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Mirrors > Home > MPE Home > Th. List > Mathboxes > lclkrlem2g | Structured version Visualization version GIF version |
Description: Lemma for lclkr 39985. Comparable hyperplanes are equal, so the kernel of the sum is closed. (Contributed by NM, 16-Jan-2015.) |
Ref | Expression |
---|---|
lclkrlem2f.h | ⊢ 𝐻 = (LHyp‘𝐾) |
lclkrlem2f.o | ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) |
lclkrlem2f.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
lclkrlem2f.v | ⊢ 𝑉 = (Base‘𝑈) |
lclkrlem2f.s | ⊢ 𝑆 = (Scalar‘𝑈) |
lclkrlem2f.q | ⊢ 𝑄 = (0g‘𝑆) |
lclkrlem2f.z | ⊢ 0 = (0g‘𝑈) |
lclkrlem2f.a | ⊢ ⊕ = (LSSum‘𝑈) |
lclkrlem2f.n | ⊢ 𝑁 = (LSpan‘𝑈) |
lclkrlem2f.f | ⊢ 𝐹 = (LFnl‘𝑈) |
lclkrlem2f.j | ⊢ 𝐽 = (LSHyp‘𝑈) |
lclkrlem2f.l | ⊢ 𝐿 = (LKer‘𝑈) |
lclkrlem2f.d | ⊢ 𝐷 = (LDual‘𝑈) |
lclkrlem2f.p | ⊢ + = (+g‘𝐷) |
lclkrlem2f.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
lclkrlem2f.b | ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2f.e | ⊢ (𝜑 → 𝐸 ∈ 𝐹) |
lclkrlem2f.g | ⊢ (𝜑 → 𝐺 ∈ 𝐹) |
lclkrlem2f.le | ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) |
lclkrlem2f.lg | ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) |
lclkrlem2f.kb | ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) |
lclkrlem2f.nx | ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) |
lclkrlem2f.x | ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2f.y | ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) |
lclkrlem2f.ne | ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺)) |
lclkrlem2f.lp | ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) |
Ref | Expression |
---|---|
lclkrlem2g | ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | lclkrlem2f.h | . . . . 5 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | lclkrlem2f.o | . . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊) | |
3 | lclkrlem2f.u | . . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | lclkrlem2f.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑈) | |
5 | lclkrlem2f.s | . . . . 5 ⊢ 𝑆 = (Scalar‘𝑈) | |
6 | lclkrlem2f.q | . . . . 5 ⊢ 𝑄 = (0g‘𝑆) | |
7 | lclkrlem2f.z | . . . . 5 ⊢ 0 = (0g‘𝑈) | |
8 | lclkrlem2f.a | . . . . 5 ⊢ ⊕ = (LSSum‘𝑈) | |
9 | lclkrlem2f.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑈) | |
10 | lclkrlem2f.f | . . . . 5 ⊢ 𝐹 = (LFnl‘𝑈) | |
11 | lclkrlem2f.j | . . . . 5 ⊢ 𝐽 = (LSHyp‘𝑈) | |
12 | lclkrlem2f.l | . . . . 5 ⊢ 𝐿 = (LKer‘𝑈) | |
13 | lclkrlem2f.d | . . . . 5 ⊢ 𝐷 = (LDual‘𝑈) | |
14 | lclkrlem2f.p | . . . . 5 ⊢ + = (+g‘𝐷) | |
15 | lclkrlem2f.k | . . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
16 | lclkrlem2f.b | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝑉 ∖ { 0 })) | |
17 | lclkrlem2f.e | . . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐹) | |
18 | lclkrlem2f.g | . . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹) | |
19 | lclkrlem2f.le | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) = ( ⊥ ‘{𝑋})) | |
20 | lclkrlem2f.lg | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑌})) | |
21 | lclkrlem2f.kb | . . . . 5 ⊢ (𝜑 → ((𝐸 + 𝐺)‘𝐵) = 𝑄) | |
22 | lclkrlem2f.nx | . . . . 5 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝐵}) ∨ ¬ 𝑌 ∈ ( ⊥ ‘{𝐵}))) | |
23 | lclkrlem2f.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 })) | |
24 | lclkrlem2f.y | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 })) | |
25 | lclkrlem2f.ne | . . . . 5 ⊢ (𝜑 → (𝐿‘𝐸) ≠ (𝐿‘𝐺)) | |
26 | lclkrlem2f.lp | . . . . 5 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ 𝐽) | |
27 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 | lclkrlem2f 39964 | . . . 4 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺))) |
28 | 1, 3, 15 | dvhlvec 39561 | . . . . 5 ⊢ (𝜑 → 𝑈 ∈ LVec) |
29 | 19, 20 | ineq12d 4172 | . . . . . . 7 ⊢ (𝜑 → ((𝐿‘𝐸) ∩ (𝐿‘𝐺)) = (( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌}))) |
30 | 29 | oveq1d 7369 | . . . . . 6 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵}))) |
31 | eqid 2736 | . . . . . . 7 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈) | |
32 | 25, 19, 20 | 3netr3d 3019 | . . . . . . 7 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ≠ ( ⊥ ‘{𝑌})) |
33 | 1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 11 | lclkrlem2c 39961 | . . . . . 6 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ 𝐽) |
