| Step | Hyp | Ref
| Expression |
| 1 | | psgnval.g |
. . . . . . . . 9
⊢ 𝐺 = (SymGrp‘𝐷) |
| 2 | | psgnval.n |
. . . . . . . . 9
⊢ 𝑁 = (pmSgn‘𝐷) |
| 3 | | eqid 2737 |
. . . . . . . . 9
⊢
(Base‘𝐺) =
(Base‘𝐺) |
| 4 | 1, 2, 3 | psgneldm 19521 |
. . . . . . . 8
⊢ (𝑃 ∈ dom 𝑁 ↔ (𝑃 ∈ (Base‘𝐺) ∧ dom (𝑃 ∖ I ) ∈ Fin)) |
| 5 | 4 | simplbi 497 |
. . . . . . 7
⊢ (𝑃 ∈ dom 𝑁 → 𝑃 ∈ (Base‘𝐺)) |
| 6 | 1, 3 | elbasfv 17253 |
. . . . . . 7
⊢ (𝑃 ∈ (Base‘𝐺) → 𝐷 ∈ V) |
| 7 | 5, 6 | syl 17 |
. . . . . 6
⊢ (𝑃 ∈ dom 𝑁 → 𝐷 ∈ V) |
| 8 | | psgnval.t |
. . . . . . 7
⊢ 𝑇 = ran (pmTrsp‘𝐷) |
| 9 | 1, 8, 2 | psgneldm2 19522 |
. . . . . 6
⊢ (𝐷 ∈ V → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤))) |
| 10 | 7, 9 | syl 17 |
. . . . 5
⊢ (𝑃 ∈ dom 𝑁 → (𝑃 ∈ dom 𝑁 ↔ ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤))) |
| 11 | 10 | ibi 267 |
. . . 4
⊢ (𝑃 ∈ dom 𝑁 → ∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤)) |
| 12 | | simpr 484 |
. . . . . . 7
⊢ (((𝑃 ∈ dom 𝑁 ∧ 𝑤 ∈ Word 𝑇) ∧ 𝑃 = (𝐺 Σg 𝑤)) → 𝑃 = (𝐺 Σg 𝑤)) |
| 13 | | eqid 2737 |
. . . . . . 7
⊢
(-1↑(♯‘𝑤)) = (-1↑(♯‘𝑤)) |
| 14 | | ovex 7464 |
. . . . . . . 8
⊢
(-1↑(♯‘𝑤)) ∈ V |
| 15 | | eqeq1 2741 |
. . . . . . . . 9
⊢ (𝑠 = (-1↑(♯‘𝑤)) → (𝑠 = (-1↑(♯‘𝑤)) ↔ (-1↑(♯‘𝑤)) =
(-1↑(♯‘𝑤)))) |
| 16 | 15 | anbi2d 630 |
. . . . . . . 8
⊢ (𝑠 = (-1↑(♯‘𝑤)) → ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑃 = (𝐺 Σg 𝑤) ∧
(-1↑(♯‘𝑤))
= (-1↑(♯‘𝑤))))) |
| 17 | 14, 16 | spcev 3606 |
. . . . . . 7
⊢ ((𝑃 = (𝐺 Σg 𝑤) ∧
(-1↑(♯‘𝑤))
= (-1↑(♯‘𝑤))) → ∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
| 18 | 12, 13, 17 | sylancl 586 |
. . . . . 6
⊢ (((𝑃 ∈ dom 𝑁 ∧ 𝑤 ∈ Word 𝑇) ∧ 𝑃 = (𝐺 Σg 𝑤)) → ∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
| 19 | 18 | ex 412 |
. . . . 5
⊢ ((𝑃 ∈ dom 𝑁 ∧ 𝑤 ∈ Word 𝑇) → (𝑃 = (𝐺 Σg 𝑤) → ∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
| 20 | 19 | reximdva 3168 |
. . . 4
⊢ (𝑃 ∈ dom 𝑁 → (∃𝑤 ∈ Word 𝑇𝑃 = (𝐺 Σg 𝑤) → ∃𝑤 ∈ Word 𝑇∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))))) |
| 21 | 11, 20 | mpd 15 |
. . 3
⊢ (𝑃 ∈ dom 𝑁 → ∃𝑤 ∈ Word 𝑇∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
| 22 | | rexcom4 3288 |
. . 3
⊢
(∃𝑤 ∈
Word 𝑇∃𝑠(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
| 23 | 21, 22 | sylib 218 |
. 2
⊢ (𝑃 ∈ dom 𝑁 → ∃𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |
| 24 | | reeanv 3229 |
. . . 4
⊢
(∃𝑤 ∈
Word 𝑇∃𝑥 ∈ Word 𝑇((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) ↔ (∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) |
| 25 | 7 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝐷 ∈ V) |
| 26 | | simplrl 777 |
. . . . . . . 8
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑤 ∈ Word 𝑇) |
| 27 | | simplrr 778 |
. . . . . . . 8
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑥 ∈ Word 𝑇) |
| 28 | | simprll 779 |
. . . . . . . . 9
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑃 = (𝐺 Σg 𝑤)) |
