mapdcv - Mathbox for Norm Megill

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Theorem mapdcv 41661

Description: Covering property of the converse of the map defined by df-mapd 41626. (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 41656	. . 3 ⊢ (𝜑 → ((𝑀‘𝑋) ⊊ (𝑀‘𝑌) ↔ 𝑋 ⊊ 𝑌))
9		mapdcv.d	. . . . . . 7 ⊢ 𝐷 = ((LCDual‘𝐾)‘𝑊)
10		eqid 2730	. . . . . . 7 ⊢ (LSubSp‘𝐷) = (LSubSp‘𝐷)
11	5	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑣 ∈ 𝑆)
13	1, 2, 3, 4, 9, 10, 11, 12	mapdcl2 41657	. . . . . 6 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → (𝑀‘𝑣) ∈ (LSubSp‘𝐷))
14	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15	1, 2, 9, 10, 5	mapdrn2 41652	. . . . . . . . . 10 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘𝐷))
16	15	eleq2d 2815	. . . . . . . . 9 ⊢ (𝜑 → (𝑓 ∈ ran 𝑀 ↔ 𝑓 ∈ (LSubSp‘𝐷)))
17	16	biimpar 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 ∈ ran 𝑀)
18	1, 2, 3, 4, 14, 17	mapdcnvcl 41653	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (^◡𝑀‘𝑓) ∈ 𝑆)
19	1, 2, 14, 17	mapdcnvid2 41658	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → (𝑀‘(^◡𝑀‘𝑓)) = 𝑓)
20	19	eqcomd 2736	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (LSubSp‘𝐷)) → 𝑓 = (𝑀‘(^◡𝑀‘𝑓)))
21		fveq2 6861	. . . . . . . 8 ⊢ (𝑣 = (^◡𝑀‘𝑓) → (𝑀‘𝑣) = (𝑀‘(^◡𝑀‘𝑓)))
22	21	rspceeqv 3614	. . . . . . 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 4057	. . . . . . . 8 ⊢ (𝑓 = (𝑀‘𝑣) → ((𝑀‘𝑋) ⊊ 𝑓 ↔ (𝑀‘𝑋) ⊊ (𝑀‘𝑣)))
25		psseq1 4056	. . . . . . . 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 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 = (𝑀‘𝑣)) → (((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
28	13, 23, 27	rexxfrd 5367	. . . . 5 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌))))
29	6	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑋 ∈ 𝑆)
30	1, 2, 3, 4, 11, 29, 12	mapdsord 41656	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ↔ 𝑋 ⊊ 𝑣))
31	7	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑣 ∈ 𝑆) → 𝑌 ∈ 𝑆)
32	1, 2, 3, 4, 11, 12, 31	mapdsord 41656	. . . . . . 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 3156	. . . . 5 ⊢ (𝜑 → (∃𝑣 ∈ 𝑆 ((𝑀‘𝑋) ⊊ (𝑀‘𝑣) ∧ (𝑀‘𝑣) ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
35	28, 34	bitrd 279	. . . 4 ⊢ (𝜑 → (∃𝑓 ∈ (LSubSp‘𝐷)((𝑀‘𝑋) ⊊ 𝑓 ∧ 𝑓 ⊊ (𝑀‘𝑌)) ↔ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌)))
36	35	notbid 318	. . 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 41593	. . 3 ⊢ (𝜑 → 𝐷 ∈ LMod)
40	1, 2, 3, 4, 9, 10, 5, 6	mapdcl2 41657	. . 3 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘𝐷))
41	1, 2, 3, 4, 9, 10, 5, 7	mapdcl2 41657	. . 3 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘𝐷))
42	10, 38, 39, 40, 41	lcvbr 39021	. 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 41111	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
45	4, 43, 44, 6, 7	lcvbr 39021	. 2 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑋 ⊊ 𝑌 ∧ ¬ ∃𝑣 ∈ 𝑆 (𝑋 ⊊ 𝑣 ∧ 𝑣 ⊊ 𝑌))))
46	37, 42, 45	3bitr4rd 312	1 ⊢ (𝜑 → (𝑋𝐶𝑌 ↔ (𝑀‘𝑋)𝐸(𝑀‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∃wrex 3054 ⊊ wpss 3918 class class class wbr 5110 ^◡ccnv 5640 ran crn 5642 ‘cfv 6514 LModclmod 20773 LSubSpclss 20844 ⋖_L clcv 39018 HLchlt 39350 LHypclh 39985 DVecHcdvh 41079 LCDualclcd 41587 mapdcmpd 41625

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 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-riotaBAD 38953

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-tpos 8208 df-undef 8255 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-er 8674 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-n0 12450 df-z 12537 df-uz 12801 df-fz 13476 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17411 df-mre 17554 df-mrc 17555 df-acs 17557 df-proset 18262 df-poset 18281 df-plt 18296 df-lub 18312 df-glb 18313 df-join 18314 df-meet 18315 df-p0 18391 df-p1 18392 df-lat 18398 df-clat 18465 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-subg 19062 df-cntz 19256 df-oppg 19285 df-lsm 19573 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-oppr 20253 df-dvdsr 20273 df-unit 20274 df-invr 20304 df-dvr 20317 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20775 df-lss 20845 df-lsp 20885 df-lvec 21017 df-lsatoms 38976 df-lshyp 38977 df-lcv 39019 df-lfl 39058 df-lkr 39086 df-ldual 39124 df-oposet 39176 df-ol 39178 df-oml 39179 df-covers 39266 df-ats 39267 df-atl 39298 df-cvlat 39322 df-hlat 39351 df-llines 39499 df-lplanes 39500 df-lvols 39501 df-lines 39502 df-psubsp 39504 df-pmap 39505 df-padd 39797 df-lhyp 39989 df-laut 39990 df-ldil 40105 df-ltrn 40106 df-trl 40160 df-tgrp 40744 df-tendo 40756 df-edring 40758 df-dveca 41004 df-disoa 41030 df-dvech 41080 df-dib 41140 df-dic 41174 df-dih 41230 df-doch 41349 df-djh 41396 df-lcdual 41588 df-mapd 41626

This theorem is referenced by: mapdcnvatN 41667 mapdat 41668

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