| Step | Hyp | Ref
| Expression |
|---|
| 1 | | baerlem3.y |
. . . . . 6
β’ (π β π β (π β { 0 })) |
| 2 | 1 | eldifad 3961 |
. . . . 5
β’ (π β π β π) |
| 3 | | baerlem3.z |
. . . . . 6
β’ (π β π β (π β { 0 })) |
| 4 | 3 | eldifad 3961 |
. . . . 5
β’ (π β π β π) |
| 5 | | baerlem3.v |
. . . . . 6
β’ π = (Baseβπ) |
| 6 | | baerlem5a.p |
. . . . . 6
β’ + =
(+gβπ) |
| 7 | | eqid 2733 |
. . . . . 6
β’
(invgβπ) = (invgβπ) |
| 8 | | baerlem3.m |
. . . . . 6
β’ β =
(-gβπ) |
| 9 | 5, 6, 7, 8 | grpsubval 18870 |
. . . . 5
β’ ((π β π β§ π β π) β (π β π) = (π +
((invgβπ)βπ))) |
| 10 | 2, 4, 9 | syl2anc 585 |
. . . 4
β’ (π β (π β π) = (π +
((invgβπ)βπ))) |
| 11 | 10 | sneqd 4641 |
. . 3
β’ (π β {(π β π)} = {(π +
((invgβπ)βπ))}) |
| 12 | 11 | fveq2d 6896 |
. 2
β’ (π β (πβ{(π β π)}) = (πβ{(π +
((invgβπ)βπ))})) |
| 13 | | baerlem3.o |
. . 3
β’ 0 =
(0gβπ) |
| 14 | | baerlem3.s |
. . 3
β’ β =
(LSSumβπ) |
| 15 | | baerlem3.n |
. . 3
β’ π = (LSpanβπ) |
| 16 | | baerlem3.w |
. . 3
β’ (π β π β LVec) |
| 17 | | baerlem3.x |
. . 3
β’ (π β π β π) |
| 18 | | lveclmod 20717 |
. . . . . 6
β’ (π β LVec β π β LMod) |
| 19 | 16, 18 | syl 17 |
. . . . 5
β’ (π β π β LMod) |
| 20 | 5, 7 | lmodvnegcl 20513 |
. . . . 5
β’ ((π β LMod β§ π β π) β ((invgβπ)βπ) β π) |
| 21 | 19, 4, 20 | syl2anc 585 |
. . . 4
β’ (π β
((invgβπ)βπ) β π) |
| 22 | | eqid 2733 |
. . . . . 6
β’
(LSubSpβπ) =
(LSubSpβπ) |
| 23 | 5, 22, 15, 19, 2, 4 | lspprcl 20589 |
. . . . . 6
β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
| 24 | | baerlem3.c |
. . . . . 6
β’ (π β Β¬ π β (πβ{π, π})) |
| 25 | 13, 22, 19, 23, 17, 24 | lssneln0 20563 |
. . . . 5
β’ (π β π β (π β { 0 })) |
| 26 | 5, 15, 16, 17, 2, 4, 24 | lspindpi 20745 |
. . . . . 6
β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
| 27 | 26 | simpld 496 |
. . . . 5
β’ (π β (πβ{π}) β (πβ{π})) |
| 28 | 5, 13, 15, 16, 25, 2, 27 | lspsnne1 20730 |
. . . 4
β’ (π β Β¬ π β (πβ{π})) |
| 29 | | baerlem3.d |
. . . . . . . 8
β’ (π β (πβ{π}) β (πβ{π})) |
| 30 | 29 | necomd 2997 |
. . . . . . 7
β’ (π β (πβ{π}) β (πβ{π})) |
| 31 | 5, 13, 15, 16, 3, 2, 30 | lspsnne1 20730 |
. . . . . 6
β’ (π β Β¬ π β (πβ{π})) |
| 32 | 5, 15, 16, 17, 4, 2, 31, 24 | lspexchn2 20744 |
. . . . 5
β’ (π β Β¬ π β (πβ{π, π})) |
| 33 | | lmodgrp 20478 |
. . . . . . . . 9
β’ (π β LMod β π β Grp) |
| 34 | 19, 33 | syl 17 |
. . . . . . . 8
β’ (π β π β Grp) |
| 35 | 34 | adantr 482 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β Grp) |
| 36 | 4 | adantr 482 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β π) |
| 37 | 5, 7 | grpinvinv 18890 |
. . . . . . 7
β’ ((π β Grp β§ π β π) β ((invgβπ)β((invgβπ)βπ)) = π) |
| 38 | 35, 36, 37 | syl2anc 585 |
