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Mirrors > Home > MPE Home > Th. List > ocv1 | Structured version Visualization version GIF version |
Description: The orthocomplement of the base set. (Contributed by Mario Carneiro, 23-Oct-2015.) |
Ref | Expression |
---|---|
ocvz.v | ⊢ 𝑉 = (Base‘𝑊) |
ocvz.o | ⊢ ⊥ = (ocv‘𝑊) |
ocvz.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
ocv1 | ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘𝑉) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ocvz.v | . . . 4 ⊢ 𝑉 = (Base‘𝑊) | |
2 | ocvz.o | . . . 4 ⊢ ⊥ = (ocv‘𝑊) | |
3 | 1, 2 | ocvss 21595 | . . 3 ⊢ ( ⊥ ‘𝑉) ⊆ 𝑉 |
4 | sseqin2 4211 | . . 3 ⊢ (( ⊥ ‘𝑉) ⊆ 𝑉 ↔ (𝑉 ∩ ( ⊥ ‘𝑉)) = ( ⊥ ‘𝑉)) | |
5 | 3, 4 | mpbi 229 | . 2 ⊢ (𝑉 ∩ ( ⊥ ‘𝑉)) = ( ⊥ ‘𝑉) |
6 | phllmod 21555 | . . . 4 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
7 | eqid 2728 | . . . . 5 ⊢ (LSubSp‘𝑊) = (LSubSp‘𝑊) | |
8 | 1, 7 | lss1 20815 | . . . 4 ⊢ (𝑊 ∈ LMod → 𝑉 ∈ (LSubSp‘𝑊)) |
9 | 6, 8 | syl 17 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑉 ∈ (LSubSp‘𝑊)) |
10 | ocvz.z | . . . 4 ⊢ 0 = (0g‘𝑊) | |
11 | 2, 7, 10 | ocvin 21599 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑉 ∈ (LSubSp‘𝑊)) → (𝑉 ∩ ( ⊥ ‘𝑉)) = { 0 }) |
12 | 9, 11 | mpdan 686 | . 2 ⊢ (𝑊 ∈ PreHil → (𝑉 ∩ ( ⊥ ‘𝑉)) = { 0 }) |
13 | 5, 12 | eqtr3id 2782 | 1 ⊢ (𝑊 ∈ PreHil → ( ⊥ ‘𝑉) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ∩ cin 3944 ⊆ wss 3945 {csn 4624 ‘cfv 6542 Basecbs 17173 0gc0g 17414 LModclmod 20736 LSubSpclss 20808 PreHilcphl 21549 ocvcocv 21585 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-plusg 17239 df-sca 17242 df-vsca 17243 df-ip 17244 df-0g 17416 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-grp 18886 df-minusg 18887 df-sbg 18888 df-ghm 19161 df-cmn 19730 df-abl 19731 df-mgp 20068 df-rng 20086 df-ur 20115 df-ring 20168 df-lmod 20738 df-lss 20809 df-lmhm 20900 df-lvec 20981 df-sra 21051 df-rgmod 21052 df-phl 21551 df-ocv 21588 |
This theorem is referenced by: css0 21614 obslbs 21657 |
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