![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > dochord | Structured version Visualization version GIF version |
Description: Ordering law for orthocomplement. (Contributed by NM, 12-Aug-2014.) |
Ref | Expression |
---|---|
doch11.h | ⊢ 𝐻 = (LHyp‘𝐾) |
doch11.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
doch11.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
doch11.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
doch11.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
doch11.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dochord | ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch11.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | 1 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
3 | doch11.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
4 | doch11.h | . . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾) | |
5 | eqid 2736 | . . . . . 6 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
6 | doch11.i | . . . . . 6 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
7 | eqid 2736 | . . . . . 6 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
8 | 4, 5, 6, 7 | dihrnss 39673 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
9 | 1, 3, 8 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
10 | 9 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
11 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → 𝑋 ⊆ 𝑌) | |
12 | doch11.o | . . . 4 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
13 | 4, 5, 7, 12 | dochss 39760 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
14 | 2, 10, 11, 13 | syl3anc 1371 | . 2 ⊢ ((𝜑 ∧ 𝑋 ⊆ 𝑌) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) |
15 | 1 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
16 | doch11.x | . . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
17 | 4, 5, 6, 7 | dihrnss 39673 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
18 | 1, 16, 17 | syl2anc 584 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
19 | 4, 6, 5, 7, 12 | dochcl 39748 | . . . . . . 7 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑋) ∈ ran 𝐼) |
20 | 1, 18, 19 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ( ⊥ ‘𝑋) ∈ ran 𝐼) |
21 | 4, 5, 6, 7 | dihrnss 39673 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ∈ ran 𝐼) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
22 | 1, 20, 21 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
23 | 22 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
24 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) | |
25 | 4, 5, 7, 12 | dochss 39760 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘𝑋) ⊆ (Base‘((DVecH‘𝐾)‘𝑊)) ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
26 | 15, 23, 24, 25 | syl3anc 1371 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) ⊆ ( ⊥ ‘( ⊥ ‘𝑌))) |
27 | 4, 6, 12 | dochoc 39762 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
28 | 1, 16, 27 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
29 | 28 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑋)) = 𝑋) |
30 | 4, 6, 12 | dochoc 39762 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
31 | 1, 3, 30 | syl2anc 584 | . . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
32 | 31 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
33 | 26, 29, 32 | 3sstr3d 3988 | . 2 ⊢ ((𝜑 ∧ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋)) → 𝑋 ⊆ 𝑌) |
34 | 14, 33 | impbida 799 | 1 ⊢ (𝜑 → (𝑋 ⊆ 𝑌 ↔ ( ⊥ ‘𝑌) ⊆ ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ⊆ wss 3908 ran crn 5632 ‘cfv 6493 Basecbs 17037 HLchlt 37744 LHypclh 38379 DVecHcdvh 39473 DIsoHcdih 39623 ocHcoch 39742 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 37347 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-tpos 8149 df-undef 8196 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-1o 8404 df-er 8606 df-map 8725 df-en 8842 df-dom 8843 df-sdom 8844 df-fin 8845 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-n0 12372 df-z 12458 df-uz 12722 df-fz 13379 df-struct 16973 df-sets 16990 df-slot 17008 df-ndx 17020 df-base 17038 df-ress 17067 df-plusg 17100 df-mulr 17101 df-sca 17103 df-vsca 17104 df-0g 17277 df-proset 18138 df-poset 18156 df-plt 18173 df-lub 18189 df-glb 18190 df-join 18191 df-meet 18192 df-p0 18268 df-p1 18269 df-lat 18275 df-clat 18342 df-mgm 18451 df-sgrp 18500 df-mnd 18511 df-submnd 18556 df-grp 18705 df-minusg 18706 df-sbg 18707 df-subg 18878 df-cntz 19050 df-lsm 19371 df-cmn 19517 df-abl 19518 df-mgp 19850 df-ur 19867 df-ring 19914 df-oppr 19996 df-dvdsr 20017 df-unit 20018 df-invr 20048 df-dvr 20059 df-drng 20134 df-lmod 20271 df-lss 20340 df-lsp 20380 df-lvec 20511 df-oposet 37570 df-ol 37572 df-oml 37573 df-covers 37660 df-ats 37661 df-atl 37692 df-cvlat 37716 df-hlat 37745 df-llines 37893 df-lplanes 37894 df-lvols 37895 df-lines 37896 df-psubsp 37898 df-pmap 37899 df-padd 38191 df-lhyp 38383 df-laut 38384 df-ldil 38499 df-ltrn 38500 df-trl 38554 df-tendo 39150 df-edring 39152 df-disoa 39424 df-dvech 39474 df-dib 39534 df-dic 39568 df-dih 39624 df-doch 39743 |
This theorem is referenced by: dochord2N 39766 dochord3 39767 doch11 39768 dochsordN 39769 dochsatshpb 39847 hdmapoc 40326 |
Copyright terms: Public domain | W3C validator |