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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 2771 | . . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
5 | dochvalr3.i | . . . . . . 7 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | eqid 2771 | . . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
7 | 3, 4, 5, 6 | dihrnss 37085 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
8 | 1, 2, 7 | syl2anc 573 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
9 | dochvalr3.n | . . . . . 6 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) | |
10 | 3, 5, 4, 6, 9 | dochcl 37160 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → (𝑁‘𝑋) ∈ ran 𝐼) |
11 | 1, 8, 10 | syl2anc 573 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ ran 𝐼) |
12 | 3, 5 | dihcnvid2 37080 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
13 | 1, 11, 12 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
14 | dochvalr3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
15 | 14, 3, 5, 9 | dochvalr 37164 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
16 | 1, 2, 15 | syl2anc 573 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
17 | 13, 16 | eqtr2d 2806 | . 2 ⊢ (𝜑 → (𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋)))) |
18 | 1 | simpld 482 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
19 | hlop 35167 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ OP) |
21 | eqid 2771 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
22 | 21, 3, 5 | dihcnvcl 37078 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
23 | 1, 2, 22 | syl2anc 573 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
24 | 21, 14 | opoccl 34999 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾)) → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
25 | 20, 23, 24 | syl2anc 573 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
26 | 21, 3, 5 | dihcnvcl 37078 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
27 | 1, 11, 26 | syl2anc 573 | . . 3 ⊢ (𝜑 → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
28 | 21, 3, 5 | dih11 37072 | . . 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 1476 | . 2 ⊢ (𝜑 → ((𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋))) ↔ ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋)))) |
30 | 17, 29 | mpbid 222 | 1 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ⊆ wss 3723 ◡ccnv 5248 ran crn 5250 ‘cfv 6029 Basecbs 16060 occoc 16153 OPcops 34977 HLchlt 35155 LHypclh 35789 DVecHcdvh 36885 DIsoHcdih 37035 ocHcoch 37154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-riotaBAD 34757 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5821 df-ord 5867 df-on 5868 df-lim 5869 df-suc 5870 df-iota 5992 df-fun 6031 df-fn 6032 df-f 6033 df-f1 6034 df-fo 6035 df-f1o 6036 df-fv 6037 df-riota 6753 df-ov 6795 df-oprab 6796 df-mpt2 6797 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-undef 7551 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11496 df-z 11581 df-uz 11890 df-fz 12530 df-struct 16062 df-ndx 16063 df-slot 16064 df-base 16066 df-sets 16067 df-ress 16068 df-plusg 16158 df-mulr 16159 df-sca 16161 df-vsca 16162 df-0g 16306 df-preset 17132 df-poset 17150 df-plt 17162 df-lub 17178 df-glb 17179 df-join 17180 df-meet 17181 df-p0 17243 df-p1 17244 df-lat 17250 df-clat 17312 df-mgm 17446 df-sgrp 17488 df-mnd 17499 df-submnd 17540 df-grp 17629 df-minusg 17630 df-sbg 17631 df-subg 17795 df-cntz 17953 df-lsm 18254 df-cmn 18398 df-abl 18399 df-mgp 18694 df-ur 18706 df-ring 18753 df-oppr 18827 df-dvdsr 18845 df-unit 18846 df-invr 18876 df-dvr 18887 df-drng 18955 df-lmod 19071 df-lss 19139 df-lsp 19181 df-lvec 19312 df-oposet 34981 df-ol 34983 df-oml 34984 df-covers 35071 df-ats 35072 df-atl 35103 df-cvlat 35127 df-hlat 35156 df-llines 35303 df-lplanes 35304 df-lvols 35305 df-lines 35306 df-psubsp 35308 df-pmap 35309 df-padd 35601 df-lhyp 35793 df-laut 35794 df-ldil 35909 df-ltrn 35910 df-trl 35965 df-tendo 36561 df-edring 36563 df-disoa 36836 df-dvech 36886 df-dib 36946 df-dic 36980 df-dih 37036 df-doch 37155 |
This theorem is referenced by: dihoml4c 37183 |
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