| 29 | | simprrl 781 |
. . . . . . . . 9
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑃 = (𝐺 Σg 𝑥)) |
| 30 | 28, 29 | eqtr3d 2779 |
. . . . . . . 8
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥)) |
| 31 | 1, 8, 25, 26, 27, 30 | psgnuni 19517 |
. . . . . . 7
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → (-1↑(♯‘𝑤)) =
(-1↑(♯‘𝑥))) |
| 32 | | simprlr 780 |
. . . . . . 7
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑠 = (-1↑(♯‘𝑤))) |
| 33 | | simprrr 782 |
. . . . . . 7
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑡 = (-1↑(♯‘𝑥))) |
| 34 | 31, 32, 33 | 3eqtr4d 2787 |
. . . . . 6
⊢ (((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) ∧ ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) → 𝑠 = 𝑡) |
| 35 | 34 | ex 412 |
. . . . 5
⊢ ((𝑃 ∈ dom 𝑁 ∧ (𝑤 ∈ Word 𝑇 ∧ 𝑥 ∈ Word 𝑇)) → (((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) → 𝑠 = 𝑡)) |
| 36 | 35 | rexlimdvva 3213 |
. . . 4
⊢ (𝑃 ∈ dom 𝑁 → (∃𝑤 ∈ Word 𝑇∃𝑥 ∈ Word 𝑇((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) → 𝑠 = 𝑡)) |
| 37 | 24, 36 | biimtrrid 243 |
. . 3
⊢ (𝑃 ∈ dom 𝑁 → ((∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) → 𝑠 = 𝑡)) |
| 38 | 37 | alrimivv 1928 |
. 2
⊢ (𝑃 ∈ dom 𝑁 → ∀𝑠∀𝑡((∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) → 𝑠 = 𝑡)) |
| 39 | | eqeq1 2741 |
. . . . . 6
⊢ (𝑠 = 𝑡 → (𝑠 = (-1↑(♯‘𝑤)) ↔ 𝑡 = (-1↑(♯‘𝑤)))) |
| 40 | 39 | anbi2d 630 |
. . . . 5
⊢ (𝑠 = 𝑡 → ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (𝑃 = (𝐺 Σg 𝑤) ∧ 𝑡 = (-1↑(♯‘𝑤))))) |
| 41 | 40 | rexbidv 3179 |
. . . 4
⊢ (𝑠 = 𝑡 → (∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑡 = (-1↑(♯‘𝑤))))) |
| 42 | | oveq2 7439 |
. . . . . . 7
⊢ (𝑤 = 𝑥 → (𝐺 Σg 𝑤) = (𝐺 Σg 𝑥)) |
| 43 | 42 | eqeq2d 2748 |
. . . . . 6
⊢ (𝑤 = 𝑥 → (𝑃 = (𝐺 Σg 𝑤) ↔ 𝑃 = (𝐺 Σg 𝑥))) |
| 44 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑤 = 𝑥 → (♯‘𝑤) = (♯‘𝑥)) |
| 45 | 44 | oveq2d 7447 |
. . . . . . 7
⊢ (𝑤 = 𝑥 → (-1↑(♯‘𝑤)) =
(-1↑(♯‘𝑥))) |
| 46 | 45 | eqeq2d 2748 |
. . . . . 6
⊢ (𝑤 = 𝑥 → (𝑡 = (-1↑(♯‘𝑤)) ↔ 𝑡 = (-1↑(♯‘𝑥)))) |
| 47 | 43, 46 | anbi12d 632 |
. . . . 5
⊢ (𝑤 = 𝑥 → ((𝑃 = (𝐺 Σg 𝑤) ∧ 𝑡 = (-1↑(♯‘𝑤))) ↔ (𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) |
| 48 | 47 | cbvrexvw 3238 |
. . . 4
⊢
(∃𝑤 ∈
Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑡 = (-1↑(♯‘𝑤))) ↔ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) |
| 49 | 41, 48 | bitrdi 287 |
. . 3
⊢ (𝑠 = 𝑡 → (∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥))))) |
| 50 | 49 | eu4 2615 |
. 2
⊢
(∃!𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ↔ (∃𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ ∀𝑠∀𝑡((∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤))) ∧ ∃𝑥 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑥) ∧ 𝑡 = (-1↑(♯‘𝑥)))) → 𝑠 = 𝑡))) |
| 51 | 23, 38, 50 | sylanbrc 583 |
1
⊢ (𝑃 ∈ dom 𝑁 → ∃!𝑠∃𝑤 ∈ Word 𝑇(𝑃 = (𝐺 Σg 𝑤) ∧ 𝑠 = (-1↑(♯‘𝑤)))) |