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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdincl | Structured version Visualization version GIF version |
Description: Closure of dual subspace intersection for the map defined by df-mapd 41230. (Contributed by NM, 12-Apr-2015.) |
Ref | Expression |
---|---|
mapdincl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdincl.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdincl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdincl.c | ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊) |
mapdincl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdincl.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) |
mapdincl.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) |
Ref | Expression |
---|---|
mapdincl | ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ ran 𝑀) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdincl.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2725 | . . . 4 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊) | |
3 | mapdincl.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | 1, 2, 3 | lcdlmod 41197 | . . 3 ⊢ (𝜑 → ((LCDual‘𝐾)‘𝑊) ∈ LMod) |
5 | mapdincl.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) | |
6 | mapdincl.m | . . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
7 | eqid 2725 | . . . . 5 ⊢ (LSubSp‘((LCDual‘𝐾)‘𝑊)) = (LSubSp‘((LCDual‘𝐾)‘𝑊)) | |
8 | 1, 6, 2, 7, 3 | mapdrn2 41256 | . . . 4 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘((LCDual‘𝐾)‘𝑊))) |
9 | 5, 8 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) |
10 | mapdincl.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) | |
11 | 10, 8 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑌 ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) |
12 | 7 | lssincl 20866 | . . 3 ⊢ ((((LCDual‘𝐾)‘𝑊) ∈ LMod ∧ 𝑋 ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)) ∧ 𝑌 ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) → (𝑋 ∩ 𝑌) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) |
13 | 4, 9, 11, 12 | syl3anc 1368 | . 2 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) |
14 | 13, 8 | eleqtrrd 2828 | 1 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ ran 𝑀) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∩ cin 3943 ran crn 5679 ‘cfv 6549 LModclmod 20760 LSubSpclss 20832 HLchlt 38954 LHypclh 39589 DVecHcdvh 40683 LCDualclcd 41191 mapdcmpd 41229 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-riotaBAD 38557 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-undef 8279 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-n0 12511 df-z 12597 df-uz 12861 df-fz 13525 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17189 df-ress 17218 df-plusg 17254 df-mulr 17255 df-sca 17257 df-vsca 17258 df-0g 17431 df-mre 17574 df-mrc 17575 df-acs 17577 df-proset 18295 df-poset 18313 df-plt 18330 df-lub 18346 df-glb 18347 df-join 18348 df-meet 18349 df-p0 18425 df-p1 18426 df-lat 18432 df-clat 18499 df-mgm 18608 df-sgrp 18687 df-mnd 18703 df-submnd 18749 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19091 df-cntz 19285 df-oppg 19314 df-lsm 19608 df-cmn 19754 df-abl 19755 df-mgp 20092 df-rng 20110 df-ur 20139 df-ring 20192 df-oppr 20290 df-dvdsr 20313 df-unit 20314 df-invr 20344 df-dvr 20357 df-drng 20643 df-lmod 20762 df-lss 20833 df-lsp 20873 df-lvec 21005 df-lsatoms 38580 df-lshyp 38581 df-lcv 38623 df-lfl 38662 df-lkr 38690 df-ldual 38728 df-oposet 38780 df-ol 38782 df-oml 38783 df-covers 38870 df-ats 38871 df-atl 38902 df-cvlat 38926 df-hlat 38955 df-llines 39103 df-lplanes 39104 df-lvols 39105 df-lines 39106 df-psubsp 39108 df-pmap 39109 df-padd 39401 df-lhyp 39593 df-laut 39594 df-ldil 39709 df-ltrn 39710 df-trl 39764 df-tgrp 40348 df-tendo 40360 df-edring 40362 df-dveca 40608 df-disoa 40634 df-dvech 40684 df-dib 40744 df-dic 40778 df-dih 40834 df-doch 40953 df-djh 41000 df-lcdual 41192 df-mapd 41230 |
This theorem is referenced by: mapdin 41267 |
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