. . . . . 6
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)β((invgβπ)βπ)) = π) |
| 39 | 19 | adantr 482 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β LMod) |
| 40 | 5, 22, 15, 19, 2, 17 | lspprcl 20589 |
. . . . . . . 8
β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
| 41 | 40 | adantr 482 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β (πβ{π, π}) β (LSubSpβπ)) |
| 42 | | simpr 486 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)βπ) β (πβ{π, π})) |
| 43 | 22, 7 | lssvnegcl 20567 |
. . . . . . 7
β’ ((π β LMod β§ (πβ{π, π}) β (LSubSpβπ) β§ ((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)β((invgβπ)βπ)) β (πβ{π, π})) |
| 44 | 39, 41, 42, 43 | syl3anc 1372 |
. . . . . 6
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)β((invgβπ)βπ)) β (πβ{π, π})) |
| 45 | 38, 44 | eqeltrrd 2835 |
. . . . 5
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β (πβ{π, π})) |
| 46 | 32, 45 | mtand 815 |
. . . 4
β’ (π β Β¬
((invgβπ)βπ) β (πβ{π, π})) |
| 47 | 5, 15, 16, 21, 17, 2, 28, 46 | lspexchn2 20744 |
. . 3
β’ (π β Β¬ π β (πβ{π, ((invgβπ)βπ)})) |
| 48 | 5, 7, 15 | lspsnneg 20617 |
. . . . 5
β’ ((π β LMod β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 49 | 19, 4, 48 | syl2anc 585 |
. . . 4
β’ (π β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 50 | 29, 49 | neeqtrrd 3016 |
. . 3
β’ (π β (πβ{π}) β (πβ{((invgβπ)βπ)})) |
| 51 | 5, 13, 7 | grpinvnzcl 18895 |
. . . 4
β’ ((π β Grp β§ π β (π β { 0 })) β
((invgβπ)βπ) β (π β { 0 })) |
| 52 | 34, 3, 51 | syl2anc 585 |
. . 3
β’ (π β
((invgβπ)βπ) β (π β { 0 })) |
| 53 | 5, 8, 13, 14, 15, 16, 17, 47, 50, 1, 52, 6 | baerlem5b 40586 |
. 2
β’ (π β (πβ{(π +
((invgβπ)βπ))}) = (((πβ{π}) β (πβ{((invgβπ)βπ)})) β© ((πβ{(π β (π +
((invgβπ)βπ)))}) β (πβ{π})))) |
| 54 | 49 | oveq2d 7425 |
. . 3
β’ (π β ((πβ{π}) β (πβ{((invgβπ)βπ)})) = ((πβ{π}) β (πβ{π}))) |
| 55 | 10 | eqcomd 2739 |
. . . . . . 7
β’ (π β (π +
((invgβπ)βπ)) = (π β π)) |
| 56 | 55 | oveq2d 7425 |
. . . . . 6
β’ (π β (π β (π +
((invgβπ)βπ))) = (π β (π β π))) |
| 57 | 56 | sneqd 4641 |
. . . . 5
β’ (π β {(π β (π +
((invgβπ)βπ)))} = {(π β (π β π))}) |
| 58 | 57 | fveq2d 6896 |
. . . 4
β’ (π β (πβ{(π β (π +
((invgβπ)βπ)))}) = (πβ{(π β (π β π))})) |
| 59 | 58 | oveq1d 7424 |
. . 3
β’ (π β ((πβ{(π β (π +
((invgβπ)βπ)))}) β (πβ{π})) = ((πβ{(π β (π β π))}) β (πβ{π}))) |
| 60 | 54, 59 | ineq12d 4214 |
. 2
β’ (π β (((πβ{π}) β (πβ{((invgβπ)βπ)})) β© ((πβ{(π β (π +
((invgβπ)βπ)))}) β (πβ{π}))) = (((πβ{π}) β (πβ{π})) β© ((πβ{(π β (π β π))}) β (πβ{π})))) |
| 61 | 12, 53, 60 | 3eqtrd 2777 |
1
β’ (π β (πβ{(π β π)}) = (((πβ{π}) β (πβ{π})) β© ((πβ{(π β (π β π))}) β (πβ{π})))) |
