![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > hlhilhillem | Structured version Visualization version GIF version |
Description: Lemma for hlhil 24791. (Contributed by NM, 23-Jun-2015.) |
Ref | Expression |
---|---|
hlhilphl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
hlhilphllem.u | ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) |
hlhilphl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
hlhilphllem.f | ⊢ 𝐹 = (Scalar‘𝑈) |
hlhilphllem.l | ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) |
hlhilphllem.v | ⊢ 𝑉 = (Base‘𝐿) |
hlhilphllem.a | ⊢ + = (+g‘𝐿) |
hlhilphllem.s | ⊢ · = ( ·𝑠 ‘𝐿) |
hlhilphllem.r | ⊢ 𝑅 = (Scalar‘𝐿) |
hlhilphllem.b | ⊢ 𝐵 = (Base‘𝑅) |
hlhilphllem.p | ⊢ ⨣ = (+g‘𝑅) |
hlhilphllem.t | ⊢ × = (.r‘𝑅) |
hlhilphllem.q | ⊢ 𝑄 = (0g‘𝑅) |
hlhilphllem.z | ⊢ 0 = (0g‘𝐿) |
hlhilphllem.i | ⊢ , = (·𝑖‘𝑈) |
hlhilphllem.j | ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) |
hlhilphllem.g | ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) |
hlhilphllem.e | ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) |
hlhilphllem.o | ⊢ 𝑂 = (ocv‘𝑈) |
hlhilphllem.c | ⊢ 𝐶 = (ClSubSp‘𝑈) |
Ref | Expression |
---|---|
hlhilhillem | ⊢ (𝜑 → 𝑈 ∈ Hil) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlhilphl.h | . . 3 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | hlhilphllem.u | . . 3 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊) | |
3 | hlhilphl.k | . . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | hlhilphllem.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑈) | |
5 | hlhilphllem.l | . . 3 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊) | |
6 | hlhilphllem.v | . . 3 ⊢ 𝑉 = (Base‘𝐿) | |
7 | hlhilphllem.a | . . 3 ⊢ + = (+g‘𝐿) | |
8 | hlhilphllem.s | . . 3 ⊢ · = ( ·𝑠 ‘𝐿) | |
9 | hlhilphllem.r | . . 3 ⊢ 𝑅 = (Scalar‘𝐿) | |
10 | hlhilphllem.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
11 | hlhilphllem.p | . . 3 ⊢ ⨣ = (+g‘𝑅) | |
12 | hlhilphllem.t | . . 3 ⊢ × = (.r‘𝑅) | |
13 | hlhilphllem.q | . . 3 ⊢ 𝑄 = (0g‘𝑅) | |
14 | hlhilphllem.z | . . 3 ⊢ 0 = (0g‘𝐿) | |
15 | hlhilphllem.i | . . 3 ⊢ , = (·𝑖‘𝑈) | |
16 | hlhilphllem.j | . . 3 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊) | |
17 | hlhilphllem.g | . . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊) | |
18 | hlhilphllem.e | . . 3 ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥)) | |
19 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 | hlhilphllem 40393 | . 2 ⊢ (𝜑 → 𝑈 ∈ PreHil) |
20 | 3 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
21 | eqid 2736 | . . . . . . 7 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
22 | hlhilphllem.o | . . . . . . 7 ⊢ 𝑂 = (ocv‘𝑈) | |
23 | eqid 2736 | . . . . . . . . . . 11 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
24 | hlhilphllem.c | . . . . . . . . . . 11 ⊢ 𝐶 = (ClSubSp‘𝑈) | |
25 | 1, 23, 2, 24, 3 | hlhillcs 40392 | . . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ran ((DIsoH‘𝐾)‘𝑊)) |
26 | 25 | eleq2d 2823 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊))) |
27 | 26 | biimpa 477 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
28 | 1, 5, 23, 6 | dihrnss 39708 | . . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉) |
29 | 3, 28 | sylan 580 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉) |
30 | 27, 29 | syldan 591 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ⊆ 𝑉) |
31 | 1, 5, 2, 20, 6, 21, 22, 30 | hlhilocv 40391 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑂‘𝑥) = (((ocH‘𝐾)‘𝑊)‘𝑥)) |
32 | 31 | oveq2d 7369 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥))) |
33 | eqid 2736 | . . . . . . . 8 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿) | |
34 | 1, 5, 2, 3, 33 | hlhillsm 40390 | . . . . . . 7 ⊢ (𝜑 → (LSSum‘𝐿) = (LSSum‘𝑈)) |
35 | 34 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (LSSum‘𝐿) = (LSSum‘𝑈)) |
