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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 2738 | . . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2738 | . . . . . . . . 9 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊) | |
3 | eqid 2738 | . . . . . . . . 9 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
4 | eqid 2738 | . . . . . . . . 9 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
5 | ocv2ss.o | . . . . . . . . 9 ⊢ ⊥ = (ocv‘𝑊) | |
6 | 1, 2, 3, 4, 5 | ocvi 20485 | . . . . . . . 8 ⊢ ((𝑥 ∈ ( ⊥ ‘𝑆) ∧ 𝑥 ∈ 𝑆) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
7 | 6 | ancoms 462 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
8 | 7 | adantl 485 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → (𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊))) |
9 | simpll 767 | . . . . . . 7 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑊 ∈ PreHil) | |
10 | ocvin.l | . . . . . . . . 9 ⊢ 𝐿 = (LSubSp‘𝑊) | |
11 | 1, 10 | lssel 19828 | . . . . . . . 8 ⊢ ((𝑆 ∈ 𝐿 ∧ 𝑥 ∈ 𝑆) → 𝑥 ∈ (Base‘𝑊)) |
12 | 11 | ad2ant2lr 748 | . . . . . . 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 20454 | . . . . . . 7 ⊢ ((𝑊 ∈ PreHil ∧ 𝑥 ∈ (Base‘𝑊)) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
15 | 9, 12, 14 | syl2anc 587 | . . . . . 6 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → ((𝑥(·𝑖‘𝑊)𝑥) = (0g‘(Scalar‘𝑊)) ↔ 𝑥 = 0 )) |
16 | 8, 15 | mpbid 235 | . . . . 5 ⊢ (((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) ∧ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) → 𝑥 = 0 ) |
17 | 16 | ex 416 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ((𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆)) → 𝑥 = 0 )) |
18 | elin 3859 | . . . 4 ⊢ (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) ↔ (𝑥 ∈ 𝑆 ∧ 𝑥 ∈ ( ⊥ ‘𝑆))) | |
19 | velsn 4532 | . . . 4 ⊢ (𝑥 ∈ { 0 } ↔ 𝑥 = 0 ) | |
20 | 17, 18, 19 | 3imtr4g 299 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑥 ∈ (𝑆 ∩ ( ⊥ ‘𝑆)) → 𝑥 ∈ { 0 })) |
21 | 20 | ssrdv 3883 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ⊆ { 0 }) |
22 | phllmod 20446 | . . 3 ⊢ (𝑊 ∈ PreHil → 𝑊 ∈ LMod) | |
23 | 1, 10 | lssss 19827 | . . . . 5 ⊢ (𝑆 ∈ 𝐿 → 𝑆 ⊆ (Base‘𝑊)) |
24 | 1, 5, 10 | ocvlss 20488 | . . . . 5 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ⊆ (Base‘𝑊)) → ( ⊥ ‘𝑆) ∈ 𝐿) |
25 | 23, 24 | sylan2 596 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → ( ⊥ ‘𝑆) ∈ 𝐿) |
26 | 10 | lssincl 19856 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
27 | 22, 26 | syl3an1 1164 | . . . 4 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿 ∧ ( ⊥ ‘𝑆) ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
28 | 25, 27 | mpd3an3 1463 | . . 3 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) |
29 | 13, 10 | lss0ss 19839 | . . 3 ⊢ ((𝑊 ∈ LMod ∧ (𝑆 ∩ ( ⊥ ‘𝑆)) ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
30 | 22, 28, 29 | syl2an2r 685 | . 2 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → { 0 } ⊆ (𝑆 ∩ ( ⊥ ‘𝑆))) |
31 | 21, 30 | eqssd 3894 | 1 ⊢ ((𝑊 ∈ PreHil ∧ 𝑆 ∈ 𝐿) → (𝑆 ∩ ( ⊥ ‘𝑆)) = { 0 }) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∩ cin 3842 ⊆ wss 3843 {csn 4516 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 Scalarcsca 16671 ·𝑖cip 16673 0gc0g 16816 LModclmod 19753 LSubSpclss 19822 PreHilcphl 20440 ocvcocv 20476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-int 4837 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-1st 7714 df-2nd 7715 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-plusg 16681 df-sca 16684 df-vsca 16685 df-ip 16686 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-minusg 18223 df-sbg 18224 df-ghm 18474 df-mgp 19359 df-ur 19371 df-ring 19418 df-lmod 19755 df-lss 19823 df-lmhm 19913 df-lvec 19994 df-sra 20063 df-rgmod 20064 df-phl 20442 df-ocv 20479 |
This theorem is referenced by: ocv1 20495 pjdm2 20527 pjff 20528 pjf2 20530 pjfo 20531 obselocv 20544 |
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