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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdcv | Structured version Visualization version GIF version |
Description: Covering property of the converse of the map defined by df-mapd 40055. (Contributed by NM, 14-Mar-2015.) |
Ref | Expression |
---|---|
mapdcv.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdcv.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdcv.u | ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) |
mapdcv.s | ⊢ 𝑆 = (LSubSp‘𝑈) |
mapdcv.c | ⊢ 𝐶 = ( ⋖L ‘𝑈) |
mapdcv.d | ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊) |
mapdcv.e | ⊢ 𝐸 = ( ⋖L ‘𝐷) |
mapdcv.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdcv.x | ⊢ (𝜑 → 𝑋 ∈ 𝑆) |
mapdcv.y | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
mapdcv | ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdcv.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdcv.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdcv.u | . . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊) | |
4 | mapdcv.s | . . . 4 ⊢ 𝑆 = (LSubSp‘𝑈) | |
5 | mapdcv.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
6 | mapdcv.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑆) | |
7 | mapdcv.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
8 | 1, 2, 3, 4, 5, 6, 7 | mapdsord 40085 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) |
9 | mapdcv.d | . . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊) | |
10 | eqid 2736 | . . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷) | |
11 | 5 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
12 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆) | |
13 | 1, 2, 3, 4, 9, 10, 11, 12 | mapdcl2 40086 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷)) |
14 | 5 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
15 | 1, 2, 9, 10, 5 | mapdrn2 40081 | . . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷)) |
16 | 15 | eleq2d 2823 | . . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷))) |
17 | 16 | biimpar 478 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀) |
18 | 1, 2, 3, 4, 14, 17 | mapdcnvcl 40082 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (◡𝑀‘𝑓) ∈ 𝑆) |
19 | 1, 2, 14, 17 | mapdcnvid2 40087 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(◡𝑀‘𝑓)) = 𝑓) |
20 | 19 | eqcomd 2742 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(◡𝑀‘𝑓))) |
21 | fveq2 6839 | . . . . . . . 8 ⊢ (𝑣 = (◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(◡𝑀‘𝑓))) | |
22 | 21 | rspceeqv 3593 | . . . . . . 7 ⊢ (((◡𝑀‘𝑓) ∈ 𝑆 ∧ 𝑓 = (𝑀‘(◡𝑀‘𝑓))) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣)) |
23 | 18, 20, 22 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣)) |
24 | psseq2 4046 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣))) | |
25 | psseq1 4045 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))) | |
26 | 24, 25 | anbi12d 631 | . . . . . . 7 ⊢ (𝑓 = (𝑀‘𝑣) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
27 | 26 | adantl 482 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
28 | 13, 23, 27 | rexxfrd 5362 | . . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
29 | 6 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆) |
30 | 1, 2, 3, 4, 11, 29, 12 | mapdsord 40085 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣)) |
31 | 7 | adantr 481 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆) |
32 | 1, 2, 3, 4, 11, 12, 31 | mapdsord 40085 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌)) |
33 | 30, 32 | anbi12d 631 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
34 | 33 | rexbidva 3171 | . . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
35 | 28, 34 | bitrd 278 | . . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
36 | 35 | notbid 317 | . . 3 ⊢ (𝜑 → (¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
37 | 8, 36 | anbi12d 631 | . 2 ⊢ (𝜑 → (((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ∧ ¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌))) ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))) |
38 | mapdcv.e | . . 3 ⊢ 𝐸 = ( ⋖L ‘𝐷) | |
39 | 1, 9, 5 | lcdlmod 40022 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
40 | 1, 2, 3, 4, 9, 10, 5, 6 | mapdcl2 40086 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷)) |
41 | 1, 2, 3, 4, 9, 10, 5, 7 | mapdcl2 40086 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷)) |
42 | 10, 38, 39, 40, 41 | lcvbr 37450 | . 2 ⊢ (𝜑 → ((𝑀‘𝑋)𝐸(𝑀‘𝑌) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ∧ ¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌))))) |
43 | mapdcv.c | . . 3 ⊢ 𝐶 = ( ⋖L ‘𝑈) | |
44 | 1, 3, 5 | dvhlmod 39540 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
45 | 4, 43, 44, 6, 7 | lcvbr 37450 | . 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))) |
46 | 37, 42, 45 | 3bitr4rd 311 | 1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∃wrex 3071 ⊊ wpss 3909 class class class wbr 5103 ◡ccnv 5630 ran crn 5632 ‘cfv 6493 LModclmod 20307 LSubSpclss 20377 ⋖L clcv 37447 HLchlt 37779 LHypclh 38414 DVecHcdvh 39508 LCDualclcd 40016 mapdcmpd 40054 |
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 5240 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 ax-riotaBAD 37382 |
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 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4864 df-int 4906 df-iun 4954 df-iin 4955 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-of 7613 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-undef 8200 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-1o 8408 df-er 8644 df-map 8763 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-n0 12410 df-z 12496 df-uz 12760 df-fz 13417 df-struct 17011 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-ress 17105 df-plusg 17138 df-mulr 17139 df-sca 17141 df-vsca 17142 df-0g 17315 df-mre 17458 df-mrc 17459 df-acs 17461 df-proset 18176 df-poset 18194 df-plt 18211 df-lub 18227 df-glb 18228 df-join 18229 df-meet 18230 df-p0 18306 df-p1 18307 df-lat 18313 df-clat 18380 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-submnd 18594 df-grp 18743 df-minusg 18744 df-sbg 18745 df-subg 18916 df-cntz 19088 df-oppg 19115 df-lsm 19409 df-cmn 19555 df-abl 19556 df-mgp 19888 df-ur 19905 df-ring 19952 df-oppr 20034 df-dvdsr 20055 df-unit 20056 df-invr 20086 df-dvr 20097 df-drng 20172 df-lmod 20309 df-lss 20378 df-lsp 20418 df-lvec 20549 df-lsatoms 37405 df-lshyp 37406 df-lcv 37448 df-lfl 37487 df-lkr 37515 df-ldual 37553 df-oposet 37605 df-ol 37607 df-oml 37608 df-covers 37695 df-ats 37696 df-atl 37727 df-cvlat 37751 df-hlat 37780 df-llines 37928 df-lplanes 37929 df-lvols 37930 df-lines 37931 df-psubsp 37933 df-pmap 37934 df-padd 38226 df-lhyp 38418 df-laut 38419 df-ldil 38534 df-ltrn 38535 df-trl 38589 df-tgrp 39173 df-tendo 39185 df-edring 39187 df-dveca 39433 df-disoa 39459 df-dvech 39509 df-dib 39569 df-dic 39603 df-dih 39659 df-doch 39778 df-djh 39825 df-lcdual 40017 df-mapd 40055 |
This theorem is referenced by: mapdcnvatN 40096 mapdat 40097 |
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