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Mirrors > Home > HSE Home > Th. List > ocsh | Structured version Visualization version Unicode version |
Description: The orthogonal complement of a subspace is a subspace. Part of Remark 3.12 of [Beran] p. 107. (Contributed by NM, 7-Aug-2000.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ocsh |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocval 28139 | . . . 4 | |
2 | ssrab2 3687 | . . . 4 | |
3 | 1, 2 | syl6eqss 3655 | . . 3 |
4 | ssel 3597 | . . . . . . 7 | |
5 | hi01 27953 | . . . . . . 7 | |
6 | 4, 5 | syl6 35 | . . . . . 6 |
7 | 6 | ralrimiv 2965 | . . . . 5 |
8 | ax-hv0cl 27860 | . . . . 5 | |
9 | 7, 8 | jctil 560 | . . . 4 |
10 | ocel 28140 | . . . 4 | |
11 | 9, 10 | mpbird 247 | . . 3 |
12 | 3, 11 | jca 554 | . 2 |
13 | ssel2 3598 | . . . . . . . . . 10 | |
14 | ax-his2 27940 | . . . . . . . . . . . . . 14 | |
15 | 14 | 3expa 1265 | . . . . . . . . . . . . 13 |
16 | oveq12 6659 | . . . . . . . . . . . . . 14 | |
17 | 00id 10211 | . . . . . . . . . . . . . 14 | |
18 | 16, 17 | syl6eq 2672 | . . . . . . . . . . . . 13 |
19 | 15, 18 | sylan9eq 2676 | . . . . . . . . . . . 12 |
20 | 19 | ex 450 | . . . . . . . . . . 11 |
21 | 20 | ancoms 469 | . . . . . . . . . 10 |
22 | 13, 21 | sylan 488 | . . . . . . . . 9 |
23 | 22 | an32s 846 | . . . . . . . 8 |
24 | 23 | ralimdva 2962 | . . . . . . 7 |
25 | 24 | imdistanda 729 | . . . . . 6 |
26 | hvaddcl 27869 | . . . . . . 7 | |
27 | 26 | anim1i 592 | . . . . . 6 |
28 | 25, 27 | syl6 35 | . . . . 5 |
29 | ocel 28140 | . . . . . . 7 | |
30 | ocel 28140 | . . . . . . 7 | |
31 | 29, 30 | anbi12d 747 | . . . . . 6 |
32 | an4 865 | . . . . . . 7 | |
33 | r19.26 3064 | . . . . . . . 8 | |
34 | 33 | anbi2i 730 | . . . . . . 7 |
35 | 32, 34 | bitr4i 267 | . . . . . 6 |
36 | 31, 35 | syl6bb 276 | . . . . 5 |
37 | ocel 28140 | . . . . 5 | |
38 | 28, 36, 37 | 3imtr4d 283 | . . . 4 |
39 | 38 | ralrimivv 2970 | . . 3 |
40 | mul01 10215 | . . . . . . . . . . . . 13 | |
41 | oveq2 6658 | . . . . . . . . . . . . . 14 | |
42 | 41 | eqeq1d 2624 | . . . . . . . . . . . . 13 |
43 | 40, 42 | syl5ibrcom 237 | . . . . . . . . . . . 12 |
44 | 43 | ad2antrl 764 | . . . . . . . . . . 11 |
45 | ax-his3 27941 | . . . . . . . . . . . . . 14 | |
46 | 45 | eqeq1d 2624 | . . . . . . . . . . . . 13 |
47 | 46 | 3expa 1265 | . . . . . . . . . . . 12 |
48 | 47 | ancoms 469 | . . . . . . . . . . 11 |
49 | 44, 48 | sylibrd 249 | . . . . . . . . . 10 |
50 | 13, 49 | sylan 488 | . . . . . . . . 9 |
51 | 50 | an32s 846 | . . . . . . . 8 |
52 | 51 | ralimdva 2962 | . . . . . . 7 |
53 | 52 | imdistanda 729 | . . . . . 6 |
54 | hvmulcl 27870 | . . . . . . 7 | |
55 | 54 | anim1i 592 | . . . . . 6 |
56 | 53, 55 | syl6 35 | . . . . 5 |
57 | 30 | anbi2d 740 | . . . . . 6 |
58 | anass 681 | . . . . . 6 | |
59 | 57, 58 | syl6bbr 278 | . . . . 5 |
60 | ocel 28140 | . . . . 5 | |
61 | 56, 59, 60 | 3imtr4d 283 | . . . 4 |
62 | 61 | ralrimivv 2970 | . . 3 |
63 | 39, 62 | jca 554 | . 2 |
64 | issh2 28066 | . 2 | |
65 | 12, 63, 64 | sylanbrc 698 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 w3a 1037 wceq 1483 wcel 1990 wral 2912 crab 2916 wss 3574 cfv 5888 (class class class)co 6650 cc 9934 cc0 9936 caddc 9939 cmul 9941 chil 27776 cva 27777 csm 27778 csp 27779 c0v 27781 csh 27785 cort 27787 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-hilex 27856 ax-hfvadd 27857 ax-hv0cl 27860 ax-hfvmul 27862 ax-hvmul0 27867 ax-hfi 27936 ax-his2 27940 ax-his3 27941 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-ltxr 10079 df-sh 28064 df-oc 28109 |
This theorem is referenced by: shocsh 28143 ocss 28144 occl 28163 spanssoc 28208 ssjo 28306 chscllem2 28497 |
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