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Theorem psgnunilem4 18556
Description: Lemma for psgnuni 18558. 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 13872 . . . . 5 (𝑊 ∈ Word 𝑇𝑊 ∈ Fin)
4 hashcl 13707 . . . . 5 (𝑊 ∈ Fin → (♯‘𝑊) ∈ ℕ0)
51, 3, 43syl 18 . . . 4 (𝜑 → (♯‘𝑊) ∈ ℕ0)
6 nn0uz 12269 . . . 4 0 = (ℤ‘0)
75, 6eleqtrdi 2923 . . 3 (𝜑 → (♯‘𝑊) ∈ (ℤ‘0))
8 fveq2 6664 . . . . . . . . 9 (𝑤 = ∅ → (♯‘𝑤) = (♯‘∅))
9 hash0 13718 . . . . . . . . 9 (♯‘∅) = 0
108, 9syl6eq 2872 . . . . . . . 8 (𝑤 = ∅ → (♯‘𝑤) = 0)
1110oveq2d 7161 . . . . . . 7 (𝑤 = ∅ → (-1↑(♯‘𝑤)) = (-1↑0))
12 neg1cn 11740 . . . . . . . 8 -1 ∈ ℂ
13 exp0 13423 . . . . . . . 8 (-1 ∈ ℂ → (-1↑0) = 1)
1412, 13ax-mp 5 . . . . . . 7 (-1↑0) = 1
1511, 14syl6eq 2872 . . . . . 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 1183 . . . . . . . . . . . . . 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 1220 . . . . . . . . . . . . 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 2822 . . . . . . . . . . . . 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 13872 . . . . . . . . . . . . . . 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 1184 . . . . . . . . . . . . . 14 (((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
27 hashnncl 13717 . . . . . . . . . . . . . . 15 (𝑤 ∈ Fin → ((♯‘𝑤) ∈ ℕ ↔ 𝑤 ≠ ∅))
2827biimpar 478 . . . . . . . . . . . . . 14 ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) → (♯‘𝑤) ∈ ℕ)
2925, 26, 28syl2anc 584 . . . . . . . . . . . . 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 1221 . . . . . . . . . . . . 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 6673 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2)))
32 oveq2 7153 . . . . . . . . . . . . . . . . . . 19 (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦))
3332eqeq1d 2823 . . . . . . . . . . . . . . . . . 18 (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3431, 33anbi12d 630 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))))
3534cbvrexvw 3451 . . . . . . . . . . . . . . . 16 (∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3635notbii 321 . . . . . . . . . . . . . . 15 (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3736biimpi 217 . . . . . . . . . . . . . 14 (¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷)))
3837adantl 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)))
3917, 18, 21, 22, 23, 29, 30, 38psgnunilem3 18555 . . . . . . . . . . . 12 ¬ ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
40 iman 402 . . . . . . . . . . . 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 232 . . . . . . . . . . 11 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
42 df-rex 3144 . . . . . . . . . . 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 219 . . . . . . . . . 10 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
44 simprl 767 . . . . . . . . . . . . . . . . 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 778 . . . . . . . . . . . . . . . . 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 512 . . . . . . . . . . . . . . . 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 13872 . . . . . . . . . . . . . . . . . 18 (𝑥 ∈ Word 𝑇𝑥 ∈ Fin)
48 hashcl 13707 . . . . . . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . . . . . . . . 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 1129 . . . . . . . . . . . . . . . . . . 19 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅)
5351, 52, 28syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((𝜑𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈ ℕ)
5453adantr 481 . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . 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 11642 . . . . . . . . . . . . . . . . . . 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 12384 . . . . . . . . . . . . . . . . . . 19 2 ∈ ℝ+
58 ltsubrp 12415 . . . . . . . . . . . . . . . . . . 19 (((♯‘𝑤) ∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) < (♯‘𝑤))
5956, 57, 58sylancl 586 . . . . . . . . . . . . . . . . . 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 5080 . . . . . . . . . . . . . . . . 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 13068 . . . . . . . . . . . . . . . . 17 ((♯‘𝑥) ∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧ (♯‘𝑤) ∈ ℕ ∧ (♯‘𝑥) < (♯‘𝑤)))
6249, 54, 60, 61syl3anbrc 1335 . . . . . . . . . . . . . . . 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 7161 . . . . . . . . . . . . . . . . 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 11742 . . . . . . . . . . . . . . . . . . 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 12003 . . . . . . . . . . . . . . . . . . 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 12075 . . . . . . . . . . . . . . . . . 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 13511 . . . . . . . . . . . . . . . . 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 13549 . . . . . . . . . . . . . . . . . . 19 (-1↑2) = 1
