mapdcv - Mathbox for Norm Megill

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Theorem mapdcv 42462

Description: Covering property of the converse of the map defined by df-mapd 42427. (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 42457	. . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌))
9		mapdcv.d	. . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊)
10		eqid 2763	. . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
11	5	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simpr 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
13	1, 2, 3, 4, 9, 10, 11, 12	mapdcl2 42458	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷))
14	5	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	1, 2, 9, 10, 5	mapdrn2 42453	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷))
16	15	eleq2d 2849	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷)))
17	16	biimpar 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀)
18	1, 2, 3, 4, 14, 17	mapdcnvcl 42454	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (^◡𝑀‘𝑓) ∈ 𝑆)
19	1, 2, 14, 17	mapdcnvid2 42459	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(^◡𝑀‘𝑓)) = 𝑓)
20	19	eqcomd 2769	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(^◡𝑀‘𝑓)))
21		fveq2 6881	. . . . . . . 8 ⊢ (𝑣 = (^◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(^◡𝑀‘𝑓)))
22	21	rspceeqv 3604	. . . . . . 7 ⊢ (((^◡𝑀‘𝑓) ∈ 𝑆 ∧ 𝑓 = (𝑀‘(^◡𝑀‘𝑓))) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
23	18, 20, 22	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → ∃𝑣 ∈ 𝑆 𝑓 = (𝑀‘𝑣))
24		psseq2 4045	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣)))
25		psseq1 4044	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → (𝑓 ⊊ (𝑀‘𝑌) ↔ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)))
26	24, 25	anbi12d 643	. . . . . . 7 ⊢ (𝑓 = (𝑀‘𝑣) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
27	26	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
28	13, 23, 27	rexxfrd 5380	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
29	6	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆)
30	1, 2, 3, 4, 11, 29, 12	mapdsord 42457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣))
31	7	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆)
32	1, 2, 3, 4, 11, 12, 31	mapdsord 42457	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑣) ⊊ (𝑀‘𝑌) ↔ 𝑣 ⊊ 𝑌))
33	30, 32	anbi12d 643	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
34	33	rexbidva 3187	. . . . 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 643	. 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 42394	. . 3 ⊢ (𝜑 → 𝐷 ∈ LMod)
40	1, 2, 3, 4, 9, 10, 5, 6	mapdcl2 42458	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷))
41	1, 2, 3, 4, 9, 10, 5, 7	mapdcl2 42458	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷))
42	10, 38, 39, 40, 41	lcvbr 39823	. 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 41912	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
45	4, 43, 44, 6, 7	lcvbr 39823	. 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

This proof depends on syntax axioms: ¬ wn 3 → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ∃wrex 3089 ⊊ wpss 3906 class class class wbr 5109 ^◡ccnv 5660 ran crn 5662 ‘cfv 6536 LModclmod 20990 LSubSpclss 21061 ⋖_L clcv 39820 HLchlt 40152 LHypclh 40786 DVecHcdvh 41880 LCDualclcd 42388 mapdcmpd 42426

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-cnex 11160 ax-resscn 11161 ax-1cn 11162 ax-icn 11163 ax-addcl 11164 ax-addrcl 11165 ax-mulcl 11166 ax-mulrcl 11167 ax-mulcom 11168 ax-addass 11169 ax-mulass 11170 ax-distr 11171 ax-i2m1 11172 ax-1ne0 11173 ax-1rid 11174 ax-rnegex 11175 ax-rrecex 11176 ax-cnre 11177 ax-pre-lttri 11178 ax-pre-lttrn 11179 ax-pre-ltadd 11180 ax-pre-mulgt0 11181 ax-riotaBAD 39755

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-iin 4959 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-sub 11447 df-neg 11448 df-nn 12238 df-2 12307 df-3 12308 df-4 12309 df-5 12310 df-6 12311 df-n0 12509 df-z 12596 df-uz 12867 df-fz 13540 df-struct 17211 df-sets 17228 df-slot 17246 df-ndx 17258 df-base 17274 df-ress 17295 df-plusg 17327 df-mulr 17328 df-sca 17330 df-vsca 17331 df-0g 17498 df-mre 17642 df-mrc 17643 df-acs 17645 df-proset 18354 df-poset 18373 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18492 df-clat 18559 df-mgm 18702 df-sgrp 18781 df-mnd 18797 df-submnd 18846 df-grp 19007 df-minusg 19008 df-sbg 19009 df-subg 19193 df-cntz 19391 df-oppg 19420 df-lsm 19710 df-cmn 19856 df-abl 19857 df-mgp 20221 df-rng 20235 df-ur 20268 df-ring 20321 df-oppr 20424 df-dvdsr 20444 df-unit 20445 df-invr 20475 df-dvr 20488 df-nzr 20619 df-rlreg 20802 df-domn 20803 df-drng 20838 df-lmod 20992 df-lss 21062 df-lsp 21102 df-lvec 21233 df-lsatoms 39778 df-lshyp 39779 df-lcv 39821 df-lfl 39860 df-lkr 39888 df-ldual 39926 df-oposet 39978 df-ol 39980 df-oml 39981 df-covers 40068 df-ats 40069 df-atl 40100 df-cvlat 40124 df-hlat 40153 df-llines 40300 df-lplanes 40301 df-lvols 40302 df-lines 40303 df-psubsp 40305 df-pmap 40306 df-padd 40598 df-lhyp 40790 df-laut 40791 df-ldil 40906 df-ltrn 40907 df-trl 40961 df-tgrp 41545 df-tendo 41557 df-edring 41559 df-dveca 41805 df-disoa 41831 df-dvech 41881 df-dib 41941 df-dic 41975 df-dih 42031 df-doch 42150 df-djh 42197 df-lcdual 42389 df-mapd 42427

This theorem is used by: mapdcnvatN 42468 mapdat 42469

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