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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 38292. (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 38322 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) |
9 | mapdcv.d | . . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊) | |
10 | eqid 2795 | . . . . . . 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 38323 | . . . . . 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 38318 | . . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷)) |
16 | 15 | eleq2d 2868 | . . . . . . . . 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 38319 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (◡𝑀‘𝑓) ∈ 𝑆) |
19 | 1, 2, 14, 17 | mapdcnvid2 38324 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(◡𝑀‘𝑓)) = 𝑓) |
20 | 19 | eqcomd 2801 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(◡𝑀‘𝑓))) |
21 | fveq2 6538 | . . . . . . . 8 ⊢ (𝑣 = (◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(◡𝑀‘𝑓))) | |
22 | 21 | rspceeqv 3577 | . . . . . . 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 3986 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣))) | |
25 | psseq1 3985 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))) | |
26 | 24, 25 | anbi12d 630 | . . . . . . 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 5201 | . . . . 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 38322 | . . . . . . 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 38322 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌)) |
33 | 30, 32 | anbi12d 630 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
34 | 33 | rexbidva 3259 | . . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
35 | 28, 34 | bitrd 280 | . . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
36 | 35 | notbid 319 | . . 3 ⊢ (𝜑 → (¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
37 | 8, 36 | anbi12d 630 | . 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 38259 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
40 | 1, 2, 3, 4, 9, 10, 5, 6 | mapdcl2 38323 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷)) |
41 | 1, 2, 3, 4, 9, 10, 5, 7 | mapdcl2 38323 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷)) |
42 | 10, 38, 39, 40, 41 | lcvbr 35688 | . 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 37777 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
45 | 4, 43, 44, 6, 7 | lcvbr 35688 | . 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))) |
46 | 37, 42, 45 | 3bitr4rd 313 | 1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2081 ∃wrex 3106 ⊊ wpss 3860 class class class wbr 4962 ◡ccnv 5442 ran crn 5444 ‘cfv 6225 LModclmod 19324 LSubSpclss 19393 ⋖L clcv 35685 HLchlt 36017 LHypclh 36651 DVecHcdvh 37745 LCDualclcd 38253 mapdcmpd 38291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-riotaBAD 35620 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-of 7267 df-om 7437 df-1st 7545 df-2nd 7546 df-tpos 7743 df-undef 7790 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-map 8258 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-n0 11746 df-z 11830 df-uz 12094 df-fz 12743 df-struct 16314 df-ndx 16315 df-slot 16316 df-base 16318 df-sets 16319 df-ress 16320 df-plusg 16407 df-mulr 16408 df-sca 16410 df-vsca 16411 df-0g 16544 df-mre 16686 df-mrc 16687 df-acs 16689 df-proset 17367 df-poset 17385 df-plt 17397 df-lub 17413 df-glb 17414 df-join 17415 df-meet 17416 df-p0 17478 df-p1 17479 df-lat 17485 df-clat 17547 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-submnd 17775 df-grp 17864 df-minusg 17865 df-sbg 17866 df-subg 18030 df-cntz 18188 df-oppg 18215 df-lsm 18491 df-cmn 18635 df-abl 18636 df-mgp 18930 df-ur 18942 df-ring 18989 df-oppr 19063 df-dvdsr 19081 df-unit 19082 df-invr 19112 df-dvr 19123 df-drng 19194 df-lmod 19326 df-lss 19394 df-lsp 19434 df-lvec 19565 df-lsatoms 35643 df-lshyp 35644 df-lcv 35686 df-lfl 35725 df-lkr 35753 df-ldual 35791 df-oposet 35843 df-ol 35845 df-oml 35846 df-covers 35933 df-ats 35934 df-atl 35965 df-cvlat 35989 df-hlat 36018 df-llines 36165 df-lplanes 36166 df-lvols 36167 df-lines 36168 df-psubsp 36170 df-pmap 36171 df-padd 36463 df-lhyp 36655 df-laut 36656 df-ldil 36771 df-ltrn 36772 df-trl 36826 df-tgrp 37410 df-tendo 37422 df-edring 37424 df-dveca 37670 df-disoa 37696 df-dvech 37746 df-dib 37806 df-dic 37840 df-dih 37896 df-doch 38015 df-djh 38062 df-lcdual 38254 df-mapd 38292 |
This theorem is referenced by: mapdcnvatN 38333 mapdat 38334 |
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