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Mathbox for Norm Megill |
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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 2735 | . . . . . . 7 ⊢ ((DVecH‘𝐾)‘𝑊) = ((DVecH‘𝐾)‘𝑊) | |
5 | dochvalr3.i | . . . . . . 7 ⊢ 𝐼 = ((DIsoH‘𝐾)‘𝑊) | |
6 | eqid 2735 | . . . . . . 7 ⊢ (Base‘((DVecH‘𝐾)‘𝑊)) = (Base‘((DVecH‘𝐾)‘𝑊)) | |
7 | 3, 4, 5, 6 | dihrnss 41261 | . . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
8 | 1, 2, 7 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) |
9 | dochvalr3.n | . . . . . 6 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) | |
10 | 3, 5, 4, 6, 9 | dochcl 41336 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ⊆ (Base‘((DVecH‘𝐾)‘𝑊))) → (𝑁‘𝑋) ∈ ran 𝐼) |
11 | 1, 8, 10 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝑁‘𝑋) ∈ ran 𝐼) |
12 | 3, 5 | dihcnvid2 41256 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
13 | 1, 11, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐼‘(◡𝐼‘(𝑁‘𝑋))) = (𝑁‘𝑋)) |
14 | dochvalr3.o | . . . . 5 ⊢ ⊥ = (oc‘𝐾) | |
15 | 14, 3, 5, 9 | dochvalr 41340 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
16 | 1, 2, 15 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑁‘𝑋) = (𝐼‘( ⊥ ‘(◡𝐼‘𝑋)))) |
17 | 13, 16 | eqtr2d 2776 | . 2 ⊢ (𝜑 → (𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋)))) |
18 | 1 | simpld 494 | . . . . 5 ⊢ (𝜑 → 𝐾 ∈ HL) |
19 | hlop 39344 | . . . . 5 ⊢ (𝐾 ∈ HL → 𝐾 ∈ OP) | |
20 | 18, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐾 ∈ OP) |
21 | eqid 2735 | . . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
22 | 21, 3, 5 | dihcnvcl 41254 | . . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ ran 𝐼) → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
23 | 1, 2, 22 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (◡𝐼‘𝑋) ∈ (Base‘𝐾)) |
24 | 21, 14 | opoccl 39176 | . . . 4 ⊢ ((𝐾 ∈ OP ∧ (◡𝐼‘𝑋) ∈ (Base‘𝐾)) → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
25 | 20, 23, 24 | syl2anc 584 | . . 3 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) ∈ (Base‘𝐾)) |
26 | 21, 3, 5 | dihcnvcl 41254 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘𝑋) ∈ ran 𝐼) → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
27 | 1, 11, 26 | syl2anc 584 | . . 3 ⊢ (𝜑 → (◡𝐼‘(𝑁‘𝑋)) ∈ (Base‘𝐾)) |
28 | 21, 3, 5 | dih11 41248 | . . 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 1370 | . 2 ⊢ (𝜑 → ((𝐼‘( ⊥ ‘(◡𝐼‘𝑋))) = (𝐼‘(◡𝐼‘(𝑁‘𝑋))) ↔ ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋)))) |
30 | 17, 29 | mpbid 232 | 1 ⊢ (𝜑 → ( ⊥ ‘(◡𝐼‘𝑋)) = (◡𝐼‘(𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ◡ccnv 5688 ran crn 5690 ‘cfv 6563 Basecbs 17245 occoc 17306 OPcops 39154 HLchlt 39332 LHypclh 39967 DVecHcdvh 41061 DIsoHcdih 41211 ocHcoch 41330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tendo 40738 df-edring 40740 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 |
This theorem is referenced by: dihoml4c 41359 |
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