| Step | Hyp | Ref
| Expression |
|---|
| 1 | | baerlem3.y |
. . . . . . 7
β’ (π β π β (π β { 0 })) |
| 2 | 1 | eldifad 3960 |
. . . . . 6
β’ (π β π β π) |
| 3 | | baerlem3.z |
. . . . . . 7
β’ (π β π β (π β { 0 })) |
| 4 | 3 | eldifad 3960 |
. . . . . 6
β’ (π β π β π) |
| 5 | | baerlem3.v |
. . . . . . 7
β’ π = (Baseβπ) |
| 6 | | baerlem5a.p |
. . . . . . 7
β’ + =
(+gβπ) |
| 7 | | eqid 2732 |
. . . . . . 7
β’
(invgβπ) = (invgβπ) |
| 8 | | baerlem3.m |
. . . . . . 7
β’ β =
(-gβπ) |
| 9 | 5, 6, 7, 8 | grpsubval 18869 |
. . . . . 6
β’ ((π β π β§ π β π) β (π β π) = (π +
((invgβπ)βπ))) |
| 10 | 2, 4, 9 | syl2anc 584 |
. . . . 5
β’ (π β (π β π) = (π +
((invgβπ)βπ))) |
| 11 | 10 | oveq2d 7424 |
. . . 4
β’ (π β (π β (π β π)) = (π β (π +
((invgβπ)βπ)))) |
| 12 | 11 | sneqd 4640 |
. . 3
β’ (π β {(π β (π β π))} = {(π β (π +
((invgβπ)βπ)))}) |
| 13 | 12 | fveq2d 6895 |
. 2
β’ (π β (πβ{(π β (π β π))}) = (πβ{(π β (π +
((invgβπ)βπ)))})) |
| 14 | | baerlem3.o |
. . 3
β’ 0 =
(0gβπ) |
| 15 | | baerlem3.s |
. . 3
β’ β =
(LSSumβπ) |
| 16 | | baerlem3.n |
. . 3
β’ π = (LSpanβπ) |
| 17 | | baerlem3.w |
. . 3
β’ (π β π β LVec) |
| 18 | | baerlem3.x |
. . 3
β’ (π β π β π) |
| 19 | | lveclmod 20716 |
. . . . . 6
β’ (π β LVec β π β LMod) |
| 20 | 17, 19 | syl 17 |
. . . . 5
β’ (π β π β LMod) |
| 21 | 5, 7 | lmodvnegcl 20512 |
. . . . 5
β’ ((π β LMod β§ π β π) β ((invgβπ)βπ) β π) |
| 22 | 20, 4, 21 | syl2anc 584 |
. . . 4
β’ (π β
((invgβπ)βπ) β π) |
| 23 | | eqid 2732 |
. . . . . 6
β’
(LSubSpβπ) =
(LSubSpβπ) |
| 24 | 5, 23, 16, 20, 2, 4 | lspprcl 20588 |
. . . . . 6
β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
| 25 | | baerlem3.c |
. . . . . 6
β’ (π β Β¬ π β (πβ{π, π})) |
| 26 | 14, 23, 20, 24, 18, 25 | lssneln0 20562 |
. . . . 5
β’ (π β π β (π β { 0 })) |
| 27 | 5, 16, 17, 18, 2, 4, 25 | lspindpi 20744 |
. . . . . 6
β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
| 28 | 27 | simpld 495 |
. . . . 5
β’ (π β (πβ{π}) β (πβ{π})) |
| 29 | 5, 14, 16, 17, 26, 2, 28 | lspsnne1 20729 |
. . . 4
β’ (π β Β¬ π β (πβ{π})) |
| 30 | | baerlem3.d |
. . . . . . . 8
β’ (π β (πβ{π}) β (πβ{π})) |
| 31 | 30 | necomd 2996 |
. . . . . . 7
β’ (π β (πβ{π}) β (πβ{π})) |
| 32 | 5, 14, 16, 17, 3, 2, 31 | lspsnne1 20729 |
. . . . . 6
β’ (π β Β¬ π β (πβ{π})) |
| 33 | 5, 16, 17, 18, 4, 2, 32, 25 | lspexchn2 20743 |
. . . . 5
β’ (π β Β¬ π β (πβ{π, π})) |
| 34 | | lmodgrp 20477 |
. . . . . . . . 9
β’ (π β LMod β π β Grp) |
| 35 | 17, 19, 34 | 3syl 18 |
. . . . . . . 8
β’ (π β π β Grp) |
| 36 | 35 | adantr 481 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β Grp) |
| 37 | 4 | adantr 481 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β π) |
