| Step | Hyp | Ref
| Expression |
|---|
| 1 | | subgacs.b |
. . . . . . . . 9
β’ π΅ = (BaseβπΊ) |
| 2 | 1 | subgss 19007 |
. . . . . . . 8
β’ (π β (SubGrpβπΊ) β π β π΅) |
| 3 | | velpw 4608 |
. . . . . . . 8
β’ (π β π« π΅ β π β π΅) |
| 4 | 2, 3 | sylibr 233 |
. . . . . . 7
β’ (π β (SubGrpβπΊ) β π β π« π΅) |
| 5 | | eleq2w 2818 |
. . . . . . . . . 10
β’ (π§ = π β (((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§ β ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 6 | 5 | raleqbi1dv 3334 |
. . . . . . . . 9
β’ (π§ = π β (βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§ β βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 7 | 6 | ralbidv 3178 |
. . . . . . . 8
β’ (π§ = π β (βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§ β βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 8 | 7 | elrab3 3685 |
. . . . . . 7
β’ (π β π« π΅ β (π β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§} β βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 9 | 4, 8 | syl 17 |
. . . . . 6
β’ (π β (SubGrpβπΊ) β (π β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§} β βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 10 | 9 | bicomd 222 |
. . . . 5
β’ (π β (SubGrpβπΊ) β (βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π β π β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§})) |
| 11 | 10 | pm5.32i 576 |
. . . 4
β’ ((π β (SubGrpβπΊ) β§ βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π ) β (π β (SubGrpβπΊ) β§ π β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§})) |
| 12 | | eqid 2733 |
. . . . 5
β’
(+gβπΊ) = (+gβπΊ) |
| 13 | | eqid 2733 |
. . . . 5
β’
(-gβπΊ) = (-gβπΊ) |
| 14 | 1, 12, 13 | isnsg3 19040 |
. . . 4
β’ (π β (NrmSGrpβπΊ) β (π β (SubGrpβπΊ) β§ βπ₯ β π΅ βπ¦ β π ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π )) |
| 15 | | elin 3965 |
. . . 4
β’ (π β ((SubGrpβπΊ) β© {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§}) β (π β (SubGrpβπΊ) β§ π β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§})) |
| 16 | 11, 14, 15 | 3bitr4i 303 |
. . 3
β’ (π β (NrmSGrpβπΊ) β π β ((SubGrpβπΊ) β© {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§})) |
| 17 | 16 | eqriv 2730 |
. 2
β’
(NrmSGrpβπΊ) =
((SubGrpβπΊ) β©
{π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§}) |
| 18 | 1 | fvexi 6906 |
. . . 4
β’ π΅ β V |
| 19 | | mreacs 17602 |
. . . 4
β’ (π΅ β V β
(ACSβπ΅) β
(Mooreβπ« π΅)) |
| 20 | 18, 19 | mp1i 13 |
. . 3
β’ (πΊ β Grp β
(ACSβπ΅) β
(Mooreβπ« π΅)) |
| 21 | 1 | subgacs 19041 |
. . 3
β’ (πΊ β Grp β
(SubGrpβπΊ) β
(ACSβπ΅)) |
| 22 | | simpl 484 |
. . . . . 6
β’ ((πΊ β Grp β§ (π₯ β π΅ β§ π¦ β π΅)) β πΊ β Grp) |
| 23 | 1, 12 | grpcl 18827 |
. . . . . . 7
β’ ((πΊ β Grp β§ π₯ β π΅ β§ π¦ β π΅) β (π₯(+gβπΊ)π¦) β π΅) |
| 24 | 23 | 3expb 1121 |
. . . . . 6
β’ ((πΊ β Grp β§ (π₯ β π΅ β§ π¦ β π΅)) β (π₯(+gβπΊ)π¦) β π΅) |
| 25 | | simprl 770 |
. . . . . 6
β’ ((πΊ β Grp β§ (π₯ β π΅ β§ π¦ β π΅)) β π₯ β π΅) |
| 26 | 1, 13 | grpsubcl 18903 |
. . . . . 6
β’ ((πΊ β Grp β§ (π₯(+gβπΊ)π¦) β π΅ β§ π₯ β π΅) β ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π΅) |
| 27 | 22, 24, 25, 26 | syl3anc 1372 |
. . . . 5
β’ ((πΊ β Grp β§ (π₯ β π΅ β§ π¦ β π΅)) β ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π΅) |
| 28 | 27 | ralrimivva 3201 |
. . . 4
β’ (πΊ β Grp β βπ₯ β π΅ βπ¦ β π΅ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π΅) |
| 29 | | acsfn1c 17606 |
. . . 4
β’ ((π΅ β V β§ βπ₯ β π΅ βπ¦ β π΅ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π΅) β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§} β (ACSβπ΅)) |
| 30 | 18, 28, 29 | sylancr 588 |
. . 3
β’ (πΊ β Grp β {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§} β (ACSβπ΅)) |
| 31 | | mreincl 17543 |
. . 3
β’
(((ACSβπ΅)
β (Mooreβπ« π΅) β§ (SubGrpβπΊ) β (ACSβπ΅) β§ {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§} β (ACSβπ΅)) β ((SubGrpβπΊ) β© {π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§}) β (ACSβπ΅)) |
| 32 | 20, 21, 30, 31 | syl3anc 1372 |
. 2
β’ (πΊ β Grp β
((SubGrpβπΊ) β©
{π§ β π« π΅ β£ βπ₯ β π΅ βπ¦ β π§ ((π₯(+gβπΊ)π¦)(-gβπΊ)π₯) β π§}) β (ACSβπ΅)) |
| 33 | 17, 32 | eqeltrid 2838 |
1
β’ (πΊ β Grp β
(NrmSGrpβπΊ) β
(ACSβπ΅)) |
