| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | psgnunilem4.w1 | . 2
⊢ (𝜑 → 𝑊 ∈ Word 𝑇) | 
| 2 |  | psgnunilem4.w2 | . 2
⊢ (𝜑 → (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) | 
| 3 |  | wrdfin 14571 | . . . . 5
⊢ (𝑊 ∈ Word 𝑇 → 𝑊 ∈ Fin) | 
| 4 |  | hashcl 14396 | . . . . 5
⊢ (𝑊 ∈ Fin →
(♯‘𝑊) ∈
ℕ0) | 
| 5 | 1, 3, 4 | 3syl 18 | . . . 4
⊢ (𝜑 → (♯‘𝑊) ∈
ℕ0) | 
| 6 |  | nn0uz 12921 | . . . 4
⊢
ℕ0 = (ℤ≥‘0) | 
| 7 | 5, 6 | eleqtrdi 2850 | . . 3
⊢ (𝜑 → (♯‘𝑊) ∈
(ℤ≥‘0)) | 
| 8 |  | fveq2 6905 | . . . . . . . . 9
⊢ (𝑤 = ∅ →
(♯‘𝑤) =
(♯‘∅)) | 
| 9 |  | hash0 14407 | . . . . . . . . 9
⊢
(♯‘∅) = 0 | 
| 10 | 8, 9 | eqtrdi 2792 | . . . . . . . 8
⊢ (𝑤 = ∅ →
(♯‘𝑤) =
0) | 
| 11 | 10 | oveq2d 7448 | . . . . . . 7
⊢ (𝑤 = ∅ →
(-1↑(♯‘𝑤))
= (-1↑0)) | 
| 12 |  | neg1cn 12381 | . . . . . . . 8
⊢ -1 ∈
ℂ | 
| 13 |  | exp0 14107 | . . . . . . . 8
⊢ (-1
∈ ℂ → (-1↑0) = 1) | 
| 14 | 12, 13 | ax-mp 5 | . . . . . . 7
⊢
(-1↑0) = 1 | 
| 15 | 11, 14 | eqtrdi 2792 | . . . . . 6
⊢ (𝑤 = ∅ →
(-1↑(♯‘𝑤))
= 1) | 
| 16 | 15 | 2a1d 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 1191 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝜑) | 
| 20 |  | psgnunilem4.d | . . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐷 ∈ 𝑉) | 
| 21 | 19, 20 | syl 17 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝐷 ∈ 𝑉) | 
| 22 |  | simpl3l 1228 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇) | 
| 23 |  | eqidd 2737 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (♯‘𝑤) = (♯‘𝑤)) | 
| 24 |  | wrdfin 14571 | . . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ Word 𝑇 → 𝑤 ∈ Fin) | 
| 25 | 22, 24 | syl 17 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin) | 
| 26 |  | simpl2 1192 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅) | 
| 27 |  | hashnncl 14406 | . . . . . . . . . . . . . . 15
⊢ (𝑤 ∈ Fin →
((♯‘𝑤) ∈
ℕ ↔ 𝑤 ≠
∅)) | 
| 28 | 27 | biimpar 477 | . . . . . . . . . . . . . 14
⊢ ((𝑤 ∈ Fin ∧ 𝑤 ≠ ∅) →
(♯‘𝑤) ∈
ℕ) | 
| 29 | 25, 26, 28 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈
ℕ) | 
| 30 |  | simpl3r 1229 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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 6914 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((♯‘𝑥) = ((♯‘𝑤) − 2) ↔ (♯‘𝑦) = ((♯‘𝑤) − 2))) | 
| 32 |  | oveq2 7440 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝐺 Σg 𝑥) = (𝐺 Σg 𝑦)) | 
| 33 | 32 | eqeq1d 2738 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((𝐺 Σg 𝑥) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) | 
| 34 | 31, 33 | anbi12d 632 | . . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) ↔ ((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg
𝑦) = ( I ↾ 𝐷)))) | 
| 35 | 34 | cbvrexvw 3237 | . . . . . . . . . . . . . . . 16
⊢
(∃𝑥 ∈
Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg
𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) | 
| 36 | 35 | notbii 320 | . . . . . . . . . . . . . . 15
⊢ (¬
∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg
𝑥) = ( I ↾ 𝐷)) ↔ ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) | 
| 37 | 36 | biimpi 216 | . . . . . . . . . . . . . 14
⊢ (¬
∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg
𝑥) = ( I ↾ 𝐷)) → ¬ ∃𝑦 ∈ Word 𝑇((♯‘𝑦) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑦) = ( I ↾ 𝐷))) | 
| 38 | 37 | adantl 481 | . . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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))) | 
| 39 | 17, 18, 21, 22, 23, 29, 30, 38 | psgnunilem3 19515 | . . . . . . . . . . . 12
⊢  ¬
((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) | 
| 40 |  | iman 401 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) ↔ ¬ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ ¬ ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) | 
| 41 | 39, 40 | mpbir 231 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥 ∈ Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) | 
