Step | Hyp | Ref
| Expression |
1 | | dib2dim.k |
. . 3
β’ (π β (πΎ β HL β§ π β π»)) |
2 | | dib2dim.h |
. . . 4
β’ π» = (LHypβπΎ) |
3 | | dib2dim.i |
. . . 4
β’ πΌ = ((DIsoBβπΎ)βπ) |
4 | 2, 3 | dibvalrel 39732 |
. . 3
β’ ((πΎ β HL β§ π β π») β Rel (πΌβ(π β¨ π))) |
5 | 1, 4 | syl 17 |
. 2
β’ (π β Rel (πΌβ(π β¨ π))) |
6 | | dib2dim.l |
. . . . . 6
β’ β€ =
(leβπΎ) |
7 | | dib2dim.j |
. . . . . 6
β’ β¨ =
(joinβπΎ) |
8 | | dib2dim.a |
. . . . . 6
β’ π΄ = (AtomsβπΎ) |
9 | | eqid 2731 |
. . . . . 6
β’
((DVecAβπΎ)βπ) = ((DVecAβπΎ)βπ) |
10 | | eqid 2731 |
. . . . . 6
β’
(LSSumβ((DVecAβπΎ)βπ)) = (LSSumβ((DVecAβπΎ)βπ)) |
11 | | eqid 2731 |
. . . . . 6
β’
((DIsoAβπΎ)βπ) = ((DIsoAβπΎ)βπ) |
12 | | dib2dim.p |
. . . . . 6
β’ (π β (π β π΄ β§ π β€ π)) |
13 | | dib2dim.q |
. . . . . 6
β’ (π β (π β π΄ β§ π β€ π)) |
14 | 6, 7, 8, 2, 9, 10,
11, 1, 12, 13 | dia2dim 39646 |
. . . . 5
β’ (π β (((DIsoAβπΎ)βπ)β(π β¨ π)) β ((((DIsoAβπΎ)βπ)βπ)(LSSumβ((DVecAβπΎ)βπ))(((DIsoAβπΎ)βπ)βπ))) |
15 | 14 | sseld 3961 |
. . . 4
β’ (π β (π β (((DIsoAβπΎ)βπ)β(π β¨ π)) β π β ((((DIsoAβπΎ)βπ)βπ)(LSSumβ((DVecAβπΎ)βπ))(((DIsoAβπΎ)βπ)βπ)))) |
16 | 15 | anim1d 611 |
. . 3
β’ (π β ((π β (((DIsoAβπΎ)βπ)β(π β¨ π)) β§ π = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ)))) β (π β ((((DIsoAβπΎ)βπ)βπ)(LSSumβ((DVecAβπΎ)βπ))(((DIsoAβπΎ)βπ)βπ)) β§ π = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ)))))) |
17 | 1 | simpld 495 |
. . . . 5
β’ (π β πΎ β HL) |
18 | 12 | simpld 495 |
. . . . 5
β’ (π β π β π΄) |
19 | 13 | simpld 495 |
. . . . 5
β’ (π β π β π΄) |
20 | | eqid 2731 |
. . . . . 6
β’
(BaseβπΎ) =
(BaseβπΎ) |
21 | 20, 7, 8 | hlatjcl 37935 |
. . . . 5
β’ ((πΎ β HL β§ π β π΄ β§ π β π΄) β (π β¨ π) β (BaseβπΎ)) |
22 | 17, 18, 19, 21 | syl3anc 1371 |
. . . 4
β’ (π β (π β¨ π) β (BaseβπΎ)) |
23 | 12 | simprd 496 |
. . . . 5
β’ (π β π β€ π) |
24 | 13 | simprd 496 |
. . . . 5
β’ (π β π β€ π) |
25 | 17 | hllatd 37932 |
. . . . . 6
β’ (π β πΎ β Lat) |
26 | 20, 8 | atbase 37857 |
. . . . . . 7
β’ (π β π΄ β π β (BaseβπΎ)) |
27 | 18, 26 | syl 17 |
. . . . . 6
β’ (π β π β (BaseβπΎ)) |
28 | 20, 8 | atbase 37857 |
. . . . . . 7
β’ (π β π΄ β π β (BaseβπΎ)) |
29 | 19, 28 | syl 17 |
. . . . . 6
β’ (π β π β (BaseβπΎ)) |
30 | 1 | simprd 496 |
. . . . . . 7
β’ (π β π β π») |
31 | 20, 2 | lhpbase 38567 |
. . . . . . 7
β’ (π β π» β π β (BaseβπΎ)) |
32 | 30, 31 | syl 17 |
. . . . . 6
β’ (π β π β (BaseβπΎ)) |
33 | 20, 6, 7 | latjle12 18368 |
. . . . . 6
β’ ((πΎ β Lat β§ (π β (BaseβπΎ) β§ π β (BaseβπΎ) β§ π β (BaseβπΎ))) β ((π β€ π β§ π β€ π) β (π β¨ π) β€ π)) |
34 | 25, 27, 29, 32, 33 | syl13anc 1372 |
. . . . 5
β’ (π β ((π β€ π β§ π β€ π) β (π β¨ π) β€ π)) |
35 | 23, 24, 34 | mpbi2and 710 |
. . . 4
β’ (π β (π β¨ π) β€ π) |
36 | | eqid 2731 |
. . . . 5
β’
((LTrnβπΎ)βπ) = ((LTrnβπΎ)βπ) |
37 | | eqid 2731 |
. . . . 5
β’ (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ))) = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ))) |
38 | 20, 6, 2, 36, 37, 11, 3 | dibopelval2 39714 |
. . . 4
β’ (((πΎ β HL β§ π β π») β§ ((π β¨ π) β (BaseβπΎ) β§ (π β¨ π) β€ π)) β (β¨π, π β© β (πΌβ(π β¨ π)) β (π β (((DIsoAβπΎ)βπ)β(π β¨ π)) β§ π = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ)))))) |
39 | 1, 22, 35, 38 | syl12anc 835 |
. . 3
β’ (π β (β¨π, π β© β (πΌβ(π β¨ π)) β (π β (((DIsoAβπΎ)βπ)β(π β¨ π)) β§ π = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ)))))) |
40 | | dib2dim.u |
. . . 4
β’ π = ((DVecHβπΎ)βπ) |
41 | | dib2dim.s |
. . . 4
β’ β =
(LSSumβπ) |
42 | 27, 23 | jca 512 |
. . . 4
β’ (π β (π β (BaseβπΎ) β§ π β€ π)) |
43 | 29, 24 | jca 512 |
. . . 4
β’ (π β (π β (BaseβπΎ) β§ π β€ π)) |
44 | 20, 6, 2, 36, 37, 9, 40, 10, 41, 11, 3, 1, 42, 43 | diblsmopel 39740 |
. . 3
β’ (π β (β¨π, π β© β ((πΌβπ) β (πΌβπ)) β (π β ((((DIsoAβπΎ)βπ)βπ)(LSSumβ((DVecAβπΎ)βπ))(((DIsoAβπΎ)βπ)βπ)) β§ π = (π β ((LTrnβπΎ)βπ) β¦ ( I βΎ (BaseβπΎ)))))) |
45 | 16, 39, 44 | 3imtr4d 293 |
. 2
β’ (π β (β¨π, π β© β (πΌβ(π β¨ π)) β β¨π, π β© β ((πΌβπ) β (πΌβπ)))) |
46 | 5, 45 | relssdv 5764 |
1
β’ (π β (πΌβ(π β¨ π)) β ((πΌβπ) β (πΌβπ))) |