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Theorem psgnunilem4 18121
Description: Lemma for psgnuni 18123. An odd-length representation of the identity is impossible, as it could be repeatedly shortened to a length of 1, but a length 1 permutation must be a transposition. (Contributed by Stefan O'Rear, 25-Aug-2015.)
Hypotheses
Ref Expression
psgnunilem4.g 𝐺 = (SymGrp‘𝐷)
psgnunilem4.t 𝑇 = ran (pmTrsp‘𝐷)
psgnunilem4.d (𝜑𝐷𝑉)
psgnunilem4.w1 (𝜑𝑊 ∈ Word 𝑇)
psgnunilem4.w2 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
Assertion
Ref Expression
psgnunilem4 (𝜑 → (-1↑(♯‘𝑊)) = 1)

Proof of Theorem psgnunilem4
Dummy variables 𝑥 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 psgnunilem4.w1 . 2 (𝜑𝑊 ∈ Word 𝑇)
2 psgnunilem4.w2 . 2 (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷))
3 wrdfin 13537 . . . . 5 (𝑊 ∈ Word 𝑇𝑊 ∈ Fin)
4 hashcl 13368 . . . . 5 (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ0)
51, 3, 43syl 18 . . . 4 (𝜑 → (♯‘𝑊) ∈ ℕ0)
6 nn0uz 11943 . . . 4 0 = (ℤ‘0)
75, 6syl6eleq 2902 . . 3 (𝜑 → (♯‘𝑊) ∈ (ℤ‘0))
8 fveq2 6411 . . . . . . . . 9 (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
9 hash0 13379 . . . . . . . . 9 (♯‘∅) = 0
108, 9syl6eq 2863 . . . . . . . 8 (𝑤 = ∅ → (♯‘𝑤) = 0)
1110oveq2d 6893 . . . . . . 7 (𝑤 = ∅ → (-1↑(♯‘𝑤)) = (-1↑0))
12 neg1cn 11409 . . . . . . . 8 -1 ∈ ℂ
13 exp0 13090 . . . . . . . 8 (-1 ∈ ℂ → (-1↑0) = 1)
1412, 13ax-mp 5 . . . . . . 7 (-1↑0) = 1
1511, 14syl6eq 2863 . . . . . 6 (𝑤 = ∅ → (-1↑(♯‘𝑤)) = 1)
16152a1d 26 . . . . 5 (𝑤 = ∅ → ((𝜑 ∧ ∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1)))
17 psgnunilem4.g . . . . . . . . . . . . 13 𝐺 = (SymGrp‘𝐷)
18 psgnunilem4.t . . . . . . . . . . . . 13 𝑇 = ran (pmTrsp‘𝐷)
19 simpl1 1235 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝜑)
20 psgnunilem4.d . . . . . . . . . . . . . 14 (𝜑𝐷𝑉)
2119, 20syl 17 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝐷𝑉)
22 simpl3l 1294 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
23 eqidd 2814 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (♯‘𝑤) = (♯‘𝑤))
24 wrdfin 13537 . . . . . . . . . . . . . . 15 (𝑤 ∈ Word 𝑇𝑤 ∈ Fin)
2522, 24syl 17 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
26 simpl2 1237 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
27 hashnncl 13378 . . . . . . . . . . . . . . 15 (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
2827biimpar 465 . . . . . . . . . . . . . 14 ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (♯‘𝑤) ∈ ℕ)
2925, 26, 28syl2anc 575 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
30 simpl3r 1296 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (𝐺 Σg 𝑤) = ( I ↾ 𝐷))
31 fveqeq2 6420 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2)))
32 oveq2 6885 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
3332eqeq1d 2815 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3431, 33anbi12d 618 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
3534cbvrexv 3368 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3635notbii 311 . . . . . . . . . . . . . . 15 (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3736biimpi 207 . . . . . . . . . . . . . 14 (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3837adantl 469 . . . . . . . . . . . . 13 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3917, 18, 21, 22, 23, 29, 30, 38psgnunilem3 18120 . . . . . . . . . . . 12 ¬ ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
40 iman 390 . . . . . . . . . . . 12 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) ↔ ¬ ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
4139, 40mpbir 222 . . . . . . . . . . 11 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
42 df-rex 3109 . . . . . . . . . . 11 (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
4341, 42sylib 209 . . . . . . . . . 10 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
44 simprl 778 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 𝑥 ∈ Word 𝑇)
45 simprrr 791 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷))
4644, 45jca 503 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
47 wrdfin 13537 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Word 𝑇𝑥 ∈ Fin)
48 hashcl 13368 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ0)
4944, 47, 483syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) ∈ ℕ0)