| 42 |  | df-rex 3070 | . . . . . . . . . . 11
⊢
(∃𝑥 ∈
Word 𝑇((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg
𝑥) = ( I ↾ 𝐷)) ↔ ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) | 
| 43 | 41, 42 | sylib 218 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → ∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) | 
| 44 |  | simprl 770 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 𝑥 ∈ Word 𝑇) | 
| 45 |  | simprrr 781 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (𝐺 Σg 𝑥) = ( I ↾ 𝐷)) | 
| 46 | 44, 45 | jca 511 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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 14571 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ Word 𝑇 → 𝑥 ∈ Fin) | 
| 48 |  | hashcl 14396 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ Fin →
(♯‘𝑥) ∈
ℕ0) | 
| 49 | 44, 47, 48 | 3syl 18 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) ∈
ℕ0) | 
| 50 |  | simp3l 1201 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Word 𝑇) | 
| 51 | 50, 24 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ∈ Fin) | 
| 52 |  | simp2 1137 | . . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → 𝑤 ≠ ∅) | 
| 53 | 51, 52, 28 | syl2anc 584 | . . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) → (♯‘𝑤) ∈
ℕ) | 
| 54 | 53 | adantr 480 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈
ℕ) | 
| 55 |  | simprrl 780 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) = ((♯‘𝑤) − 2)) | 
| 56 | 54 | nnred 12282 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈
ℝ) | 
| 57 |  | 2rp 13040 | . . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℝ+ | 
| 58 |  | ltsubrp 13072 | . . . . . . . . . . . . . . . . . . 19
⊢
(((♯‘𝑤)
∈ ℝ ∧ 2 ∈ ℝ+) → ((♯‘𝑤) − 2) <
(♯‘𝑤)) | 
| 59 | 56, 57, 58 | sylancl 586 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) →
((♯‘𝑤) −
2) < (♯‘𝑤)) | 
| 60 | 55, 59 | eqbrtrd 5164 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑥) < (♯‘𝑤)) | 
| 61 |  | elfzo0 13741 | . . . . . . . . . . . . . . . . 17
⊢
((♯‘𝑥)
∈ (0..^(♯‘𝑤)) ↔ ((♯‘𝑥) ∈ ℕ0 ∧
(♯‘𝑤) ∈
ℕ ∧ (♯‘𝑥) < (♯‘𝑤))) | 
| 62 | 49, 54, 60, 61 | syl3anbrc 1343 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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))) | 
| 64 | 63 | com13 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))) | 
| 65 | 46, 62, 64 | sylc 65 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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)) | 
| 66 | 55 | oveq2d 7448 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) →
(-1↑(♯‘𝑥))
= (-1↑((♯‘𝑤) − 2))) | 
| 67 | 12 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ∈
ℂ) | 
| 68 |  | neg1ne0 12383 | . . . . . . . . . . . . . . . . . . 19
⊢ -1 ≠
0 | 
| 69 | 68 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → -1 ≠
0) | 
| 70 |  | 2z 12651 | . . . . . . . . . . . . . . . . . . 19
⊢ 2 ∈
ℤ | 
| 71 | 70 | a1i 11 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → 2 ∈
ℤ) | 
| 72 | 54 | nnzd 12642 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) → (♯‘𝑤) ∈
ℤ) | 
| 73 | 67, 69, 71, 72 | expsubd 14198 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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 14236 | . . . . . . . . . . . . . . . . . . 19
⊢
(-1↑2) = 1 | 
| 75 | 74 | oveq2i 7443 | . . . . . . . . . . . . . . . . . 18
⊢
((-1↑(♯‘𝑤)) / (-1↑2)) =
((-1↑(♯‘𝑤)) / 1) | 
| 76 |  | m1expcl 14128 | . . . . . . . . . . . . . . . . . . . . 21
⊢
((♯‘𝑤)
∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℤ) | 
| 77 | 76 | zcnd 12725 | . . . . . . . . . . . . . . . . . . . 20
⊢
((♯‘𝑤)
∈ ℤ → (-1↑(♯‘𝑤)) ∈ ℂ) | 
| 78 | 72, 77 | syl 17 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) →
(-1↑(♯‘𝑤))
∈ ℂ) | 
| 79 | 78 | div1d 12036 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) →
((-1↑(♯‘𝑤)) / 1) = (-1↑(♯‘𝑤))) | 
