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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdcnvcl | Structured version Visualization version GIF version |
Description: Closure of the converse of the map defined by df-mapd 40433. (Contributed by NM, 13-Mar-2015.) |
Ref | Expression |
---|---|
mapdcnvcl.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdcnvcl.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdcnvcl.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdcnvcl.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
mapdcnvcl.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdcnvcl.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) |
Ref | Expression |
---|---|
mapdcnvcl | ⊢ (𝜑 → (◡𝑀‘𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdcnvcl.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | eqid 2733 | . . . 4 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊) | |
3 | mapdcnvcl.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
4 | mapdcnvcl.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
5 | mapdcnvcl.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑈) | |
6 | eqid 2733 | . . . 4 ⊢ (LFnl‘𝑈) = (LFnl‘𝑈) | |
7 | eqid 2733 | . . . 4 ⊢ (LKer‘𝑈) = (LKer‘𝑈) | |
8 | eqid 2733 | . . . 4 ⊢ (LDual‘𝑈) = (LDual‘𝑈) | |
9 | eqid 2733 | . . . 4 ⊢ (LSubSp‘(LDual‘𝑈)) = (LSubSp‘(LDual‘𝑈)) | |
10 | eqid 2733 | . . . 4 ⊢ {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)} = {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)} | |
11 | mapdcnvcl.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
12 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 | mapd1o 40456 | . . 3 ⊢ (𝜑 → 𝑀:𝑆–1-1-onto→((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)})) |
13 | f1of1 6828 | . . 3 ⊢ (𝑀:𝑆–1-1-onto→((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)}) → 𝑀:𝑆–1-1→((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)})) | |
14 | f1f1orn 6840 | . . 3 ⊢ (𝑀:𝑆–1-1→((LSubSp‘(LDual‘𝑈)) ∩ 𝒫 {𝑔 ∈ (LFnl‘𝑈) ∣ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘((LKer‘𝑈)‘𝑔))) = ((LKer‘𝑈)‘𝑔)}) → 𝑀:𝑆–1-1-onto→ran 𝑀) | |
15 | 12, 13, 14 | 3syl 18 | . 2 ⊢ (𝜑 → 𝑀:𝑆–1-1-onto→ran 𝑀) |
16 | mapdcnvcl.x | . 2 ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) | |
17 | f1ocnvdm 7277 | . 2 ⊢ ((𝑀:𝑆–1-1-onto→ran 𝑀 ∧ 𝑋 ∈ ran 𝑀) → (◡𝑀‘𝑋) ∈ 𝑆) | |
18 | 15, 16, 17 | syl2anc 585 | 1 ⊢ (𝜑 → (◡𝑀‘𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3433 ∩ cin 3945 𝒫 cpw 4600 ◡ccnv 5673 ran crn 5675 –1-1→wf1 6536 –1-1-onto→wf1o 6538 ‘cfv 6539 LSubSpclss 20529 LFnlclfn 37864 LKerclk 37892 LDualcld 37930 HLchlt 38157 LHypclh 38792 DVecHcdvh 39886 ocHcoch 40155 mapdcmpd 40432 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5283 ax-sep 5297 ax-nul 5304 ax-pow 5361 ax-pr 5425 ax-un 7719 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 37760 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4527 df-pw 4602 df-sn 4627 df-pr 4629 df-tp 4631 df-op 4633 df-uni 4907 df-int 4949 df-iun 4997 df-iin 4998 df-br 5147 df-opab 5209 df-mpt 5230 df-tr 5264 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6296 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6491 df-fun 6541 df-fn 6542 df-f 6543 df-f1 6544 df-fo 6545 df-f1o 6546 df-fv 6547 df-riota 7359 df-ov 7406 df-oprab 7407 df-mpo 7408 df-of 7664 df-om 7850 df-1st 7969 df-2nd 7970 df-tpos 8205 df-undef 8252 df-frecs 8260 df-wrecs 8291 df-recs 8365 df-rdg 8404 df-1o 8460 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11245 df-mnf 11246 df-xr 11247 df-ltxr 11248 df-le 11249 df-sub 11441 df-neg 11442 df-nn 12208 df-2 12270 df-3 12271 df-4 12272 df-5 12273 df-6 12274 df-n0 12468 df-z 12554 df-uz 12818 df-fz 13480 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17140 df-ress 17169 df-plusg 17205 df-mulr 17206 df-sca 17208 df-vsca 17209 df-0g 17382 df-mre 17525 df-mrc 17526 df-acs 17528 df-proset 18243 df-poset 18261 df-plt 18278 df-lub 18294 df-glb 18295 df-join 18296 df-meet 18297 df-p0 18373 df-p1 18374 df-lat 18380 df-clat 18447 df-mgm 18556 df-sgrp 18605 df-mnd 18621 df-submnd 18667 df-grp 18817 df-minusg 18818 df-sbg 18819 df-subg 18996 df-cntz 19174 df-oppg 19202 df-lsm 19496 df-cmn 19642 df-abl 19643 df-mgp 19979 df-ur 19996 df-ring 20048 df-oppr 20138 df-dvdsr 20159 df-unit 20160 df-invr 20190 df-dvr 20203 df-drng 20305 df-lmod 20460 df-lss 20530 df-lsp 20570 df-lvec 20701 df-lsatoms 37783 df-lshyp 37784 df-lcv 37826 df-lfl 37865 df-lkr 37893 df-ldual 37931 df-oposet 37983 df-ol 37985 df-oml 37986 df-covers 38073 df-ats 38074 df-atl 38105 df-cvlat 38129 df-hlat 38158 df-llines 38306 df-lplanes 38307 df-lvols 38308 df-lines 38309 df-psubsp 38311 df-pmap 38312 df-padd 38604 df-lhyp 38796 df-laut 38797 df-ldil 38912 df-ltrn 38913 df-trl 38967 df-tgrp 39551 df-tendo 39563 df-edring 39565 df-dveca 39811 df-disoa 39837 df-dvech 39887 df-dib 39947 df-dic 39981 df-dih 40037 df-doch 40156 df-djh 40203 df-mapd 40433 |
This theorem is referenced by: mapdcnvordN 40466 mapdcv 40468 mapdin 40470 mapdlsm 40472 mapdcnvatN 40474 hdmaprnlem3N 40658 hdmaprnlem9N 40665 |
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