Step | Hyp | Ref
| Expression |
1 | | hdmap1l6.h |
. . . . . 6
β’ π» = (LHypβπΎ) |
2 | | hdmap1l6.c |
. . . . . 6
β’ πΆ = ((LCDualβπΎ)βπ) |
3 | | hdmap1l6.k |
. . . . . 6
β’ (π β (πΎ β HL β§ π β π»)) |
4 | 1, 2, 3 | lcdlmod 40084 |
. . . . 5
β’ (π β πΆ β LMod) |
5 | | hdmap1l6.u |
. . . . . 6
β’ π = ((DVecHβπΎ)βπ) |
6 | | hdmap1l6.v |
. . . . . 6
β’ π = (Baseβπ) |
7 | | hdmap1l6c.o |
. . . . . 6
β’ 0 =
(0gβπ) |
8 | | hdmap1l6.n |
. . . . . 6
β’ π = (LSpanβπ) |
9 | | hdmap1l6.d |
. . . . . 6
β’ π· = (BaseβπΆ) |
10 | | hdmap1l6.l |
. . . . . 6
β’ πΏ = (LSpanβπΆ) |
11 | | hdmap1l6.m |
. . . . . 6
β’ π = ((mapdβπΎ)βπ) |
12 | | hdmap1l6.i |
. . . . . 6
β’ πΌ = ((HDMap1βπΎ)βπ) |
13 | | hdmap1l6.f |
. . . . . 6
β’ (π β πΉ β π·) |
14 | | hdmap1l6.mn |
. . . . . 6
β’ (π β (πβ(πβ{π})) = (πΏβ{πΉ})) |
15 | 1, 5, 3 | dvhlvec 39601 |
. . . . . . . . 9
β’ (π β π β LVec) |
16 | | hdmap1l6d.w |
. . . . . . . . . 10
β’ (π β π€ β (π β { 0 })) |
17 | 16 | eldifad 3927 |
. . . . . . . . 9
β’ (π β π€ β π) |
18 | | hdmap1l6cl.x |
. . . . . . . . . 10
β’ (π β π β (π β { 0 })) |
19 | 18 | eldifad 3927 |
. . . . . . . . 9
β’ (π β π β π) |
20 | | hdmap1l6d.y |
. . . . . . . . . 10
β’ (π β π β (π β { 0 })) |
21 | 20 | eldifad 3927 |
. . . . . . . . 9
β’ (π β π β π) |
22 | | hdmap1l6d.wn |
. . . . . . . . 9
β’ (π β Β¬ π€ β (πβ{π, π})) |
23 | 6, 8, 15, 17, 19, 21, 22 | lspindpi 20609 |
. . . . . . . 8
β’ (π β ((πβ{π€}) β (πβ{π}) β§ (πβ{π€}) β (πβ{π}))) |
24 | 23 | simpld 496 |
. . . . . . 7
β’ (π β (πβ{π€}) β (πβ{π})) |
25 | 24 | necomd 3000 |
. . . . . 6
β’ (π β (πβ{π}) β (πβ{π€})) |
26 | 1, 5, 6, 7, 8, 2, 9, 10, 11, 12, 3, 13, 14, 25, 18, 17 | hdmap1cl 40296 |
. . . . 5
β’ (π β (πΌββ¨π, πΉ, π€β©) β π·) |
27 | | hdmap1l6.a |
. . . . . 6
β’ β =
(+gβπΆ) |
28 | | hdmap1l6.q |
. . . . . 6
β’ π = (0gβπΆ) |
29 | 9, 27, 28 | lmod0vrid 20369 |
. . . . 5
β’ ((πΆ β LMod β§ (πΌββ¨π, πΉ, π€β©) β π·) β ((πΌββ¨π, πΉ, π€β©) β π) = (πΌββ¨π, πΉ, π€β©)) |
30 | 4, 26, 29 | syl2anc 585 |
. . . 4
β’ (π β ((πΌββ¨π, πΉ, π€β©) β π) = (πΌββ¨π, πΉ, π€β©)) |
31 | 30 | adantr 482 |
. . 3
β’ ((π β§ (π + π) = 0 ) β ((πΌββ¨π, πΉ, π€β©) β π) = (πΌββ¨π, πΉ, π€β©)) |
32 | | oteq3 4846 |
. . . . . 6
β’ ((π + π) = 0 β β¨π, πΉ, (π + π)β© = β¨π, πΉ, 0 β©) |
33 | 32 | fveq2d 6851 |
. . . . 5
β’ ((π + π) = 0 β (πΌββ¨π, πΉ, (π + π)β©) = (πΌββ¨π, πΉ, 0 β©)) |
34 | 1, 5, 6, 7, 2, 9, 28, 12, 3, 13, 19 | hdmap1val0 40291 |
. . . . 5
β’ (π β (πΌββ¨π, πΉ, 0 β©) = π) |
35 | 33, 34 | sylan9eqr 2799 |
. . . 4
β’ ((π β§ (π + π) = 0 ) β (πΌββ¨π, πΉ, (π + π)β©) = π) |
36 | 35 | oveq2d 7378 |
