mapdin - Mathbox for Norm Megill

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Theorem mapdin 41641

Description: Subspace intersection is preserved by the map defined by df-mapd 41604. Part of property (e) in [Baer] p. 40. (Contributed by NM, 12-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdin.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdin.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdin.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdin.s	⊢ 𝑆 = (LSubSp‘𝑈)
mapdin.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdin.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)
mapdin.y	⊢ (𝜑 → 𝑌 ∈ 𝑆)

**Assertion**
Ref	Expression
mapdin	⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) = ((𝑀‘𝑋) ∩ (𝑀‘𝑌)))

**Proof of Theorem mapdin**
Step	Hyp	Ref	Expression
1		inss1 4188	. . . 4 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑋
2		mapdin.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdin.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdin.s	. . . . 5 ⊢ 𝑆 = (LSubSp‘𝑈)
5		mapdin.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
6		mapdin.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	2, 3, 6	dvhlmod 41089	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
8		mapdin.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
9		mapdin.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑆)
10	4	lssincl 20868	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑆 ∧ 