mapdcv - Mathbox for Norm Megill

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Theorem mapdcv 37420

Description: Covering property of the converse of the map defined by df-mapd 37385. (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 37415	. . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌))
9		mapdcv.d	. . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊)
10		eqid 2748	. . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
11	5	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simpr 479	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
13	1, 2, 3, 4, 9, 10, 11, 12	mapdcl2 37416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷))
14	5	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	1, 2, 9, 10, 5	mapdrn2 37411	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷))
16	15	eleq2d 2813	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷)))
17	16	biimpar 503	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀)
18	1, 2, 3, 4, 14, 17	mapdcnvcl 37412	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (^◡𝑀‘𝑓) ∈ 𝑆)
19	1, 2, 14, 17	mapdcnvid2 37417	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(^◡𝑀‘𝑓)) = 𝑓)
20	19	eqcomd 2754	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(^◡𝑀‘𝑓)))
21		fveq2 6340	. . . . . . . . 9 ⊢ (𝑣 = (^◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(^◡𝑀‘𝑓)))
22	21	eqeq2d 2758	. . . . . . . 8 ⊢ (𝑣 = (^◡𝑀‘𝑓) → (𝑓 = (𝑀‘𝑣) ↔ 𝑓 = (𝑀‘(^◡𝑀‘𝑓))))
23	22	rspcev 3437	. . . . . . 7 ⊢ (((^◡𝑀‘𝑓) ∈ 𝑆 ∧ 𝑓 = (𝑀‘(^◡𝑀‘𝑓))) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
24	18, 20, 23	syl2anc 696	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
25		psseq2 3825	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣)))
26		psseq1 3824	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))
27	25, 26	anbi12d 749	. . . . . . 7 ⊢ (𝑓 = (𝑀‘𝑣) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
28	27	adantl 473	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
29	13, 24, 28	rexxfrd 5018	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
30	6	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆)
31	1, 2, 3, 4, 11, 30, 12	mapdsord 37415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣))
32	7	adantr 472	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆)
33	1, 2, 3, 4, 11, 12, 32	mapdsord 37415	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌))
34	31, 33	anbi12d 749	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
35	34	rexbidva 3175	. . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
36	29, 35	bitrd 268	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
37	36	notbid 307	. . 3 ⊢ (𝜑 → (¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
38	8, 37	anbi12d 749	. 2 ⊢ (𝜑 → (((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ∧ ¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌))) ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))))
39		mapdcv.e	. . 3 ⊢ 𝐸 = ( ⋖_L ‘𝐷)
40	1, 9, 5	lcdlmod 37352	. . 3 ⊢ (𝜑 → 𝐷 ∈ LMod)
41	1, 2, 3, 4, 9, 10, 5, 6	mapdcl2 37416	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷))
42	1, 2, 3, 4, 9, 10, 5, 7	mapdcl2 37416	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷))
43	10, 39, 40, 41, 42	lcvbr 34780	. 2 ⊢ (𝜑 → ((𝑀‘𝑋)𝐸(𝑀‘𝑌) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ∧ ¬ ∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)))))
44		mapdcv.c	. . 3 ⊢ 𝐶 = ( ⋖_L ‘𝑈)
45	1, 3, 5	dvhlmod 36870	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
46	4, 44, 45, 6, 7	lcvbr 34780	. 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))))
47	38, 43, 46	3bitr4rd 301	1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ∃wrex 3039 ⊊ wpss 3704 class class class wbr 4792 ^◡ccnv 5253 ran crn 5255 ‘cfv 6037 LModclmod 19036 LSubSpclss 19105 ⋖_L clcv 34777 HLchlt 35109 LHypclh 35742 DVecHcdvh 36838 LCDualclcd 37346 mapdcmpd 37384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-riotaBAD 34711

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-tpos 7509 df-undef 7556 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7899 df-map 8013 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-sca 16130 df-vsca 16131 df-0g 16275 df-mre 16419 df-mrc 16420 df-acs 16422 df-preset 17100 df-poset 17118 df-plt 17130 df-lub 17146 df-glb 17147 df-join 17148 df-meet 17149 df-p0 17211 df-p1 17212 df-lat 17218 df-clat 17280 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-grp 17597 df-minusg 17598 df-sbg 17599 df-subg 17763 df-cntz 17921 df-oppg 17947 df-lsm 18222 df-cmn 18366 df-abl 18367 df-mgp 18661 df-ur 18673 df-ring 18720 df-oppr 18794 df-dvdsr 18812 df-unit 18813 df-invr 18843 df-dvr 18854 df-drng 18922 df-lmod 19038 df-lss 19106 df-lsp 19145 df-lvec 19276 df-lsatoms 34735 df-lshyp 34736 df-lcv 34778 df-lfl 34817 df-lkr 34845 df-ldual 34883 df-oposet 34935 df-ol 34937 df-oml 34938 df-covers 35025 df-ats 35026 df-atl 35057 df-cvlat 35081 df-hlat 35110 df-llines 35256 df-lplanes 35257 df-lvols 35258 df-lines 35259 df-psubsp 35261 df-pmap 35262 df-padd 35554 df-lhyp 35746 df-laut 35747 df-ldil 35862 df-ltrn 35863 df-trl 35918 df-tgrp 36502 df-tendo 36514 df-edring 36516 df-dveca 36762 df-disoa 36789 df-dvech 36839 df-dib 36899 df-dic 36933 df-dih 36989 df-doch 37108 df-djh 37155 df-lcdual 37347 df-mapd 37385

This theorem is referenced by: mapdcnvatN 37426 mapdat 37427

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