36 | 35 | oveqd 7370 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝑈)(𝑂‘𝑥))) |
37 | eqid 2736 | . . . . . . 7 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿) | |
38 | 3 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
39 | 1, 5, 23, 37 | dihrnlss 39707 | . . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿)) |
40 | 3, 39 | sylan 580 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿)) |
41 | 1, 23, 5, 6, 21, 38, 29 | dochoccl 39799 | . . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥)) |
42 | 41 | biimpd 228 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥)) |
43 | 42 | ex 413 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))) |
44 | 43 | pm2.43d 53 | . . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥)) |
45 | 44 | imp 407 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥) |
46 | 1, 21, 5, 6, 37, 33, 38, 40, 45 | dochexmid 39898 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉) |
47 | 27, 46 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉) |
48 | 32, 36, 47 | 3eqtr3d 2784 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = 𝑉) |
49 | 1, 2, 3, 5, 6 | hlhilbase 40366 | . . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑈)) |
50 | 49 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑉 = (Base‘𝑈)) |
51 | 48, 50 | eqtrd 2776 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈)) |
52 | 51 | ralrimiva 3141 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈)) |
53 | eqid 2736 | . . 3 ⊢ (Base‘𝑈) = (Base‘𝑈) | |
54 | eqid 2736 | . . 3 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈) | |
55 | 53, 54, 22, 24 | ishil2 21110 | . 2 ⊢ (𝑈 ∈ Hil ↔ (𝑈 ∈ PreHil ∧ ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))) |
56 | 19, 52, 55 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑈 ∈ Hil) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ⊆ wss 3908 ran crn 5632 ‘cfv 6493 (class class class)co 7353 ∈ cmpo 7355 Basecbs 17075 +gcplusg 17125 .rcmulr 17126 Scalarcsca 17128 ·𝑠 cvsca 17129 ·𝑖cip 17130 0gc0g 17313 LSSumclsm 19407 LSubSpclss 20377 PreHilcphl 21013 ocvcocv 21049 ClSubSpccss 21050 Hilchil 21092 HLchlt 37779 LHypclh 38414 DVecHcdvh 39508 DIsoHcdih 39658 ocHcoch 39777 HDMapchdma 40222 HGMapchg 40313 HLHilchlh 40362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-riotaBAD 37382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-undef 8200 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-starv 17140 df-sca 17141 df-vsca 17142 df-ip 17143 df-0g 17315 df-mre 17458 df-mrc 17459 df-acs 17461 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-p1 18307 df-lat 18313 df-clat 18380 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-mhm 18593 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-subg 18916 df-ghm 18997 df-cntz 19088 df-oppg 19115 df-lsm 19409 df-pj1 19410 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-ring 19952 df-oppr 20034 df-dvdsr 20055 df-unit 20056 df-invr 20086 df-dvr 20097 df-rnghom 20131 df-drng 20172 df-subrg 20205 df-staf 20289 df-srng 20290 df-lmod 20309 df-lss 20378 df-lsp 20418 df-lmhm 20468 df-lvec 20549 df-sra 20618 df-rgmod 20619 df-phl 21015 df-ocv 21052 df-css 21053 df-pj 21094 df-hil 21095 df-lsatoms 37405 df-lshyp 37406 df-lcv 37448 df-lfl 37487 df-lkr 37515 df-ldual 37553 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 df-lvols 37930 df-lines 37931 df-psubsp 37933 df-pmap 37934 df-padd 38226 df-lhyp 38418 df-laut 38419 df-ldil 38534 df-ltrn 38535 df-trl 38589 df-tgrp 39173 df-tendo 39185 df-edring 39187 df-dveca 39433 df-disoa 39459 df-dvech 39509 df-dib 39569 df-dic 39603 df-dih 39659 df-doch 39778 df-djh 39825 df-lcdual 40017 df-mapd 40055 df-hvmap 40187 df-hdmap1 40223 df-hdmap 40224 df-hgmap 40314 df-hlhil 40363 |
This theorem is referenced by: hlathil 40395 |
Copyright terms: Public domain | W3C validator |