . . . . 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 763 . . . . . . . . . . 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 725 . . . . . . . . . 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 765 . . . . . . . . . . 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 725 . . . . . . . . . 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 1186 . . . . . . . . . . 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 725 . . . . . . . . . 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 1187 . . . . . . . . . . 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 725 . . . . . . . . . 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 1188 . . . . . . . . . . 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 725 . . . . . . . . . 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 6672 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
142 oveq2 7153 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
143142eqeq1d 2820 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
144141, 143anbi12d 630 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
145144cbvrexvw 3448 . . . . . . . . . . . 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 329 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147146ad2antrr 722 . . . . . . . . . 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 18552 . . . . . . . . 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 3282 . . . . . . . 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 12065 . . . . 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 200 . . . 4 (𝐿 ∈ ℕ0 → ¬ (𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )))
154153con2i 141 . . 3 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
15521, 154exlimddv 1927 . 2 (𝜑 → ¬ 𝐿 ∈ ℕ0)
1564, 155pm2.65i 195 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  cdif 3930  c0 4288   class class class wbr 5057   I cid 5452  dom cdm 5548  ran crn 5549  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  0cc0 10525  1c1 10526   + caddc 10528   < clt 10663  cmin 10858  cn 11626  2c2 11680  0cn0 11885  ..^cfzo 13021  chash 13678  Word cword 13849   Σg cgsu 16702  SymGrpcsymg 18433  pmTrspcpmtr 18498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-xor 1496  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-ot 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-card 9356  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-xnn0 11956  df-z 11970  df-uz 12232  df-fz 12881  df-fzo 13022  df-seq 13358  df-hash 13679  df-word 13850  df-lsw 13903  df-concat 13911  df-s1 13938  df-substr 13991  df-pfx 14021  df-splice 14100  df-s2 14198  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-sets 16478  df-ress 16479  df-plusg 16566  df-tset 16572  df-0g 16703  df-gsum 16704  df-mgm 17840  df-sgrp 17889  df-mnd 17900  df-submnd 17945  df-grp 18044  df-minusg 18045  df-subg 18214  df-symg 18434  df-pmtr 18499
This theorem is referenced by:  psgnunilem4  18554
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