50 simp3l 1251 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
5150, 24syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
52 simp2 1160 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
5351, 52, 28syl2anc 575 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
5453adantr 468 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℕ)
55 simprrl 790 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) = ((♯‘𝑤) − 2))
5654nnred 11323 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℝ)
57 2rp 12054 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ+
58 ltsubrp 12083 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) < (♯‘𝑤))
5956, 57, 58sylancl 576 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((♯‘𝑤) − 2) < (♯‘𝑤))
6055, 59eqbrtrd 4873 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) < (♯‘𝑤))
61 elfzo0 12736 . . . . . . . . . . . . . . . . 17 ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧ (♯‘𝑤) ∈ ℕ ∧ (♯‘𝑥) < (♯‘𝑤)))
6249, 54, 60, 61syl3anbrc 1436 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) ∈ (0..^(♯‘𝑤)))
63 id 22 . . . . . . . . . . . . . . . . 17 (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)))
6463com13 88 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑥)) = 1)))
6546, 62, 64sylc 65 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑥)) = 1))
6655oveq2d 6893 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(♯‘𝑥)) = (-1↑((♯‘𝑤) − 2)))
6712a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ∈ ℂ)
68 neg1ne0 11411 . . . . . . . . . . . . . . . . . . 19 -1 ≠ 0
6968a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ≠ 0)
70 2z 11678 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
7170a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 2 ∈ ℤ)
7254nnzd 11750 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈ ℤ)
7367, 69, 71, 72expsubd 13245 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑((♯‘𝑤) − 2)) = ((-1↑(♯‘𝑤)) / (-1↑2)))
74 neg1sqe1 13185 . . . . . . . . . . . . . . . . . . 19 (-1↑2) = 1
7574oveq2i 6888 . . . . . . . . . . . . . . . . . 18 ((-1↑(♯‘𝑤)) / (-1↑2)) = ((-1↑(♯‘𝑤)) / 1)
76 m1expcl 13109 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ)
7776zcnd 11752 . . . . . . . . . . . . . . . . . . . 20 ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℂ)
7872, 77syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(♯‘𝑤)) ∈ ℂ)
7978div1d 11081 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(♯‘𝑤)) / 1) = (-1↑(♯‘𝑤)))
8075, 79syl5eq 2859 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(♯‘𝑤)) / (-1↑2)) = (-1↑(♯‘𝑤)))
8166, 73, 803eqtrd 2851 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (-1↑(♯‘𝑥)) = (-1↑(♯‘𝑤)))
8281eqeq1d 2815 . . . . . . . . . . . . . . 15 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((-1↑(♯‘𝑥)) = 1 ↔ (-1↑(♯‘𝑤)) = 1))
8365, 82sylibd 230 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑤)) = 1))
8483ex 399 . . . . . . . . . . . . 13 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ((𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑤)) = 1)))
8584com23 86 . . . . . . . . . . . 12 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → ((𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(♯‘𝑤)) = 1)))
8685alimdv 2007 . . . . . . . . . . 11 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → ∀𝑥((𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(♯‘𝑤)) = 1)))
87 19.23v 2034 . . . . . . . . . . 11 (∀𝑥((𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(♯‘𝑤)) = 1) ↔ (∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(♯‘𝑤)) = 1))
8886, 87syl6ib 242 . . . . . . . . . 10 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (-1↑(♯‘𝑤)) = 1)))
8943, 88mpid 44 . . . . . . . . 9 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑤)) = 1))
90893exp 1141 . . . . . . . 8 (𝜑 → (𝑤 ≠ ∅ → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → (-1↑(♯‘𝑤)) = 1))))
9190com34 91 . . . . . . 7 (𝜑 → (𝑤 ≠ ∅ → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1))))
9291com12 32 . . . . . 6 (𝑤 ≠ ∅ → (𝜑 → (∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1))))
9392impd 398 . . . . 5 (𝑤 ≠ ∅ → ((𝜑 ∧ ∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1)))
9416, 93pm2.61ine 3068 . . . 4 ((𝜑 ∧ ∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1))
