Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
||
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 38765. (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 38795 | . . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌)) |
9 | mapdcv.d | . . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊) | |
10 | eqid 2824 | . . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷) | |
11 | 5 | adantr 483 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
12 | simpr 487 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆) | |
13 | 1, 2, 3, 4, 9, 10, 11, 12 | mapdcl2 38796 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷)) |
14 | 5 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
15 | 1, 2, 9, 10, 5 | mapdrn2 38791 | . . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷)) |
16 | 15 | eleq2d 2901 | . . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷))) |
17 | 16 | biimpar 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀) |
18 | 1, 2, 3, 4, 14, 17 | mapdcnvcl 38792 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (◡𝑀‘𝑓) ∈ 𝑆) |
19 | 1, 2, 14, 17 | mapdcnvid2 38797 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(◡𝑀‘𝑓)) = 𝑓) |
20 | 19 | eqcomd 2830 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(◡𝑀‘𝑓))) |
21 | fveq2 6673 | . . . . . . . 8 ⊢ (𝑣 = (◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(◡𝑀‘𝑓))) | |
22 | 21 | rspceeqv 3641 | . . . . . . 7 ⊢ (((◡𝑀‘𝑓) ∈ 𝑆 ∧ 𝑓 = (𝑀‘(◡𝑀‘𝑓))) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣)) |
23 | 18, 20, 22 | syl2anc 586 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣)) |
24 | psseq2 4068 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣))) | |
25 | psseq1 4067 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))) | |
26 | 24, 25 | anbi12d 632 | . . . . . . 7 ⊢ (𝑓 = (𝑀‘𝑣) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
27 | 26 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
28 | 13, 23, 27 | rexxfrd 5313 | . . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))) |
29 | 6 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆) |
30 | 1, 2, 3, 4, 11, 29, 12 | mapdsord 38795 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣)) |
31 | 7 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆) |
32 | 1, 2, 3, 4, 11, 12, 31 | mapdsord 38795 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌)) |
33 | 30, 32 | anbi12d 632 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
34 | 33 | rexbidva 3299 | . . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
35 | 28, 34 | bitrd 281 | . . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
36 | 35 | notbid 320 | . . 3 ⊢ (𝜑 → (¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))) |
37 | 8, 36 | anbi12d 632 | . 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 38732 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
40 | 1, 2, 3, 4, 9, 10, 5, 6 | mapdcl2 38796 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷)) |
41 | 1, 2, 3, 4, 9, 10, 5, 7 | mapdcl2 38796 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷)) |
42 | 10, 38, 39, 40, 41 | lcvbr 36161 | . 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 38250 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
45 | 4, 43, 44, 6, 7 | lcvbr 36161 | . 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))) |
46 | 37, 42, 45 | 3bitr4rd 314 | 1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ∃wrex 3142 ⊊ wpss 3940 class class class wbr 5069 ◡ccnv 5557 ran crn 5559 ‘cfv 6358 LModclmod 19637 LSubSpclss 19706 ⋖L clcv 36158 HLchlt 36490 LHypclh 37124 DVecHcdvh 38218 LCDualclcd 38726 mapdcmpd 38764 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-riotaBAD 36093 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-iin 4925 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-of 7412 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-undef 7942 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-sca 16584 df-vsca 16585 df-0g 16718 df-mre 16860 df-mrc 16861 df-acs 16863 df-proset 17541 df-poset 17559 df-plt 17571 df-lub 17587 df-glb 17588 df-join 17589 df-meet 17590 df-p0 17652 df-p1 17653 df-lat 17659 df-clat 17721 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-submnd 17960 df-grp 18109 df-minusg 18110 df-sbg 18111 df-subg 18279 df-cntz 18450 df-oppg 18477 df-lsm 18764 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-invr 19425 df-dvr 19436 df-drng 19507 df-lmod 19639 df-lss 19707 df-lsp 19747 df-lvec 19878 df-lsatoms 36116 df-lshyp 36117 df-lcv 36159 df-lfl 36198 df-lkr 36226 df-ldual 36264 df-oposet 36316 df-ol 36318 df-oml 36319 df-covers 36406 df-ats 36407 df-atl 36438 df-cvlat 36462 df-hlat 36491 df-llines 36638 df-lplanes 36639 df-lvols 36640 df-lines 36641 df-psubsp 36643 df-pmap 36644 df-padd 36936 df-lhyp 37128 df-laut 37129 df-ldil 37244 df-ltrn 37245 df-trl 37299 df-tgrp 37883 df-tendo 37895 df-edring 37897 df-dveca 38143 df-disoa 38169 df-dvech 38219 df-dib 38279 df-dic 38313 df-dih 38369 df-doch 38488 df-djh 38535 df-lcdual 38727 df-mapd 38765 |
This theorem is referenced by: mapdcnvatN 38806 mapdat 38807 |
Copyright terms: Public domain | W3C validator |