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Theorem psgnunilem3 18122
Description: Lemma for psgnuni 18125. 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 2850 . . 3 (𝜑𝐿 ∈ ℕ)
43nnnn0d 11552 . 2 (𝜑𝐿 ∈ ℕ0)
5 psgnunilem3.w1 . . . . . . 7 (𝜑𝑊 ∈ Word 𝑇)
6 wrdf 13505 . . . . . . 7 (𝑊 ∈ Word 𝑇𝑊:(0..^(♯‘𝑊))⟶𝑇)
75, 6syl 17 . . . . . 6 (𝜑𝑊:(0..^(♯‘𝑊))⟶𝑇)
8 0nn0 11508 . . . . . . . . 9 0 ∈ ℕ0
98a1i 11 . . . . . . . 8 (𝜑 → 0 ∈ ℕ0)
103nngt0d 11265 . . . . . . . 8 (𝜑 → 0 < 𝐿)
11 elfzo0 12716 . . . . . . . 8 (0 ∈ (0..^𝐿) ↔ (0 ∈ ℕ0𝐿 ∈ ℕ ∧ 0 < 𝐿))
129, 3, 10, 11syl3anbrc 1427 . . . . . . 7 (𝜑 → 0 ∈ (0..^𝐿))
131oveq2d 6808 . . . . . . 7 (𝜑 → (0..^(♯‘𝑊)) = (0..^𝐿))
1412, 13eleqtrrd 2852 . . . . . 6 (𝜑 → 0 ∈ (0..^(♯‘𝑊)))
157, 14ffvelrnd 6503 . . . . 5 (𝜑 → (𝑊‘0) ∈ 𝑇)
16 eqid 2770 . . . . . 6 (pmTrsp‘𝐷) = (pmTrsp‘𝐷)
17 psgnunilem3.t . . . . . 6 𝑇 = ran (pmTrsp‘𝐷)
1816, 17pmtrfmvdn0 18088 . . . . 5 ((𝑊‘0) ∈ 𝑇 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
1915, 18syl 17 . . . 4 (𝜑 → dom ((𝑊‘0) ∖ I ) ≠ ∅)
20 n0 4076 . . . 4 (dom ((𝑊‘0) ∖ I ) ≠ ∅ ↔ ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
2119, 20sylib 208 . . 3 (𝜑 → ∃𝑒 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
22 fzonel 12690 . . . . . . . 8 ¬ 𝐿 ∈ (0..^𝐿)
23 simpr1 1232 . . . . . . . 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 188 . . . . . . 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 3147 . . . . 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 2837 . . . . . . . . . 10 (𝑎 = 0 → (𝑎 ∈ (0..^𝐿) ↔ 0 ∈ (0..^𝐿)))
28 fveq2 6332 . . . . . . . . . . . . 13 (𝑎 = 0 → (𝑤𝑎) = (𝑤‘0))
2928difeq1d 3876 . . . . . . . . . . . 12 (𝑎 = 0 → ((𝑤𝑎) ∖ I ) = ((𝑤‘0) ∖ I ))
3029dmeqd 5464 . . . . . . . . . . 11 (𝑎 = 0 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘0) ∖ I ))
3130eleq2d 2835 . . . . . . . . . 10 (𝑎 = 0 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘0) ∖ I )))
32 oveq2 6800 . . . . . . . . . . 11 (𝑎 = 0 → (0..^𝑎) = (0..^0))
3332raleqdv 3292 . . . . . . . . . 10 (𝑎 = 0 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
3427, 31, 333anbi123d 1546 . . . . . . . . 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 606 . . . . . . . 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 3199 . . . . . . 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 329 . . . . . 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 2837 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑎 ∈ (0..^𝐿) ↔ 𝑏 ∈ (0..^𝐿)))
39 fveq2 6332 . . . . . . . . . . . . . 14 (𝑎 = 𝑏 → (𝑤𝑎) = (𝑤𝑏))
4039difeq1d 3876 . . . . . . . . . . . . 13 (𝑎 = 𝑏 → ((𝑤𝑎) ∖ I ) = ((𝑤𝑏) ∖ I ))
4140dmeqd 5464 . . . . . . . . . . . 12 (𝑎 = 𝑏 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝑏) ∖ I ))
4241eleq2d 2835 . . . . . . . . . . 11 (𝑎 = 𝑏 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝑏) ∖ I )))
43 oveq2 6800 . . . . . . . . . . . 12 (𝑎 = 𝑏 → (0..^𝑎) = (0..^𝑏))
4443raleqdv 3292 . . . . . . . . . . 11 (𝑎 = 𝑏 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
4538, 42, 443anbi123d 1546 . . . . . . . . . 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 606 . . . . . . . . 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 3199 . . . . . . . 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 6800 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
4948eqeq1d 2772 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
50 fveq2 6332 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
5150eqeq1d 2772 . . . . . . . . . . 11 (𝑤 = 𝑥 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑥) = 𝐿))
5249, 51anbi12d 608 . . . . . . . . . 10 (𝑤 = 𝑥 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿)))
53 fveq1 6331 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑤𝑏) = (𝑥𝑏))
5453difeq1d 3876 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → ((𝑤𝑏) ∖ I ) = ((𝑥𝑏) ∖ I ))
