Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilocv | Structured version Visualization version GIF version |
Description: The orthocomplement for the final constructed Hilbert space. (Contributed by NM, 23-Jun-2015.) (Revised by Mario Carneiro, 29-Jun-2015.) |
Ref | Expression |
---|---|
hlhil0.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhil0.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhil0.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhil0.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilocv.v | ⊢ 𝑉 = (Base‘𝐿) |
hlhilocv.n | ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) |
hlhilocv.o | ⊢ 𝑂 = (ocv‘𝑈) |
hlhilocv.x | ⊢ (𝜑 → 𝑋 ⊆ 𝑉) |
Ref | Expression |
---|---|
hlhilocv | ⊢ (𝜑 → (𝑂‘𝑋) = (𝑁‘𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhil0.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhil0.u | . . . . 5 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
3 | hlhil0.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | hlhil0.l | . . . . 5 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
5 | hlhilocv.v | . . . . 5 ⊢ 𝑉 = (Base‘𝐿) | |
6 | 1, 2, 3, 4, 5 | hlhilbase 37228 | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | rabeq 3192 | . . . 4 ⊢ (𝑉 = (Base‘𝑈) → {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))} = {𝑦 ∈ (Base‘𝑈) ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))}) | |
8 | 6, 7 | syl 17 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))} = {𝑦 ∈ (Base‘𝑈) ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))}) |
9 | eqid 2622 | . . . . . . 7 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
10 | 3 | ad2antrr 762 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | eqid 2622 | . . . . . . 7 ⊢ (·𝑖‘𝑈) = (·𝑖‘𝑈) | |
12 | simplr 792 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉) | |
13 | hlhilocv.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
14 | 13 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉) |
15 | 14 | sselda 3603 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉) |
16 | 1, 4, 5, 9, 2, 10, 11, 12, 15 | hlhilipval 37241 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝑦(·𝑖‘𝑈)𝑧) = ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦)) |
17 | eqid 2622 | . . . . . . . . 9 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿) | |
18 | eqid 2622 | . . . . . . . . 9 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
19 | eqid 2622 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿)) | |
20 | 1, 4, 17, 2, 18, 3, 19 | hlhils0 37237 | . . . . . . . 8 ⊢ (𝜑 → (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝑈))) |
21 | 20 | eqcomd 2628 | . . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝐿))) |
22 | 21 | ad2antrr 762 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝐿))) |
23 | 16, 22 | eqeq12d 2637 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → ((𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈)) ↔ ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿)))) |
24 | 23 | ralbidva 2985 | . . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → (∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈)) ↔ ∀𝑧 ∈ 𝑋 ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿)))) |
25 | 24 | rabbidva 3188 | . . 3 ⊢ (𝜑 → {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))} = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿))}) |
26 | 8, 25 | eqtr3d 2658 | . 2 ⊢ (𝜑 → {𝑦 ∈ (Base‘𝑈) ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))} = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿))}) |
27 | 13, 6 | sseqtrd 3641 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
28 | eqid 2622 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | eqid 2622 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
30 | hlhilocv.o | . . . 4 ⊢ 𝑂 = (ocv‘𝑈) | |
31 | 28, 11, 18, 29, 30 | ocvval 20011 | . . 3 ⊢ (𝑋 ⊆ (Base‘𝑈) → (𝑂‘𝑋) = {𝑦 ∈ (Base‘𝑈) ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))}) |
32 | 27, 31 | syl 17 | . 2 ⊢ (𝜑 → (𝑂‘𝑋) = {𝑦 ∈ (Base‘𝑈) ∣ ∀𝑧 ∈ 𝑋 (𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈))}) |
33 | hlhilocv.n | . . 3 ⊢ 𝑁 = ((ocH‘𝐾)‘𝑊) | |
34 | 1, 4, 5, 17, 19, 33, 9, 3, 13 | hdmapoc 37223 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿))}) |
35 | 26, 32, 34 | 3eqtr4d 2666 | 1 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 {crab 2916 ⊆ wss 3574 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 Scalarcsca 15944 ·𝑖cip 15946 0gc0g 16100 ocvcocv 20004 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 ocHcoch 36636 HDMapchdma 37082 HLHilchlh 37224 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-ocv 20007 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hvmap 37046 df-hdmap1 37083 df-hdmap 37084 df-hlhil 37225 |
This theorem is referenced by: hlhillcs 37250 hlhilhillem 37252 |
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