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Theorem psgnunilem4 17857
Description: Lemma for psgnuni 17859. 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 13278 . . . . 5 (𝑊 ∈ Word 𝑇𝑊 ∈ Fin)
4 hashcl 13103 . . . . 5 (𝑊 ∈ Fin → (#‘𝑊) ∈ ℕ0)
51, 3, 43syl 18 . . . 4 (𝜑 → (#‘𝑊) ∈ ℕ0)
6 nn0uz 11682 . . . 4 0 = (ℤ‘0)
75, 6syl6eleq 2708 . . 3 (𝜑 → (#‘𝑊) ∈ (ℤ‘0))
8 fveq2 6158 . . . . . . . . 9 (𝑤 = ∅ → (#‘𝑤) = (#‘∅))
9 hash0 13114 . . . . . . . . 9 (#‘∅) = 0
108, 9syl6eq 2671 . . . . . . . 8 (𝑤 = ∅ → (#‘𝑤) = 0)
1110oveq2d 6631 . . . . . . 7 (𝑤 = ∅ → (-1↑(#‘𝑤)) = (-1↑0))
12 neg1cn 11084 . . . . . . . 8 -1 ∈ ℂ
13 exp0 12820 . . . . . . . 8 (-1 ∈ ℂ → (-1↑0) = 1)
1412, 13ax-mp 5 . . . . . . 7 (-1↑0) = 1
1511, 14syl6eq 2671 . . . . . 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 1062 . . . . . . . . . . . . . 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 1114 . . . . . . . . . . . . 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 2622 . . . . . . . . . . . . 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 13278 . . . . . . . . . . . . . . 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 1063 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
27 hashnncl 13113 . . . . . . . . . . . . . . 15 (𝑤 ∈ Fin → ((#‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
2827biimpar 502 . . . . . . . . . . . . . 14 ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (#‘𝑤) ∈ ℕ)
2925, 26, 28syl2anc 692 . . . . . . . . . . . . 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 1115 . . . . . . . . . . . . 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 fveq2 6158 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (#‘𝑥) = (#‘𝑦))
3231eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((#‘𝑥) = ((#‘𝑤) − 2) ↔ (#‘𝑦) = ((#‘𝑤) − 2)))
33 oveq2 6623 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
3433eqeq1d 2623 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3532, 34anbi12d 746 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
3635cbvrexv 3164 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3736notbii 310 . . . . . . . . . . . . . . 15 (¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3837biimpi 206 . . . . . . . . . . . . . 14 (¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((#‘𝑦) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3938adantl 482 . . . . . . . . . . . . 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)))
4017, 18, 21, 22, 23, 29, 30, 39psgnunilem3 17856 . . . . . . . . . . . 12 ¬ ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
41 iman 440 . . . . . . . . . . . 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))))
4240, 41mpbir 221 . . . . . . . . . . 11 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
43 df-rex 2914 . . . . . . . . . . 11 (∃𝑥 ∈ Word 𝑇((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
4442, 43sylib 208 . . . . . . . . . 10 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
45 simprl 793 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 𝑥 ∈ Word 𝑇)
46 simprrr 804 . . . . . . . . . . . . . . . . 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))
4745, 46jca 554 . . . . . . . . . . . . . . . 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)))
48 wrdfin 13278 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Word 𝑇𝑥 ∈ Fin)
49 hashcl 13103 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ0)
5045, 48, 493syl 18 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) ∈ ℕ0)
51 simp3l 1087 . . . . . . . . . . . . . . . . . . . 20 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇)
5251, 24syl 17 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin)
53 simp2 1060 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
5452, 53, 28syl2anc 692 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (#‘𝑤) ∈ ℕ)
5554adantr 481 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℕ)
56 simprrl 803 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) = ((#‘𝑤) − 2))
5755nnred 10995 . . . . . . . . . . . . . . . . . . 19 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℝ)
58 2rp 11797 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ+
59 ltsubrp 11826 . . . . . . . . . . . . . . . . . . 19 (((#‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((#‘𝑤) − 2) < (#‘𝑤))
6057, 58, 59sylancl 693 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → ((#‘𝑤) − 2) < (#‘𝑤))
6156, 60eqbrtrd 4645 . . . . . . . . . . . . . . . . 17 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) < (#‘𝑤))
62 elfzo0 12465 . . . . . . . . . . . . . . . . 17 ((#‘𝑥) ∈ (0..^(#‘𝑤)) ↔ ((#‘𝑥) ∈ ℕ0 ∧ (#‘𝑤) ∈ ℕ ∧ (#‘𝑥) < (#‘𝑤)))
6350, 55, 61, 62syl3anbrc 1244 . . . . . . . . . . . . . . . 16 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑥) ∈ (0..^(#‘𝑤)))
64 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)))
6564com13 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)))
6647, 63, 65sylc 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))
6756oveq2d 6631 . . . . . . . . . . . . . . . . 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)))
6812a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ∈ ℂ)
69 neg1ne0 11086 . . . . . . . . . . . . . . . . . . 19 -1 ≠ 0
7069a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ≠ 0)
71 2z 11369 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℤ
7271a1i 11 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 2 ∈ ℤ)
7355nnzd 11441 . . . . . . . . . . . . . . . . . 18 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((#‘𝑥) = ((#‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (#‘𝑤) ∈ ℤ)
7468, 70, 72, 73expsubd 12975 . . . . . . . . . . . . . . . . 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)))
75 neg1sqe1 12915 . . . . . . . . . . . . . . . . . . 19 (-1↑2) = 1
7675oveq2i 6626 . . . . . . . . . . . . . . . . . 18 ((-1↑(#‘𝑤)) / (-1↑2)) = ((-1↑(#‘𝑤)) / 1)
77 m1expcl 12839 . . . . . . . . . . . . . . . . . . . . 21 ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ ℤ)
7877zcnd 11443 . . . . . . . . . . . . . . . . . . . 20 ((#‘𝑤) ∈ ℤ → (-1↑(#‘𝑤)) ∈ ℂ)
7973, 78syl 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)
8079div1d 10753 . . . . . . . . . . . . . . . . . 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)))
8176, 80syl5eq 2667 . . . . . . . . . . . . . . . . 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)))
8267, 74, 813eqtrd 2659 . . . . . . . . . . . . . . . 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)))
8382eqeq1d 2623 . . . . . . . . . . . . . . 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))
8466, 83sylibd 229 . . . . . . . . . . . . . 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))
8584ex 450 . . . . . . . . . . . . 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)))
8685com23 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)))
8786alimdv 1842 . . . . . . . . . . 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)))
88 19.23v 1899 . . . . . . . . . . 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))
8987, 88syl6ib 241 . . . . . . . . . 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)))
9044, 89mpid 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))
91903exp 1261 . . . . . . . 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))))
9291com34 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))))
9392com12 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))))
9493impd 447 . . . . 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)))
9516, 94pm2.61ine 2873 . . . 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))
96953adant2 1078 . . 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))
97 eleq1 2686 . . . . 5 (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇𝑥 ∈ Word 𝑇))
98 oveq2 6623 . . . . . 6 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
9998eqeq1d 2623 . . . . 5 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
10097, 99anbi12d 746 . . . 4 (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
101 fveq2 6158 . . . . . 6 (𝑤 = 𝑥 → (#‘𝑤) = (#‘𝑥))
102101oveq2d 6631 . . . . 5 (𝑤 = 𝑥 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑥)))
103102eqeq1d 2623 . . . 4 (𝑤 = 𝑥 → ((-1↑(#‘𝑤)) = 1 ↔ (-1↑(#‘𝑥)) = 1))
104100, 103imbi12d 334 . . 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)))
105 eleq1 2686 . . . . 5 (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇𝑊 ∈ Word 𝑇))
106 oveq2 6623 . . . . . 6 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
107106eqeq1d 2623 . . . . 5 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
108105, 107anbi12d 746 . . . 4 (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
109 fveq2 6158 . . . . . 6 (𝑤 = 𝑊 → (#‘𝑤) = (#‘𝑊))
110109oveq2d 6631 . . . . 5 (𝑤 = 𝑊 → (-1↑(#‘𝑤)) = (-1↑(#‘𝑊)))
111110eqeq1d 2623 . . . 4 (𝑤 = 𝑊 → ((-1↑(#‘𝑤)) = 1 ↔ (-1↑(#‘𝑊)) = 1))
112108, 111imbi12d 334 . . 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)))
1131, 7, 96, 104, 112, 101, 109uzindi 12737 . 2 (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(#‘𝑊)) = 1))
1141, 2, 113mp2and 714 1 (𝜑 → (-1↑(#‘𝑊)) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wne 2790  wrex 2909  c0 3897   class class class wbr 4623   I cid 4994  ran crn 5085  cres 5086  cfv 5857  (class class class)co 6615  Fincfn 7915  cc 9894  cr 9895  0cc0 9896  1c1 9897   < clt 10034  cmin 10226  -cneg 10227   / cdiv 10644  cn 10980  2c2 11030  0cn0 11252  cz 11337  cuz 11647  +crp 11792  ...cfz 12284  ..^cfzo 12422  cexp 12816  #chash 13073  Word cword 13246   Σg cgsu 16041  SymGrpcsymg 17737  pmTrspcpmtr 17801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-xor 1462  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-ot 4164  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-isom 5866  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-2o 7521  df-oadd 7524  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-card 8725  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-3 11040  df-4 11041  df-5 11042  df-6 11043  df-7 11044  df-8 11045  df-9 11046  df-n0 11253  df-z 11338  df-uz 11648  df-rp 11793  df-fz 12285  df-fzo 12423  df-seq 12758  df-exp 12817  df-hash 13074  df-word 13254  df-lsw 13255  df-concat 13256  df-s1 13257  df-substr 13258  df-splice 13259  df-s2 13546  df-struct 15802  df-ndx 15803  df-slot 15804  df-base 15805  df-sets 15806  df-ress 15807  df-plusg 15894  df-tset 15900  df-0g 16042  df-gsum 16043  df-mgm 17182  df-sgrp 17224  df-mnd 17235  df-submnd 17276  df-grp 17365  df-minusg 17366  df-subg 17531  df-symg 17738  df-pmtr 17802
This theorem is referenced by:  psgnuni  17859
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