5554dmeqd 5464 . . . . . . . . . . . 12 (𝑤 = 𝑥 → dom ((𝑤𝑏) ∖ I ) = dom ((𝑥𝑏) ∖ I ))
5655eleq2d 2835 . . . . . . . . . . 11 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑏) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑏) ∖ I )))
57 fveq1 6331 . . . . . . . . . . . . . . . . 17 (𝑤 = 𝑥 → (𝑤𝑐) = (𝑥𝑐))
5857difeq1d 3876 . . . . . . . . . . . . . . . 16 (𝑤 = 𝑥 → ((𝑤𝑐) ∖ I ) = ((𝑥𝑐) ∖ I ))
5958dmeqd 5464 . . . . . . . . . . . . . . 15 (𝑤 = 𝑥 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑥𝑐) ∖ I ))
6059eleq2d 2835 . . . . . . . . . . . . . 14 (𝑤 = 𝑥 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6160notbid 307 . . . . . . . . . . . . 13 (𝑤 = 𝑥 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
6261ralbidv 3134 . . . . . . . . . . . 12 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I )))
63 fveq2 6332 . . . . . . . . . . . . . . . . 17 (𝑐 = 𝑑 → (𝑥𝑐) = (𝑥𝑑))
6463difeq1d 3876 . . . . . . . . . . . . . . . 16 (𝑐 = 𝑑 → ((𝑥𝑐) ∖ I ) = ((𝑥𝑑) ∖ I ))
6564dmeqd 5464 . . . . . . . . . . . . . . 15 (𝑐 = 𝑑 → dom ((𝑥𝑐) ∖ I ) = dom ((𝑥𝑑) ∖ I ))
6665eleq2d 2835 . . . . . . . . . . . . . 14 (𝑐 = 𝑑 → (𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6766notbid 307 . . . . . . . . . . . . 13 (𝑐 = 𝑑 → (¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
6867cbvralv 3319 . . . . . . . . . . . 12 (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))
6962, 68syl6bb 276 . . . . . . . . . . 11 (𝑤 = 𝑥 → (∀𝑐 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I )))
7056, 693anbi23d 1549 . . . . . . . . . 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 ))))
7152, 70anbi12d 608 . . . . . . . . 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 )))))
7271cbvrexv 3320 . . . . . . . 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 ))))
7347, 72syl6bb 276 . . . . . . 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 )))))
7473imbi2d 329 . . . . . 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 ))))))
75 eleq1 2837 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑎 ∈ (0..^𝐿) ↔ (𝑏 + 1) ∈ (0..^𝐿)))
76 fveq2 6332 . . . . . . . . . . . . 13 (𝑎 = (𝑏 + 1) → (𝑤𝑎) = (𝑤‘(𝑏 + 1)))
7776difeq1d 3876 . . . . . . . . . . . 12 (𝑎 = (𝑏 + 1) → ((𝑤𝑎) ∖ I ) = ((𝑤‘(𝑏 + 1)) ∖ I ))
7877dmeqd 5464 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤‘(𝑏 + 1)) ∖ I ))
7978eleq2d 2835 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤‘(𝑏 + 1)) ∖ I )))
80 oveq2 6800 . . . . . . . . . . 11 (𝑎 = (𝑏 + 1) → (0..^𝑎) = (0..^(𝑏 + 1)))
8180raleqdv 3292 . . . . . . . . . 10 (𝑎 = (𝑏 + 1) → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^(𝑏 + 1)) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
8275, 79, 813anbi123d 1546 . . . . . . . . 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 ))))
8382anbi2d 606 . . . . . . . 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 )))))
8483rexbidv 3199 . . . . . . 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 )))))
8584imbi2d 329 . . . . . 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 ))))))
86 eleq1 2837 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑎 ∈ (0..^𝐿) ↔ 𝐿 ∈ (0..^𝐿)))
87 fveq2 6332 . . . . . . . . . . . . 13 (𝑎 = 𝐿 → (𝑤𝑎) = (𝑤𝐿))
8887difeq1d 3876 . . . . . . . . . . . 12 (𝑎 = 𝐿 → ((𝑤𝑎) ∖ I ) = ((𝑤𝐿) ∖ I ))
8988dmeqd 5464 . . . . . . . . . . 11 (𝑎 = 𝐿 → dom ((𝑤𝑎) ∖ I ) = dom ((𝑤𝐿) ∖ I ))
9089eleq2d 2835 . . . . . . . . . 10 (𝑎 = 𝐿 → (𝑒 ∈ dom ((𝑤𝑎) ∖ I ) ↔ 𝑒 ∈ dom ((𝑤𝐿) ∖ I )))
91 oveq2 6800 . . . . . . . . . . 11 (𝑎 = 𝐿 → (0..^𝑎) = (0..^𝐿))
9291raleqdv 3292 . . . . . . . . . 10 (𝑎 = 𝐿 → (∀𝑐 ∈ (0..^𝑎) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^𝐿) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I )))
9386, 90, 923anbi123d 1546 . . . . . . . . 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 ))))
9493anbi2d 606 . . . . . . . 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 )))))
