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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 40088. (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 40118 | . . 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 40119 | . . . . . 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 40114 | . . . . . . . . . 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 40115 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (◡𝑀‘𝑓) ∈ 𝑆) |
19 | 1, 2, 14, 17 | mapdcnvid2 40120 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(◡𝑀‘𝑓)) = 𝑓) |
20 | 19 | eqcomd 2742 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(◡𝑀‘𝑓))) |
21 | fveq2 6842 | . . . . . . . 8 ⊢ (𝑣 = (◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(◡𝑀‘𝑓))) | |
22 | 21 | rspceeqv 3595 | . . . . . . 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 4048 | . . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣))) | |
25 | psseq1 4047 | . . . . . . . 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 5364 | . . . . 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 40118 | . . . . . . 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 40118 | . . . . . . 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 3173 | . . . . 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 40055 | . . 3 ⊢ (𝜑 → 𝐷 ∈ LMod) |
40 | 1, 2, 3, 4, 9, 10, 5, 6 | mapdcl2 40119 | . . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷)) |
41 | 1, 2, 3, 4, 9, 10, 5, 7 | mapdcl2 40119 | . . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷)) |
42 | 10, 38, 39, 40, 41 | lcvbr 37483 | . 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 39573 | . . 3 ⊢ (𝜑 → 𝑈 ∈ LMod) |
45 | 4, 43, 44, 6, 7 | lcvbr 37483 | . 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 3073 ⊊ wpss 3911 class class class wbr 5105 ◡ccnv 5632 ran crn 5634 ‘cfv 6496 LModclmod 20322 LSubSpclss 20392 ⋖L clcv 37480 HLchlt 37812 LHypclh 38447 DVecHcdvh 39541 LCDualclcd 40049 mapdcmpd 40087 |
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 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415 |
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 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lshyp 37439 df-lcv 37481 df-lfl 37520 df-lkr 37548 df-ldual 37586 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858 df-lcdual 40050 df-mapd 40088 |
This theorem is referenced by: mapdcnvatN 40129 mapdat 40130 |
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