| Step | Hyp | Ref
| Expression |
|---|
| 1 | | simpl 483 |
. . . 4
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β NrmMod) |
| 2 | | simpr1 1194 |
. . . 4
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β πΎ) |
| 3 | | nlmngp 24193 |
. . . . . . 7
β’ (π β NrmMod β π β NrmGrp) |
| 4 | 3 | adantr 481 |
. . . . . 6
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β NrmGrp) |
| 5 | | ngpgrp 24107 |
. . . . . 6
β’ (π β NrmGrp β π β Grp) |
| 6 | 4, 5 | syl 17 |
. . . . 5
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β Grp) |
| 7 | | simpr2 1195 |
. . . . 5
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β π) |
| 8 | | simpr3 1196 |
. . . . 5
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β π) |
| 9 | | nlmdsdi.v |
. . . . . 6
β’ π = (Baseβπ) |
| 10 | | eqid 2732 |
. . . . . 6
β’
(-gβπ) = (-gβπ) |
| 11 | 9, 10 | grpsubcl 18902 |
. . . . 5
β’ ((π β Grp β§ π β π β§ π β π) β (π(-gβπ)π) β π) |
| 12 | 6, 7, 8, 11 | syl3anc 1371 |
. . . 4
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β (π(-gβπ)π) β π) |
| 13 | | eqid 2732 |
. . . . 5
β’
(normβπ) =
(normβπ) |
| 14 | | nlmdsdi.s |
. . . . 5
β’ Β· = (
Β·π βπ) |
| 15 | | nlmdsdi.f |
. . . . 5
β’ πΉ = (Scalarβπ) |
| 16 | | nlmdsdi.k |
. . . . 5
β’ πΎ = (BaseβπΉ) |
| 17 | | nlmdsdi.a |
. . . . 5
β’ π΄ = (normβπΉ) |
| 18 | 9, 13, 14, 15, 16, 17 | nmvs 24192 |
. . . 4
β’ ((π β NrmMod β§ π β πΎ β§ (π(-gβπ)π) β π) β ((normβπ)β(π Β· (π(-gβπ)π))) = ((π΄βπ) Β· ((normβπ)β(π(-gβπ)π)))) |
| 19 | 1, 2, 12, 18 | syl3anc 1371 |
. . 3
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((normβπ)β(π Β· (π(-gβπ)π))) = ((π΄βπ) Β· ((normβπ)β(π(-gβπ)π)))) |
| 20 | | nlmlmod 24194 |
. . . . . 6
β’ (π β NrmMod β π β LMod) |
| 21 | 20 | adantr 481 |
. . . . 5
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β π β LMod) |
| 22 | 9, 14, 15, 16, 10, 21, 2, 7, 8 | lmodsubdi 20528 |
. . . 4
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β (π Β· (π(-gβπ)π)) = ((π Β· π)(-gβπ)(π Β· π))) |
| 23 | 22 | fveq2d 6895 |
. . 3
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((normβπ)β(π Β· (π(-gβπ)π))) = ((normβπ)β((π Β· π)(-gβπ)(π Β· π)))) |
| 24 | 19, 23 | eqtr3d 2774 |
. 2
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((π΄βπ) Β· ((normβπ)β(π(-gβπ)π))) = ((normβπ)β((π Β· π)(-gβπ)(π Β· π)))) |
| 25 | | nlmdsdi.d |
. . . . 5
β’ π· = (distβπ) |
| 26 | 13, 9, 10, 25 | ngpds 24112 |
. . . 4
β’ ((π β NrmGrp β§ π β π β§ π β π) β (ππ·π) = ((normβπ)β(π(-gβπ)π))) |
| 27 | 4, 7, 8, 26 | syl3anc 1371 |
. . 3
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β (ππ·π) = ((normβπ)β(π(-gβπ)π))) |
| 28 | 27 | oveq2d 7424 |
. 2
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((π΄βπ) Β· (ππ·π)) = ((π΄βπ) Β· ((normβπ)β(π(-gβπ)π)))) |
| 29 | 9, 15, 14, 16 | lmodvscl 20488 |
. . . 4
β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
| 30 | 21, 2, 7, 29 | syl3anc 1371 |
. . 3
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β (π Β· π) β π) |
| 31 | 9, 15, 14, 16 | lmodvscl 20488 |
. . . 4
β’ ((π β LMod β§ π β πΎ β§ π β π) β (π Β· π) β π) |
| 32 | 21, 2, 8, 31 | syl3anc 1371 |
. . 3
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β (π Β· π) β π) |
| 33 | 13, 9, 10, 25 | ngpds 24112 |
. . 3
β’ ((π β NrmGrp β§ (π Β· π) β π β§ (π Β· π) β π) β ((π Β· π)π·(π Β· π)) = ((normβπ)β((π Β· π)(-gβπ)(π Β· π)))) |
| 34 | 4, 30, 32, 33 | syl3anc 1371 |
. 2
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((π Β· π)π·(π Β· π)) = ((normβπ)β((π Β· π)(-gβπ)(π Β· π)))) |
| 35 | 24, 28, 34 | 3eqtr4d 2782 |
1
β’ ((π β NrmMod β§ (π β πΎ β§ π β π β§ π β π)) β ((π΄βπ) Β· (ππ·π)) = ((π Β· π)π·(π Β· π))) |