| 38 | 5, 7 | grpinvinv 18889 |
. . . . . . 7
β’ ((π β Grp β§ π β π) β ((invgβπ)β((invgβπ)βπ)) = π) |
| 39 | 36, 37, 38 | syl2anc 584 |
. . . . . 6
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)β((invgβπ)βπ)) = π) |
| 40 | 20 | adantr 481 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β LMod) |
| 41 | 5, 23, 16, 20, 2, 18 | lspprcl 20588 |
. . . . . . . 8
β’ (π β (πβ{π, π}) β (LSubSpβπ)) |
| 42 | 41 | adantr 481 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β (πβ{π, π}) β (LSubSpβπ)) |
| 43 | | simpr 485 |
. . . . . . 7
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)βπ) β (πβ{π, π})) |
| 44 | 23, 7 | lssvnegcl 20566 |
. . . . . . 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})) |
| 45 | 40, 42, 43, 44 | syl3anc 1371 |
. . . . . 6
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β ((invgβπ)β((invgβπ)βπ)) β (πβ{π, π})) |
| 46 | 39, 45 | eqeltrrd 2834 |
. . . . 5
β’ ((π β§
((invgβπ)βπ) β (πβ{π, π})) β π β (πβ{π, π})) |
| 47 | 33, 46 | mtand 814 |
. . . 4
β’ (π β Β¬
((invgβπ)βπ) β (πβ{π, π})) |
| 48 | 5, 16, 17, 22, 18, 2, 29, 47 | lspexchn2 20743 |
. . 3
β’ (π β Β¬ π β (πβ{π, ((invgβπ)βπ)})) |
| 49 | 5, 7, 16 | lspsnneg 20616 |
. . . . 5
β’ ((π β LMod β§ π β π) β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 50 | 20, 4, 49 | syl2anc 584 |
. . . 4
β’ (π β (πβ{((invgβπ)βπ)}) = (πβ{π})) |
| 51 | 30, 50 | neeqtrrd 3015 |
. . 3
β’ (π β (πβ{π}) β (πβ{((invgβπ)βπ)})) |
| 52 | 5, 14, 7 | grpinvnzcl 18894 |
. . . 4
β’ ((π β Grp β§ π β (π β { 0 })) β
((invgβπ)βπ) β (π β { 0 })) |
| 53 | 35, 3, 52 | syl2anc 584 |
. . 3
β’ (π β
((invgβπ)βπ) β (π β { 0 })) |
| 54 | 5, 8, 14, 15, 16, 17, 18, 48, 51, 1, 53, 6 | baerlem5a 40580 |
. 2
β’ (π β (πβ{(π β (π +
((invgβπ)βπ)))}) = (((πβ{(π β π)}) β (πβ{((invgβπ)βπ)})) β© ((πβ{(π β
((invgβπ)βπ))}) β (πβ{π})))) |
| 55 | 50 | oveq2d 7424 |
. . 3
β’ (π β ((πβ{(π β π)}) β (πβ{((invgβπ)βπ)})) = ((πβ{(π β π)}) β (πβ{π}))) |
| 56 | 5, 6, 8, 7, 35, 18, 4 | grpsubinv 18895 |
. . . . . 6
β’ (π β (π β
((invgβπ)βπ)) = (π + π)) |
| 57 | 56 | sneqd 4640 |
. . . . 5
β’ (π β {(π β
((invgβπ)βπ))} = {(π + π)}) |
| 58 | 57 | fveq2d 6895 |
. . . 4
β’ (π β (πβ{(π β
((invgβπ)βπ))}) = (πβ{(π + π)})) |
| 59 | 58 | oveq1d 7423 |
. . 3
β’ (π β ((πβ{(π β
((invgβπ)βπ))}) β (πβ{π})) = ((πβ{(π + π)}) β (πβ{π}))) |
| 60 | 55, 59 | ineq12d 4213 |
. 2
β’ (π β (((πβ{(π β π)}) β (πβ{((invgβπ)βπ)})) β© ((πβ{(π β
((invgβπ)βπ))}) β (πβ{π}))) = (((πβ{(π β π)}) β (πβ{π})) β© ((πβ{(π + π)}) β (πβ{π})))) |
| 61 | 13, 54, 60 | 3eqtrd 2776 |
1
β’ (π β (πβ{(π β (π β π))}) = (((πβ{(π β π)}) β (πβ{π})) β© ((πβ{(π + π)}) β (πβ{π})))) |