. . 3
β’ ((π β§ (π + π) = 0 ) β ((πΌββ¨π, πΉ, π€β©) β (πΌββ¨π, πΉ, (π + π)β©)) = ((πΌββ¨π, πΉ, π€β©) β π)) |
37 | | oveq2 7370 |
. . . . . 6
β’ ((π + π) = 0 β (π€ + (π + π)) = (π€ + 0 )) |
38 | 1, 5, 3 | dvhlmod 39602 |
. . . . . . 7
β’ (π β π β LMod) |
39 | | hdmap1l6.p |
. . . . . . . 8
β’ + =
(+gβπ) |
40 | 6, 39, 7 | lmod0vrid 20369 |
. . . . . . 7
β’ ((π β LMod β§ π€ β π) β (π€ + 0 ) = π€) |
41 | 38, 17, 40 | syl2anc 585 |
. . . . . 6
β’ (π β (π€ + 0 ) = π€) |
42 | 37, 41 | sylan9eqr 2799 |
. . . . 5
β’ ((π β§ (π + π) = 0 ) β (π€ + (π + π)) = π€) |
43 | 42 | oteq3d 4849 |
. . . 4
β’ ((π β§ (π + π) = 0 ) β β¨π, πΉ, (π€ + (π + π))β© = β¨π, πΉ, π€β©) |
44 | 43 | fveq2d 6851 |
. . 3
β’ ((π β§ (π + π) = 0 ) β (πΌββ¨π, πΉ, (π€ + (π + π))β©) = (πΌββ¨π, πΉ, π€β©)) |
45 | 31, 36, 44 | 3eqtr4rd 2788 |
. 2
β’ ((π β§ (π + π) = 0 ) β (πΌββ¨π, πΉ, (π€ + (π + π))β©) = ((πΌββ¨π, πΉ, π€β©) β (πΌββ¨π, πΉ, (π + π)β©))) |
46 | | hdmap1l6.s |
. . 3
β’ β =
(-gβπ) |
47 | | hdmap1l6.r |
. . 3
β’ π
= (-gβπΆ) |
48 | 3 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (πΎ β HL β§ π β π»)) |
49 | 13 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β πΉ β π·) |
50 | 18 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β π β (π β { 0 })) |
51 | 14 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (πβ(πβ{π})) = (πΏβ{πΉ})) |
52 | 16 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β π€ β (π β { 0 })) |
53 | | hdmap1l6d.z |
. . . . . . 7
β’ (π β π β (π β { 0 })) |
54 | 53 | eldifad 3927 |
. . . . . 6
β’ (π β π β π) |
55 | 6, 39 | lmodvacl 20352 |
. . . . . 6
β’ ((π β LMod β§ π β π β§ π β π) β (π + π) β π) |
56 | 38, 21, 54, 55 | syl3anc 1372 |
. . . . 5
β’ (π β (π + π) β π) |
57 | 56 | anim1i 616 |
. . . 4
β’ ((π β§ (π + π) β 0 ) β ((π + π) β π β§ (π + π) β 0 )) |
58 | | eldifsn 4752 |
. . . 4
β’ ((π + π) β (π β { 0 }) β ((π + π) β π β§ (π + π) β 0 )) |
59 | 57, 58 | sylibr 233 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (π + π) β (π β { 0 })) |
60 | | hdmap1l6d.yz |
. . . . . . 7
β’ (π β (πβ{π}) = (πβ{π})) |
61 | | hdmap1l6d.xn |
. . . . . . . . 9
β’ (π β Β¬ π β (πβ{π, π})) |
62 | 6, 8, 15, 19, 21, 54, 61 | lspindpi 20609 |
. . . . . . . 8
β’ (π β ((πβ{π}) β (πβ{π}) β§ (πβ{π}) β (πβ{π}))) |
63 | 62 | simpld 496 |
. . . . . . 7
β’ (π β (πβ{π}) β (πβ{π})) |
64 | 6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22 | mapdindp1 40212 |
. . . . . 6
β’ (π β (πβ{π}) β (πβ{(π + π)})) |
65 | 6, 39, 7, 8, 15, 18, 20, 53, 16, 60, 63, 22 | mapdindp2 40213 |
. . . . . 6
β’ (π β Β¬ π€ β (πβ{π, (π + π)})) |
