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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochord3 | Structured version Visualization version GIF version |
Description: Ordering law for orthocomplement. (Contributed by NM, 9-Mar-2015.) |
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 |
---|---|
dochord3 | ⊢ (𝜑 → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | doch11.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | doch11.i | . . 3 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
3 | doch11.o | . . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
4 | doch11.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
5 | doch11.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
6 | doch11.y | . . . . . 6 ⊢ (𝜑 → 𝑌 ∈ ran 𝐼) | |
7 | eqid 2738 | . . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
8 | eqid 2738 | . . . . . . 7 ⊢ (LSubSp‘((DVecH‘𝐾)‘𝑊)) = (LSubSp‘((DVecH‘𝐾)‘𝑊)) | |
9 | 1, 7, 2, 8 | dihrnlss 39278 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → 𝑌 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
10 | 4, 6, 9 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊))) |
11 | eqid 2738 | . . . . . 6 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
12 | 11, 8 | lssss 20187 | . . . . 5 ⊢ (𝑌 ∈ (LSubSp‘((DVecH‘𝐾)‘𝑊)) → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
13 | 10, 12 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
14 | 1, 2, 7, 11, 3 | dochcl 39354 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → ( ⊥ ‘𝑌) ∈ ran 𝐼) |
15 | 4, 13, 14 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘𝑌) ∈ ran 𝐼) |
16 | 1, 2, 3, 4, 5, 15 | dochord 39371 | . 2 ⊢ (𝜑 → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ ( ⊥ ‘( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋))) |
17 | 1, 2, 3 | dochoc 39368 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑌 ∈ ran 𝐼) → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
18 | 4, 6, 17 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑌)) = 𝑌) |
19 | 18 | sseq1d 3953 | . 2 ⊢ (𝜑 → (( ⊥ ‘( ⊥ ‘𝑌)) ⊆ ( ⊥ ‘𝑋) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋))) |
20 | 16, 19 | bitrd 278 | 1 ⊢ (𝜑 → (𝑋 ⊆ ( ⊥ ‘𝑌) ↔ 𝑌 ⊆ ( ⊥ ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ⊆ wss 3888 ran crn 5587 ‘cfv 6428 Basecbs 16901 LSubSpclss 20182 HLchlt 37351 LHypclh 37985 DVecHcdvh 39079 DIsoHcdih 39229 ocHcoch 39348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5210 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7580 ax-cnex 10916 ax-resscn 10917 ax-1cn 10918 ax-icn 10919 ax-addcl 10920 ax-addrcl 10921 ax-mulcl 10922 ax-mulrcl 10923 ax-mulcom 10924 ax-addass 10925 ax-mulass 10926 ax-distr 10927 ax-i2m1 10928 ax-1ne0 10929 ax-1rid 10930 ax-rnegex 10931 ax-rrecex 10932 ax-cnre 10933 ax-pre-lttri 10934 ax-pre-lttrn 10935 ax-pre-ltadd 10936 ax-pre-mulgt0 10937 ax-riotaBAD 36954 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-tp 4568 df-op 4570 df-uni 4842 df-int 4882 df-iun 4928 df-iin 4929 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5486 df-eprel 5492 df-po 5500 df-so 5501 df-fr 5541 df-we 5543 df-xp 5592 df-rel 5593 df-cnv 5594 df-co 5595 df-dm 5596 df-rn 5597 df-res 5598 df-ima 5599 df-pred 6197 df-ord 6264 df-on 6265 df-lim 6266 df-suc 6267 df-iota 6386 df-fun 6430 df-fn 6431 df-f 6432 df-f1 6433 df-fo 6434 df-f1o 6435 df-fv 6436 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7705 df-1st 7822 df-2nd 7823 df-tpos 8031 df-undef 8078 df-frecs 8086 df-wrecs 8117 df-recs 8191 df-rdg 8230 df-1o 8286 df-er 8487 df-map 8606 df-en 8723 df-dom 8724 df-sdom 8725 df-fin 8726 df-pnf 11000 df-mnf 11001 df-xr 11002 df-ltxr 11003 df-le 11004 df-sub 11196 df-neg 11197 df-nn 11963 df-2 12025 df-3 12026 df-4 12027 df-5 12028 df-6 12029 df-n0 12223 df-z 12309 df-uz 12572 df-fz 13229 df-struct 16837 df-sets 16854 df-slot 16872 df-ndx 16884 df-base 16902 df-ress 16931 df-plusg 16964 df-mulr 16965 df-sca 16967 df-vsca 16968 df-0g 17141 df-proset 18002 df-poset 18020 df-plt 18037 df-lub 18053 df-glb 18054 df-join 18055 df-meet 18056 df-p0 18132 df-p1 18133 df-lat 18139 df-clat 18206 df-mgm 18315 df-sgrp 18364 df-mnd 18375 df-submnd 18420 df-grp 18569 df-minusg 18570 df-sbg 18571 df-subg 18741 df-cntz 18912 df-lsm 19230 df-cmn 19377 df-abl 19378 df-mgp 19710 df-ur 19727 df-ring 19774 df-oppr 19851 df-dvdsr 19872 df-unit 19873 df-invr 19903 df-dvr 19914 df-drng 19982 df-lmod 20114 df-lss 20183 df-lsp 20223 df-lvec 20354 df-oposet 37177 df-ol 37179 df-oml 37180 df-covers 37267 df-ats 37268 df-atl 37299 df-cvlat 37323 df-hlat 37352 df-llines 37499 df-lplanes 37500 df-lvols 37501 df-lines 37502 df-psubsp 37504 df-pmap 37505 df-padd 37797 df-lhyp 37989 df-laut 37990 df-ldil 38105 df-ltrn 38106 df-trl 38160 df-tendo 38756 df-edring 38758 df-disoa 39030 df-dvech 39080 df-dib 39140 df-dic 39174 df-dih 39230 df-doch 39349 |
This theorem is referenced by: dochsncom 39383 |
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