34 | 30, 33 | eqeltrd 2838 | . . . . 5 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ∈ 𝐽) |
35 | 11, 28, 34, 26 | lshpcmp 37439 | . . . 4 ⊢ (𝜑 → ((((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ⊆ (𝐿‘(𝐸 + 𝐺)) ↔ (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = (𝐿‘(𝐸 + 𝐺)))) |
36 | 27, 35 | mpbid 231 | . . 3 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) = (𝐿‘(𝐸 + 𝐺))) |
37 | eqid 2736 | . . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊) | |
38 | 1, 2, 3, 4, 7, 8, 9, 31, 15, 16, 23, 24, 32, 22, 37 | lclkrlem2d 39962 | . . . 4 ⊢ (𝜑 → ((( ⊥ ‘{𝑋}) ∩ ( ⊥ ‘{𝑌})) ⊕ (𝑁‘{𝐵})) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
39 | 30, 38 | eqeltrd 2838 | . . 3 ⊢ (𝜑 → (((𝐿‘𝐸) ∩ (𝐿‘𝐺)) ⊕ (𝑁‘{𝐵})) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
40 | 36, 39 | eqeltrrd 2839 | . 2 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) |
41 | 1, 3, 37, 4 | dihrnss 39730 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉) |
42 | 15, 40, 41 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐸 + 𝐺)) ⊆ 𝑉) |
43 | 1, 37, 3, 4, 2, 15, 42 | dochoccl 39821 | . 2 ⊢ (𝜑 → ((𝐿‘(𝐸 + 𝐺)) ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺)))) |
44 | 40, 43 | mpbid 231 | 1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝐸 + 𝐺)))) = (𝐿‘(𝐸 + 𝐺))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∨ wo 845 = wceq 1541 ∈ wcel 2106 ≠ wne 2942 ∖ cdif 3906 ∩ cin 3908 ⊆ wss 3909 {csn 4585 ran crn 5633 ‘cfv 6494 (class class class)co 7354 Basecbs 17080 +gcplusg 17130 Scalarcsca 17133 0gc0g 17318 LSSumclsm 19412 LSpanclspn 20428 LSAtomsclsa 37425 LSHypclsh 37426 LFnlclfn 37508 LKerclk 37536 LDualcld 37574 HLchlt 37801 LHypclh 38436 DVecHcdvh 39530 DIsoHcdih 39680 ocHcoch 39799 |
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 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7669 ax-cnex 11104 ax-resscn 11105 ax-1cn 11106 ax-icn 11107 ax-addcl 11108 ax-addrcl 11109 ax-mulcl 11110 ax-mulrcl 11111 ax-mulcom 11112 ax-addass 11113 ax-mulass 11114 ax-distr 11115 ax-i2m1 11116 ax-1ne0 11117 ax-1rid 11118 ax-rnegex 11119 ax-rrecex 11120 ax-cnre 11121 ax-pre-lttri 11122 ax-pre-lttrn 11123 ax-pre-ltadd 11124 ax-pre-mulgt0 11125 ax-riotaBAD 37404 |
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 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5530 df-eprel 5536 df-po 5544 df-so 5545 df-fr 5587 df-we 5589 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6252 df-ord 6319 df-on 6320 df-lim 6321 df-suc 6322 df-iota 6446 df-fun 6496 df-fn 6497 df-f 6498 df-f1 6499 df-fo 6500 df-f1o 6501 df-fv 6502 df-riota 7310 df-ov 7357 df-oprab 7358 df-mpo 7359 df-of 7614 df-om 7800 df-1st 7918 df-2nd 7919 df-tpos 8154 df-undef 8201 df-frecs 8209 df-wrecs 8240 df-recs 8314 df-rdg 8353 df-1o 8409 df-er 8645 df-map 8764 df-en 8881 df-dom 8882 df-sdom 8883 df-fin 8884 df-pnf 11188 df-mnf 11189 df-xr 11190 df-ltxr 11191 df-le 11192 df-sub 11384 df-neg 11385 df-nn 12151 df-2 12213 df-3 12214 df-4 12215 df-5 12216 df-6 12217 df-n0 12411 df-z 12497 df-uz 12761 df-fz 13422 df-struct 17016 df-sets 17033 df-slot 17051 df-ndx 17063 df-base 17081 df-ress 17110 df-plusg 17143 df-mulr 17144 df-sca 17146 df-vsca 17147 df-0g 17320 df-mre 17463 df-mrc 17464 df-acs 17466 df-proset 18181 df-poset 18199 df-plt 18216 df-lub 18232 df-glb 18233 df-join 18234 df-meet 18235 df-p0 18311 df-p1 18312 df-lat 18318 df-clat 18385 df-mgm 18494 df-sgrp 18543 df-mnd 18554 df-submnd 18599 df-grp 18748 df-minusg 18749 df-sbg 18750 df-subg 18921 df-cntz 19093 df-oppg 19120 df-lsm 19414 df-cmn 19560 df-abl 19561 df-mgp 19893 df-ur 19910 df-ring 19962 df-oppr 20045 df-dvdsr 20066 df-unit 20067 df-invr 20097 df-dvr 20108 df-drng 20183 df-lmod 20320 df-lss 20389 df-lsp 20429 df-lvec 20560 df-lsatoms 37427 df-lshyp 37428 df-lcv 37470 df-lfl 37509 df-lkr 37537 df-ldual 37575 df-oposet 37627 df-ol 37629 df-oml 37630 df-covers 37717 df-ats 37718 df-atl 37749 df-cvlat 37773 df-hlat 37802 df-llines 37950 df-lplanes 37951 df-lvols 37952 df-lines 37953 df-psubsp 37955 df-pmap 37956 df-padd 38248 df-lhyp 38440 df-laut 38441 df-ldil 38556 df-ltrn 38557 df-trl 38611 df-tgrp 39195 df-tendo 39207 df-edring 39209 df-dveca 39455 df-disoa 39481 df-dvech 39531 df-dib 39591 df-dic 39625 df-dih 39681 df-doch 39800 df-djh 39847 |
This theorem is referenced by: lclkrlem2h 39966 |
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