Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 39087 | . . . 4 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
7 | rabeq 3483 | . . . 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 2821 | . . . . . . 7 ⊢ ((HDMap‘𝐾)‘𝑊) = ((HDMap‘𝐾)‘𝑊) | |
10 | 3 | ad2antrr 724 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
11 | eqid 2821 | . . . . . . 7 ⊢ (·𝑖‘𝑈) = (·𝑖‘𝑈) | |
12 | simplr 767 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑦 ∈ 𝑉) | |
13 | hlhilocv.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ⊆ 𝑉) | |
14 | 13 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑉) → 𝑋 ⊆ 𝑉) |
15 | 14 | sselda 3967 | . . . . . . 7 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑉) |
16 | 1, 4, 5, 9, 2, 10, 11, 12, 15 | hlhilipval 39100 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (𝑦(·𝑖‘𝑈)𝑧) = ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦)) |
17 | eqid 2821 | . . . . . . . . 9 ⊢ (Scalar‘𝐿) = (Scalar‘𝐿) | |
18 | eqid 2821 | . . . . . . . . 9 ⊢ (Scalar‘𝑈) = (Scalar‘𝑈) | |
19 | eqid 2821 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝐿)) | |
20 | 1, 4, 17, 2, 18, 3, 19 | hlhils0 39096 | . . . . . . . 8 ⊢ (𝜑 → (0g‘(Scalar‘𝐿)) = (0g‘(Scalar‘𝑈))) |
21 | 20 | eqcomd 2827 | . . . . . . 7 ⊢ (𝜑 → (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝐿))) |
22 | 21 | ad2antrr 724 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝐿))) |
23 | 16, 22 | eqeq12d 2837 | . . . . 5 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝑉) ∧ 𝑧 ∈ 𝑋) → ((𝑦(·𝑖‘𝑈)𝑧) = (0g‘(Scalar‘𝑈)) ↔ ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿)))) |
24 | 23 | ralbidva 3196 | . . . 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 3478 | . . 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 2858 | . 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 4007 | . . 3 ⊢ (𝜑 → 𝑋 ⊆ (Base‘𝑈)) |
28 | eqid 2821 | . . . 4 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
29 | eqid 2821 | . . . 4 ⊢ (0g‘(Scalar‘𝑈)) = (0g‘(Scalar‘𝑈)) | |
30 | hlhilocv.o | . . . 4 ⊢ 𝑂 = (ocv‘𝑈) | |
31 | 28, 11, 18, 29, 30 | ocvval 20811 | . . 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 39082 | . 2 ⊢ (𝜑 → (𝑁‘𝑋) = {𝑦 ∈ 𝑉 ∣ ∀𝑧 ∈ 𝑋 ((((HDMap‘𝐾)‘𝑊)‘𝑧)‘𝑦) = (0g‘(Scalar‘𝐿))}) |
35 | 26, 32, 34 | 3eqtr4d 2866 | 1 ⊢ (𝜑 → (𝑂‘𝑋) = (𝑁‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3138 {crab 3142 ⊆ wss 3936 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 Scalarcsca 16568 ·𝑖cip 16570 0gc0g 16713 ocvcocv 20804 HLchlt 36501 LHypclh 37135 DVecHcdvh 38229 ocHcoch 38498 HDMapchdma 38943 HLHilchlh 39083 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-riotaBAD 36104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-tpos 7892 df-undef 7939 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-n0 11899 df-z 11983 df-uz 12245 df-fz 12894 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-0g 16715 df-mre 16857 df-mrc 16858 df-acs 16860 df-proset 17538 df-poset 17556 df-plt 17568 df-lub 17584 df-glb 17585 df-join 17586 df-meet 17587 df-p0 17649 df-p1 17650 df-lat 17656 df-clat 17718 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-grp 18106 df-minusg 18107 df-sbg 18108 df-subg 18276 df-cntz 18447 df-oppg 18474 df-lsm 18761 df-cmn 18908 df-abl 18909 df-mgp 19240 df-ur 19252 df-ring 19299 df-oppr 19373 df-dvdsr 19391 df-unit 19392 df-invr 19422 df-dvr 19433 df-drng 19504 df-lmod 19636 df-lss 19704 df-lsp 19744 df-lvec 19875 df-ocv 20807 df-lsatoms 36127 df-lshyp 36128 df-lcv 36170 df-lfl 36209 df-lkr 36237 df-ldual 36275 df-oposet 36327 df-ol 36329 df-oml 36330 df-covers 36417 df-ats 36418 df-atl 36449 df-cvlat 36473 df-hlat 36502 df-llines 36649 df-lplanes 36650 df-lvols 36651 df-lines 36652 df-psubsp 36654 df-pmap 36655 df-padd 36947 df-lhyp 37139 df-laut 37140 df-ldil 37255 df-ltrn 37256 df-trl 37310 df-tgrp 37894 df-tendo 37906 df-edring 37908 df-dveca 38154 df-disoa 38180 df-dvech 38230 df-dib 38290 df-dic 38324 df-dih 38380 df-doch 38499 df-djh 38546 df-lcdual 38738 df-mapd 38776 df-hvmap 38908 df-hdmap1 38944 df-hdmap 38945 df-hlhil 39084 |
This theorem is referenced by: hlhillcs 39109 hlhilhillem 39111 |
Copyright terms: Public domain | W3C validator |