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Mirrors > Home > MPE Home > Th. List > Mathboxes > dochvalr3 | Structured version Visualization version GIF version |
Description: Orthocomplement of a closed subspace. (Contributed by NM, 15-Jan-2015.) |
Ref | Expression |
---|---|
dochvalr3.o | ⊢ ⊥ = (oc‘𝐾) |
dochvalr3.h | ⊢ 𝐻 = (LHyp‘𝐾) |
dochvalr3.i | ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) |
dochvalr3.n | ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) |
dochvalr3.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
dochvalr3.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) |
Ref | Expression |
---|---|
dochvalr3 | ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dochvalr3.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
2 | dochvalr3.x | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ran 𝐼) | |
3 | dochvalr3.h | . . . . . . 7 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | eqid 2778 | . . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
5 | dochvalr3.i | . . . . . . 7 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | eqid 2778 | . . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
7 | 3, 4, 5, 6 | dihrnss 37859 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
8 | 1, 2, 7 | syl2anc 576 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
9 | dochvalr3.n | . . . . . 6 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) | |
10 | 3, 5, 4, 6, 9 | dochcl 37934 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → (𝑁‘𝑋) ∈ ran 𝐼) |
11 | 1, 8, 10 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ ran 𝐼) |
12 | 3, 5 | dihcnvid2 37854 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
13 | 1, 11, 12 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
14 | dochvalr3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
15 | 14, 3, 5, 9 | dochvalr 37938 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
16 | 1, 2, 15 | syl2anc 576 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
17 | 13, 16 | eqtr2d 2815 | . 2 ⊢ (𝜑 → (𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋)))) |
18 | 1 | simpld 487 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
19 | hlop 35943 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ OP) |
21 | eqid 2778 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
22 | 21, 3, 5 | dihcnvcl 37852 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
23 | 1, 2, 22 | syl2anc 576 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
24 | 21, 14 | opoccl 35775 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾)) → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
25 | 20, 23, 24 | syl2anc 576 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
26 | 21, 3, 5 | dihcnvcl 37852 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
27 | 1, 11, 26 | syl2anc 576 | . . 3 ⊢ (𝜑 → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
28 | 21, 3, 5 | dih11 37846 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾) ∧ (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) → ((𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋))) ↔ ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋)))) |
29 | 1, 25, 27, 28 | syl3anc 1351 | . 2 ⊢ (𝜑 → ((𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋))) ↔ ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋)))) |
30 | 17, 29 | mpbid 224 | 1 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 = wceq 1507 ∈ wcel 2050 ⊆ wss 3831 ◡ccnv 5407 ran crn 5409 ‘cfv 6190 Basecbs 16342 occoc 16432 OPcops 35753 HLchlt 35931 LHypclh 36565 DVecHcdvh 37659 DIsoHcdih 37809 ocHcoch 37928 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-riotaBAD 35534 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-tpos 7697 df-undef 7744 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-map 8210 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-nn 11442 df-2 11506 df-3 11507 df-4 11508 df-5 11509 df-6 11510 df-n0 11711 df-z 11797 df-uz 12062 df-fz 12712 df-struct 16344 df-ndx 16345 df-slot 16346 df-base 16348 df-sets 16349 df-ress 16350 df-plusg 16437 df-mulr 16438 df-sca 16440 df-vsca 16441 df-0g 16574 df-proset 17399 df-poset 17417 df-plt 17429 df-lub 17445 df-glb 17446 df-join 17447 df-meet 17448 df-p0 17510 df-p1 17511 df-lat 17517 df-clat 17579 df-mgm 17713 df-sgrp 17755 df-mnd 17766 df-submnd 17807 df-grp 17897 df-minusg 17898 df-sbg 17899 df-subg 18063 df-cntz 18221 df-lsm 18525 df-cmn 18671 df-abl 18672 df-mgp 18966 df-ur 18978 df-ring 19025 df-oppr 19099 df-dvdsr 19117 df-unit 19118 df-invr 19148 df-dvr 19159 df-drng 19230 df-lmod 19361 df-lss 19429 df-lsp 19469 df-lvec 19600 df-oposet 35757 df-ol 35759 df-oml 35760 df-covers 35847 df-ats 35848 df-atl 35879 df-cvlat 35903 df-hlat 35932 df-llines 36079 df-lplanes 36080 df-lvols 36081 df-lines 36082 df-psubsp 36084 df-pmap 36085 df-padd 36377 df-lhyp 36569 df-laut 36570 df-ldil 36685 df-ltrn 36686 df-trl 36740 df-tendo 37336 df-edring 37338 df-disoa 37610 df-dvech 37660 df-dib 37720 df-dic 37754 df-dih 37810 df-doch 37929 |
This theorem is referenced by: dihoml4c 37957 |
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