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Mirrors > Home > MPE Home > Th. List > ocvin | Structured version Visualization version GIF version |
Description: An orthocomplement has trivial intersection with the original subspace. (Contributed by Mario Carneiro, 16-Oct-2015.) |
Ref | Expression |
---|---|
ocv2ss.o | ⊢ ⊥ = (ocv‘𝑊) |
ocvin.l | ⊢ 𝐿 = (LSubSp‘𝑊) |
ocvin.z | ⊢ 0 = (0g‘𝑊) |
Ref | Expression |
---|---|
ocvin | ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2724 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2724 | . . . . . . . . 9 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2724 | . . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2724 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | ocv2ss.o | . . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ocvi 21530 | . . . . . . . 8 ⊢ ((𝑥 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
7 | 6 | ancoms 458 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
8 | 7 | adantl 481 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
9 | simpll 764 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil) | |
10 | ocvin.l | . . . . . . . . 9 ⊢ 𝐿 = (LSubSp‘𝑊) | |
11 | 1, 10 | lssel 20774 | . . . . . . . 8 ⊢ ((𝑆 ∈ 𝐿 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑊)) |
12 | 11 | ad2ant2lr 745 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 ∈ (Base‘𝑊)) |
13 | ocvin.z | . . . . . . . 8 ⊢ 0 = (0g‘𝑊) | |
14 | 3, 2, 1, 4, 13 | ipeq0 21499 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
15 | 9, 12, 14 | syl2anc 583 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
16 | 8, 15 | mpbid 231 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 = 0 ) |
17 | 16 | ex 412 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → 𝑥 = 0 )) |
18 | elin 3956 | . . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) | |
19 | velsn 4636 | . . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) | |
20 | 17, 18, 19 | 3imtr4g 296 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) → 𝑥 ∈ { 0 })) |
21 | 20 | ssrdv 3980 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ⊆ { 0 }) |
22 | phllmod 21491 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
23 | 1, 10 | lssss 20773 | . . . . 5 ⊢ (𝑆 ∈ 𝐿 → 𝑆 ⊆ (Base‘𝑊)) |
24 | 1, 5, 10 | ocvlss 21533 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ (Base‘𝑊)) → ( ⊥ ‘𝑆) ∈ 𝐿) |
25 | 23, 24 | sylan2 592 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ( ⊥ ‘𝑆) ∈ 𝐿) |
26 | 10 | lssincl 20802 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
27 | 22, 26 | syl3an1 1160 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
28 | 25, 27 | mpd3an3 1458 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
29 | 13, 10 | lss0ss 20786 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
30 | 22, 28, 29 | syl2an2r 682 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
31 | 21, 30 | eqssd 3991 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∩ cin 3939 ⊆ wss 3940 {csn 4620 ‘cfv 6533 (class class class)co 7401 Basecbs 17143 Scalarcsca 17199 ·𝑖cip 17201 0gc0g 17384 LModclmod 20696 LSubSpclss 20768 PreHilcphl 21485 ocvcocv 21521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-sets 17096 df-slot 17114 df-ndx 17126 df-base 17144 df-plusg 17209 df-sca 17212 df-vsca 17213 df-ip 17214 df-0g 17386 df-mgm 18563 df-sgrp 18642 df-mnd 18658 df-grp 18856 df-minusg 18857 df-sbg 18858 df-ghm 19129 df-cmn 19692 df-abl 19693 df-mgp 20030 df-rng 20048 df-ur 20077 df-ring 20130 df-lmod 20698 df-lss 20769 df-lmhm 20860 df-lvec 20941 df-sra 21011 df-rgmod 21012 df-phl 21487 df-ocv 21524 |
This theorem is referenced by: ocv1 21540 pjdm2 21574 pjff 21575 pjf2 21577 pjfo 21578 obselocv 21591 |
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