95943adant2 1154 . . 3 ((𝜑 ∧ (♯‘𝑤) ∈ (0...(♯‘𝑊)) ∧ ∀𝑥((♯‘𝑥) ∈ (0..^(♯‘𝑤)) → ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1))) → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1))
96 eleq1 2880 . . . . 5 (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇𝑥 ∈ Word 𝑇))
97 oveq2 6885 . . . . . 6 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
9897eqeq1d 2815 . . . . 5 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
9996, 98anbi12d 618 . . . 4 (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
100 fveq2 6411 . . . . . 6 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
101100oveq2d 6893 . . . . 5 (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑥)))
102101eqeq1d 2815 . . . 4 (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑥)) = 1))
10399, 102imbi12d 335 . . 3 (𝑤 = 𝑥 → (((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1) ↔ ((𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑥)) = 1)))
104 eleq1 2880 . . . . 5 (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇𝑊 ∈ Word 𝑇))
105 oveq2 6885 . . . . . 6 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
106105eqeq1d 2815 . . . . 5 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
107104, 106anbi12d 618 . . . 4 (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
108 fveq2 6411 . . . . . 6 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
109108oveq2d 6893 . . . . 5 (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊)))
110109eqeq1d 2815 . . . 4 (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑊)) = 1))
111107, 110imbi12d 335 . . 3 (𝑤 = 𝑊 → (((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑤)) = 1) ↔ ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑊)) = 1)))
1121, 7, 95, 103, 111, 100, 108uzindi 13008 . 2 (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑊)) = 1))
1131, 2, 112mp2and 682 1 (𝜑 → (-1↑(♯‘𝑊)) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100  wal 1635   = wceq 1637  wex 1859  wcel 2157  wne 2985  wrex 3104  c0 4123   class class class wbr 4851   I cid 5225  ran crn 5319  cres 5320  cfv 6104  (class class class)co 6877  Fincfn 8195  cc 10222  cr 10223  0cc0 10224  1c1 10225   < clt 10362  cmin 10554  -cneg 10555   / cdiv 10972  cn 11308  2c2 11359  0cn0 11562  cz 11646  cuz 11907  +crp 12049  ...cfz 12552  ..^cfzo 12692  cexp 13086  chash 13340  Word cword 13505   Σg cgsu 16309  SymGrpcsymg 18001  pmTrspcpmtr 18065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103  ax-un 7182  ax-cnex 10280  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300  ax-pre-mulgt0 10301
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-xor 1619  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-nel 3089  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-pss 3792  df-nul 4124  df-if 4287  df-pw 4360  df-sn 4378  df-pr 4380  df-tp 4382  df-op 4384  df-ot 4386  df-uni 4638  df-int 4677  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-tr 4954  df-id 5226  df-eprel 5231  df-po 5239  df-so 5240  df-fr 5277  df-se 5278  df-we 5279  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-ord 5946  df-on 5947  df-lim 5948  df-suc 5949  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-isom 6113  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-om 7299  df-1st 7401  df-2nd 7402  df-wrecs 7645  df-recs 7707  df-rdg 7745  df-1o 7799  df-2o 7800  df-oadd 7803  df-er 7982  df-map 8097  df-en 8196  df-dom 8197  df-sdom 8198  df-fin 8199  df-card 9051  df-pnf 10364  df-mnf 10365  df-xr 10366  df-ltxr 10367  df-le 10368  df-sub 10556  df-neg 10557  df-div 10973  df-nn 11309  df-2 11367  df-3 11368  df-4 11369  df-5 11370  df-6 11371  df-7 11372  df-8 11373  df-9 11374  df-n0 11563  df-z 11647  df-uz 11908  df-rp 12050  df-fz 12553  df-fzo 12693  df-seq 13028  df-exp 13087  df-hash 13341  df-word 13513  df-lsw 13514  df-concat 13515  df-s1 13516  df-substr 13517  df-splice 13518  df-s2 13820  df-struct 16073  df-ndx 16074  df-slot 16075  df-base 16077  df-sets 16078  df-ress 16079  df-plusg 16169  df-tset 16175  df-0g 16310  df-gsum 16311  df-mgm 17450  df-sgrp 17492  df-mnd 17503  df-submnd 17544  df-grp 17633  df-minusg 17634  df-subg 17796  df-symg 18002  df-pmtr 18066
This theorem is referenced by:  psgnuni  18123
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