mapdcv - Mathbox for Norm Megill

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Theorem mapdcv 39411

Description: Covering property of the converse of the map defined by df-mapd 39376. (Contributed by NM, 14-Mar-2015.)

**Hypotheses**
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)

**Assertion**
Ref	Expression
mapdcv	⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌)))

Proof of Theorem mapdcv Dummy variables 𝑣 𝑓 are mutually distinct and distinct from all other variables.

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 39406	. . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌))
9		mapdcv.d	. . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊)
10		eqid 2737	. . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
11	5	adantr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
13	1, 2, 3, 4, 9, 10, 11, 12	mapdcl2 39407	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷))
14	5	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	1, 2, 9, 10, 5	mapdrn2 39402	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷))
16	15	eleq2d 2823	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷)))
17	16	biimpar 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀)
18	1, 2, 3, 4, 14, 17	mapdcnvcl 39403	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (^◡𝑀‘𝑓) ∈ 𝑆)
19	1, 2, 14, 17	mapdcnvid2 39408	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(^◡𝑀‘𝑓)) = 𝑓)
20	19	eqcomd 2743	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(^◡𝑀‘𝑓)))
21		fveq2 6717	. . . . . . . 8 ⊢ (𝑣 = (^◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(^◡𝑀‘𝑓)))
22	21	rspceeqv 3552	. . . . . . 7 ⊢ (((^◡𝑀‘𝑓) ∈ 𝑆 ∧ 𝑓 = (𝑀‘(^◡𝑀‘𝑓))) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
23	18, 20, 22	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
24		psseq2 4003	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣)))
25		psseq1 4002	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))
26	24, 25	anbi12d 634	. . . . . . 7 ⊢ (𝑓 = (𝑀‘𝑣) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
27	26	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
28	13, 23, 27	rexxfrd 5302	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
29	6	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆)
30	1, 2, 3, 4, 11, 29, 12	mapdsord 39406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣))
31	7	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆)
32	1, 2, 3, 4, 11, 12, 31	mapdsord 39406	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌))
33	30, 32	anbi12d 634	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
34	33	rexbidva 3215	. . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
35	28, 34	bitrd 282	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
36	35	notbid 321	. . 3 ⊢ (𝜑 → (¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
37	8, 36	anbi12d 634	. 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 39343	. . 3 ⊢ (𝜑 → 𝐷 ∈ LMod)
40	1, 2, 3, 4, 9, 10, 5, 6	mapdcl2 39407	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷))
41	1, 2, 3, 4, 9, 10, 5, 7	mapdcl2 39407	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷))
42	10, 38, 39, 40, 41	lcvbr 36772	. 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 38861	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
45	4, 43, 44, 6, 7	lcvbr 36772	. 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))))
46	37, 42, 45	3bitr4rd 315	1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 ⊊ wpss 3867 class class class wbr 5053 ^◡ccnv 5550 ran crn 5552 ‘cfv 6380 LModclmod 19899 LSubSpclss 19968 ⋖_L clcv 36769 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 LCDualclcd 39337 mapdcmpd 39375

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-of 7469 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-mre 17089 df-mrc 17090 df-acs 17092 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-oppg 18738 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-lshyp 36728 df-lcv 36770 df-lfl 36809 df-lkr 36837 df-ldual 36875 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tgrp 38494 df-tendo 38506 df-edring 38508 df-dveca 38754 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980 df-doch 39099 df-djh 39146 df-lcdual 39338 df-mapd 39376

This theorem is referenced by: mapdcnvatN 39417 mapdat 39418

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