9594rexbidv 3199 . . . . . . 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 )))))
9695imbi2d 329 . . . . . 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 ))))))
975adantr 466 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑊 ∈ Word 𝑇)
98 psgnunilem3.w3 . . . . . . . . 9 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
9998, 1jca 495 . . . . . . . 8 (𝜑 → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
10099adantr 466 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿))
10112adantr 466 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 0 ∈ (0..^𝐿))
102 simpr 471 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → 𝑒 ∈ dom ((𝑊‘0) ∖ I ))
103 ral0 4215 . . . . . . . . . 10 𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
104 fzo0 12699 . . . . . . . . . . 11 (0..^0) = ∅
105104raleqi 3290 . . . . . . . . . 10 (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ) ↔ ∀𝑐 ∈ ∅ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
106103, 105mpbir 221 . . . . . . . . 9 𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )
107106a1i 11 . . . . . . . 8 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I ))
108101, 102, 1073jca 1121 . . . . . . 7 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → (0 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑊‘0) ∖ I ) ∧ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
109 oveq2 6800 . . . . . . . . . . 11 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
110109eqeq1d 2772 . . . . . . . . . 10 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
111 fveq2 6332 . . . . . . . . . . 11 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
112111eqeq1d 2772 . . . . . . . . . 10 (𝑤 = 𝑊 → ((♯‘𝑤) = 𝐿 ↔ (♯‘𝑊) = 𝐿))
113110, 112anbi12d 608 . . . . . . . . 9 (𝑤 = 𝑊 → (((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ∧ (♯‘𝑤) = 𝐿) ↔ ((𝐺 Σg 𝑊) = ( I ↾ 𝐷) ∧ (♯‘𝑊) = 𝐿)))
114 fveq1 6331 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → (𝑤‘0) = (𝑊‘0))
115114difeq1d 3876 . . . . . . . . . . . 12 (𝑤 = 𝑊 → ((𝑤‘0) ∖ I ) = ((𝑊‘0) ∖ I ))
116115dmeqd 5464 . . . . . . . . . . 11 (𝑤 = 𝑊 → dom ((𝑤‘0) ∖ I ) = dom ((𝑊‘0) ∖ I ))
117116eleq2d 2835 . . . . . . . . . 10 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤‘0) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊‘0) ∖ I )))
118 fveq1 6331 . . . . . . . . . . . . . . 15 (𝑤 = 𝑊 → (𝑤𝑐) = (𝑊𝑐))
119118difeq1d 3876 . . . . . . . . . . . . . 14 (𝑤 = 𝑊 → ((𝑤𝑐) ∖ I ) = ((𝑊𝑐) ∖ I ))
120119dmeqd 5464 . . . . . . . . . . . . 13 (𝑤 = 𝑊 → dom ((𝑤𝑐) ∖ I ) = dom ((𝑊𝑐) ∖ I ))
121120eleq2d 2835 . . . . . . . . . . . 12 (𝑤 = 𝑊 → (𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
122121notbid 307 . . . . . . . . . . 11 (𝑤 = 𝑊 → (¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
123122ralbidv 3134 . . . . . . . . . 10 (𝑤 = 𝑊 → (∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑤𝑐) ∖ I ) ↔ ∀𝑐 ∈ (0..^0) ¬ 𝑒 ∈ dom ((𝑊𝑐) ∖ I )))
124117, 1233anbi23d 1549 . . . . . . . . 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 ))))
125113, 124anbi12d 608 . . . . . . . 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 )))))
126125rspcev 3458 . . . . . . 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 ))))
12797, 100, 108, 126syl12anc 1473 . . . . . 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 ))))
128 psgnunilem3.g . . . . . . . . . 10 𝐺 = (SymGrp‘𝐷)
129 psgnunilem3.d . . . . . . . . . . 11 (𝜑𝐷𝑉)
130129ad2antrr 697 . . . . . . . . . 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)
131 simprl 746 . . . . . . . . . 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)
132 simpll 742 . . . . . . . . . . 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))
133132ad2antll 700 . . . . . . . . . 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))