7574oveq2i 7156 . . . . . . . . . . . . . . . . . 18 ((-1↑(♯‘𝑤)) / (-1↑2)) = ((-1↑(♯‘𝑤)) / 1)
76 m1expcl 13442 . . . . . . . . . . . . . . . . . . . . 21 ((♯‘𝑤) ∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ)
7776zcnd 12077 . . . . . . . . . . . . . . . . . . . 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 11397 . . . . . . . . . . . . . . . . . 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 2868 . . . . . . . . . . . . . . . . 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 2860 . . . . . . . . . . . . . . . 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 2823 . . . . . . . . . . . . . . 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 240 . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . 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 1908 . . . . . . . . . . 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 1934 . . . . . . . . . . 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 252 . . . . . . . . . 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 1111 . . . . . . . 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 411 . . . . 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 3100 . . . 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 1123 . . 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 2900 . . . . 5 (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇𝑥 ∈ Word 𝑇))
97 oveq2 7153 . . . . . 6 (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥))
9897eqeq1d 2823 . . . . 5 (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))
9996, 98anbi12d 630 . . . 4 (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))))
100 fveq2 6664 . . . . . 6 (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥))
101100oveq2d 7161 . . . . 5 (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑥)))
102101eqeq1d 2823 . . . 4 (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑥)) = 1))
10399, 102imbi12d 346 . . 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 2900 . . . . 5 (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇𝑊 ∈ Word 𝑇))
105 oveq2 7153 . . . . . 6 (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊))
106105eqeq1d 2823 . . . . 5 (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))
107104, 106anbi12d 630 . . . 4 (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))))
108 fveq2 6664 . . . . . 6 (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊))
109108oveq2d 7161 . . . . 5 (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) = (-1↑(♯‘𝑊)))
110109eqeq1d 2823 . . . 4 (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔ (-1↑(♯‘𝑊)) = 1))
111107, 110imbi12d 346 . . 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 13340 . 2 (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) → (-1↑(♯‘𝑊)) = 1))
1131, 2, 112mp2and 695 1 (𝜑 → (-1↑(♯‘𝑊)) = 1)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1079  wal 1526   = wceq 1528  wex 1771  wcel 2105  wne 3016  wrex 3139  c0 4290   class class class wbr 5058   I cid 5453  ran crn 5550  cres 5551  cfv 6349  (class class class)co 7145  Fincfn 8498  cc 10524  cr 10525  0cc0 10526  1c1 10527   < clt 10664  cmin 10859  -cneg 10860   / cdiv 11286  cn 11627  2c2 11681  0cn0 11886  cz 11970  cuz 12232  +crp 12379  ...cfz 12882  ..^cfzo 13023  cexp 13419  chash 13680  Word cword 13851   Σg cgsu 16704  SymGrpcsymg 18435  pmTrspcpmtr 18500
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 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7450  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-ot 4568  df-uni 4833  df-int 4870  df-iun 4914  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-se 5509  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-isom 6358  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-1o 8093  df-2o 8094  df-oadd 8097  df-er 8279  df-map 8398  df-en 8499  df-dom 8500  df-sdom 8501  df-fin 8502  df-card 9357  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11628  df-2 11689  df-3 11690  df-4 11691  df-5 11692  df-6 11693  df-7 11694  df-8 11695  df-9 11696  df-n0 11887  df-xnn0 11957  df-z 11971  df-uz 12233  df-rp 12380  df-fz 12883  df-fzo 13024  df-seq 13360  df-exp 13420  df-hash 13681  df-word 13852  df-lsw 13905  df-concat 13913  df-s1 13940  df-substr 13993  df-pfx 14023  df-splice 14102  df-s2 14200  df-struct 16475  df-ndx 16476  df-slot 16477  df-base 16479  df-sets 16480  df-ress 16481  df-plusg 16568  df-tset 16574  df-0g 16705  df-gsum 16706  df-mgm 17842  df-sgrp 17891  df-mnd 17902  df-submnd 17947  df-grp 18046  df-minusg 18047  df-subg 18216  df-symg 18436  df-pmtr 18501
This theorem is referenced by:  psgnuni  18558
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