66 | 6, 7, 8, 15, 18, 56, 17, 64, 65 | lspindp1 20610 |
. . . . 5
β’ (π β ((πβ{π€}) β (πβ{(π + π)}) β§ Β¬ π β (πβ{π€, (π + π)}))) |
67 | 66 | simprd 497 |
. . . 4
β’ (π β Β¬ π β (πβ{π€, (π + π)})) |
68 | 67 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β Β¬ π β (πβ{π€, (π + π)})) |
69 | 23 | simprd 497 |
. . . . . . . . 9
β’ (π β (πβ{π€}) β (πβ{π})) |
70 | 6, 7, 8, 15, 16, 21, 69 | lspsnne1 20594 |
. . . . . . . 8
β’ (π β Β¬ π€ β (πβ{π})) |
71 | | eqid 2737 |
. . . . . . . . . 10
β’
(LSSumβπ) =
(LSSumβπ) |
72 | 6, 8, 71, 38, 21, 54 | lsmpr 20566 |
. . . . . . . . 9
β’ (π β (πβ{π, π}) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
73 | 60 | oveq2d 7378 |
. . . . . . . . 9
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = ((πβ{π})(LSSumβπ)(πβ{π}))) |
74 | | eqid 2737 |
. . . . . . . . . . . . 13
β’
(LSubSpβπ) =
(LSubSpβπ) |
75 | 6, 74, 8 | lspsncl 20454 |
. . . . . . . . . . . 12
β’ ((π β LMod β§ π β π) β (πβ{π}) β (LSubSpβπ)) |
76 | 38, 21, 75 | syl2anc 585 |
. . . . . . . . . . 11
β’ (π β (πβ{π}) β (LSubSpβπ)) |
77 | 74 | lsssubg 20434 |
. . . . . . . . . . 11
β’ ((π β LMod β§ (πβ{π}) β (LSubSpβπ)) β (πβ{π}) β (SubGrpβπ)) |
78 | 38, 76, 77 | syl2anc 585 |
. . . . . . . . . 10
β’ (π β (πβ{π}) β (SubGrpβπ)) |
79 | 71 | lsmidm 19452 |
. . . . . . . . . 10
β’ ((πβ{π}) β (SubGrpβπ) β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
80 | 78, 79 | syl 17 |
. . . . . . . . 9
β’ (π β ((πβ{π})(LSSumβπ)(πβ{π})) = (πβ{π})) |
81 | 72, 73, 80 | 3eqtr2d 2783 |
. . . . . . . 8
β’ (π β (πβ{π, π}) = (πβ{π})) |
82 | 70, 81 | neleqtrrd 2861 |
. . . . . . 7
β’ (π β Β¬ π€ β (πβ{π, π})) |
83 | 6, 39, 8, 38, 21, 54, 17, 82 | lspindp4 20614 |
. . . . . 6
β’ (π β Β¬ π€ β (πβ{π, (π + π)})) |
84 | 6, 8, 15, 17, 21, 56, 83 | lspindpi 20609 |
. . . . 5
β’ (π β ((πβ{π€}) β (πβ{π}) β§ (πβ{π€}) β (πβ{(π + π)}))) |
85 | 84 | simprd 497 |
. . . 4
β’ (π β (πβ{π€}) β (πβ{(π + π)})) |
86 | 85 | adantr 482 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (πβ{π€}) β (πβ{(π + π)})) |
87 | | eqidd 2738 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (πΌββ¨π, πΉ, π€β©) = (πΌββ¨π, πΉ, π€β©)) |
88 | | eqidd 2738 |
. . 3
β’ ((π β§ (π + π) β 0 ) β (πΌββ¨π, πΉ, (π + π)β©) = (πΌββ¨π, πΉ, (π + π)β©)) |
89 | 1, 5, 6, 39, 46, 7, 8, 2, 9, 27, 47, 28, 10, 11, 12, 48, 49, 50, 51, 52, 59, 68, 86, 87, 88 | hdmap1l6a 40301 |
. 2
β’ ((π β§ (π + π) β 0 ) β (πΌββ¨π, πΉ, (π€ + (π + π))β©) = ((πΌββ¨π, πΉ, π€β©) β (πΌββ¨π, πΉ, (π + π)β©))) |
90 | 45, 89 | pm2.61dane 3033 |
1
β’ (π β (πΌββ¨π, πΉ, (π€ + (π + π))β©) = ((πΌββ¨π, πΉ, π€β©) β (πΌββ¨π, πΉ, (π + π)β©))) |