134 simplr 744 . . . . . . . . . . 11 ((((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ∧ (♯‘𝑥) = 𝐿) ∧ (𝑏 ∈ (0..^𝐿) ∧ 𝑒 ∈ dom ((𝑥𝑏) ∖ I ) ∧ ∀𝑑 ∈ (0..^𝑏) ¬ 𝑒 ∈ dom ((𝑥𝑑) ∖ I ))) → (♯‘𝑥) = 𝐿)
135134ad2antll 700 . . . . . . . . . 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)
136 simpr1 1232 . . . . . . . . . . 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))
137136ad2antll 700 . . . . . . . . . 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))
138 simpr2 1234 . . . . . . . . . . 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 ))
139138ad2antll 700 . . . . . . . . . 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 ))
140 simpr3 1236 . . . . . . . . . . 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 ))
141140ad2antll 700 . . . . . . . . . 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 ))
142 psgnunilem3.in . . . . . . . . . . . 12 (𝜑 → ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
143 fveq2 6332 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (♯‘𝑥) = (♯‘𝑦))
144143eqeq1d 2772 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((♯‘𝑥) = (𝐿 − 2) ↔ (♯‘𝑦) = (𝐿 − 2)))
145 oveq2 6800 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
146145eqeq1d 2772 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
147144, 146anbi12d 608 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
148147cbvrexv 3320 . . . . . . . . . . . 12 (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = (𝐿 − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
149142, 148sylnib 317 . . . . . . . . . . 11 (𝜑 → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = (𝐿 − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
150149ad2antrr 697 . . . . . . . . . 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)))
151128, 17, 130, 131, 133, 135, 137, 139, 141, 150psgnunilem2 18121 . . . . . . . . 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 ))))
152151rexlimdvaa 3179 . . . . . . . 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 )))))
153152a2i 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 )))))
154153a1i 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 ))))))
15537, 74, 85, 96, 127, 154nn0ind 11673 . . . . 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 )))))
15626, 155mtoi 190 . . . 4 (𝐿 ∈ ℕ0 → ¬ (𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )))
157156con2i 136 . . 3 ((𝜑𝑒 ∈ dom ((𝑊‘0) ∖ I )) → ¬ 𝐿 ∈ ℕ0)
15821, 157exlimddv 2014 . 2 (𝜑 → ¬ 𝐿 ∈ ℕ0)
1594, 158pm2.65i 185 1 ¬ 𝜑
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  w3a 1070   = wceq 1630  wex 1851  wcel 2144  wne 2942  wral 3060  wrex 3061  cdif 3718  c0 4061   class class class wbr 4784   I cid 5156  dom cdm 5249  ran crn 5250  cres 5251  wf 6027  cfv 6031  (class class class)co 6792  0cc0 10137  1c1 10138   + caddc 10140   < clt 10275  cmin 10467  cn 11221  2c2 11271  0cn0 11493  ..^cfzo 12672  chash 13320  Word cword 13486   Σg cgsu 16308  SymGrpcsymg 18003  pmTrspcpmtr 18067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-xor 1612  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-ot 4323  df-uni 4573  df-int 4610  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-om 7212  df-1st 7314  df-2nd 7315  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-2o 7713  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-card 8964  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-7 11285  df-8 11286  df-9 11287  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-fzo 12673  df-seq 13008  df-hash 13321  df-word 13494  df-lsw 13495  df-concat 13496  df-s1 13497  df-substr 13498  df-splice 13499  df-s2 13801  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-tset 16167  df-0g 16309  df-gsum 16310  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-grp 17632  df-minusg 17633  df-subg 17798  df-symg 18004  df-pmtr 18068
This theorem is referenced by:  psgnunilem4  18123
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