| 80 | 75, 79 | eqtrid 2788 | . . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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))) | 
| 81 | 66, 73, 80 | 3eqtrd 2780 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷))) ∧ (𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) →
(-1↑(♯‘𝑥))
= (-1↑(♯‘𝑤))) | 
| 82 | 81 | eqeq1d 2738 | . . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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)) | 
| 83 | 65, 82 | sylibd 239 | . . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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)) | 
| 84 | 83 | ex 412 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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))) | 
| 85 | 84 | com23 86 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑤 ≠ ∅ ∧ (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))) | 
| 86 | 85 | alimdv 1915 | . . . . . . . . . . 11
⊢ ((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)) → ∀𝑥((𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) →
(-1↑(♯‘𝑤))
= 1))) | 
| 87 |  | 19.23v 1941 | . . . . . . . . . . 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)) | 
| 88 | 86, 87 | imbitrdi 251 | . . . . . . . . . 10
⊢ ((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)) → (∃𝑥(𝑥 ∈ Word 𝑇 ∧ ((♯‘𝑥) = ((♯‘𝑤) − 2) ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) →
(-1↑(♯‘𝑤))
= 1))) | 
| 89 | 43, 88 | mpid 44 | . . . . . . . . 9
⊢ ((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)) | 
| 90 | 89 | 3exp 1119 | . . . . . . . 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)))) | 
| 91 | 90 | com34 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)))) | 
| 92 | 91 | com12 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)))) | 
| 93 | 92 | impd 410 | . . . . 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))) | 
| 94 | 16, 93 | pm2.61ine 3024 | . . . 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)) | 
| 95 | 94 | 3adant2 1131 | . . 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 2828 | . . . . 5
⊢ (𝑤 = 𝑥 → (𝑤 ∈ Word 𝑇 ↔ 𝑥 ∈ Word 𝑇)) | 
| 97 |  | oveq2 7440 | . . . . . 6
⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥)) | 
| 98 | 97 | eqeq1d 2738 | . . . . 5
⊢ (𝑤 = 𝑥 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑥) = ( I ↾ 𝐷))) | 
| 99 | 96, 98 | anbi12d 632 | . . . 4
⊢ (𝑤 = 𝑥 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑥 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑥) = ( I ↾ 𝐷)))) | 
| 100 |  | fveq2 6905 | . . . . . 6
⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥)) | 
| 101 | 100 | oveq2d 7448 | . . . . 5
⊢ (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) =
(-1↑(♯‘𝑥))) | 
| 102 | 101 | eqeq1d 2738 | . . . 4
⊢ (𝑤 = 𝑥 → ((-1↑(♯‘𝑤)) = 1 ↔
(-1↑(♯‘𝑥))
= 1)) | 
| 103 | 99, 102 | imbi12d 344 | . . 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 2828 | . . . . 5
⊢ (𝑤 = 𝑊 → (𝑤 ∈ Word 𝑇 ↔ 𝑊 ∈ Word 𝑇)) | 
| 105 |  | oveq2 7440 | . . . . . 6
⊢ (𝑤 = 𝑊 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑊)) | 
| 106 | 105 | eqeq1d 2738 | . . . . 5
⊢ (𝑤 = 𝑊 → ((𝐺 Σg 𝑤) = ( I ↾ 𝐷) ↔ (𝐺 Σg 𝑊) = ( I ↾ 𝐷))) | 
| 107 | 104, 106 | anbi12d 632 | . . . 4
⊢ (𝑤 = 𝑊 → ((𝑤 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑤) = ( I ↾ 𝐷)) ↔ (𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)))) | 
| 108 |  | fveq2 6905 | . . . . . 6
⊢ (𝑤 = 𝑊 → (♯‘𝑤) = (♯‘𝑊)) | 
| 109 | 108 | oveq2d 7448 | . . . . 5
⊢ (𝑤 = 𝑊 → (-1↑(♯‘𝑤)) =
(-1↑(♯‘𝑊))) | 
| 110 | 109 | eqeq1d 2738 | . . . 4
⊢ (𝑤 = 𝑊 → ((-1↑(♯‘𝑤)) = 1 ↔
(-1↑(♯‘𝑊))
= 1)) | 
| 111 | 107, 110 | imbi12d 344 | . . 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))) | 
| 112 | 1, 7, 95, 103, 111, 100, 108 | uzindi 14024 | . 2
⊢ (𝜑 → ((𝑊 ∈ Word 𝑇 ∧ (𝐺 Σg 𝑊) = ( I ↾ 𝐷)) →
(-1↑(♯‘𝑊))
= 1)) | 
| 113 | 1, 2, 112 | mp2and 699 | 1
⊢ (𝜑 →
(-1↑(♯‘𝑊))
= 1) |