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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 2820 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2820 | . . . . . . . . 9 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2820 | . . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2820 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | ocv2ss.o | . . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ocvi 20791 | . . . . . . . 8 ⊢ ((𝑥 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
7 | 6 | ancoms 461 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
8 | 7 | adantl 484 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
9 | simpll 765 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil) | |
10 | ocvin.l | . . . . . . . . 9 ⊢ 𝐿 = (LSubSp‘𝑊) | |
11 | 1, 10 | lssel 19687 | . . . . . . . 8 ⊢ ((𝑆 ∈ 𝐿 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑊)) |
12 | 11 | ad2ant2lr 746 | . . . . . . 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 20760 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
15 | 9, 12, 14 | syl2anc 586 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
16 | 8, 15 | mpbid 234 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 = 0 ) |
17 | 16 | ex 415 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → 𝑥 = 0 )) |
18 | elin 4152 | . . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) | |
19 | velsn 4564 | . . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) | |
20 | 17, 18, 19 | 3imtr4g 298 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) → 𝑥 ∈ { 0 })) |
21 | 20 | ssrdv 3956 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ⊆ { 0 }) |
22 | phllmod 20752 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
23 | 1, 10 | lssss 19686 | . . . . 5 ⊢ (𝑆 ∈ 𝐿 → 𝑆 ⊆ (Base‘𝑊)) |
24 | 1, 5, 10 | ocvlss 20794 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ (Base‘𝑊)) → ( ⊥ ‘𝑆) ∈ 𝐿) |
25 | 23, 24 | sylan2 594 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ( ⊥ ‘𝑆) ∈ 𝐿) |
26 | 10 | lssincl 19715 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
27 | 22, 26 | syl3an1 1159 | . . . 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 19698 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
30 | 22, 28, 29 | syl2an2r 683 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
31 | 21, 30 | eqssd 3967 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∩ cin 3918 ⊆ wss 3919 {csn 4548 ‘cfv 6336 (class class class)co 7137 Basecbs 16461 Scalarcsca 16546 ·𝑖cip 16548 0gc0g 16691 LModclmod 19612 LSubSpclss 19681 PreHilcphl 20746 ocvcocv 20782 |
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 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-er 8270 df-en 8491 df-dom 8492 df-sdom 8493 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-nn 11620 df-2 11682 df-3 11683 df-4 11684 df-5 11685 df-6 11686 df-7 11687 df-8 11688 df-ndx 16464 df-slot 16465 df-base 16467 df-sets 16468 df-plusg 16556 df-sca 16559 df-vsca 16560 df-ip 16561 df-0g 16693 df-mgm 17830 df-sgrp 17879 df-mnd 17890 df-grp 18084 df-minusg 18085 df-sbg 18086 df-ghm 18334 df-mgp 19218 df-ur 19230 df-ring 19277 df-lmod 19614 df-lss 19682 df-lmhm 19772 df-lvec 19853 df-sra 19922 df-rgmod 19923 df-phl 20748 df-ocv 20785 |
This theorem is referenced by: ocv1 20801 pjdm2 20833 pjff 20834 pjf2 20836 pjfo 20837 obselocv 20850 |
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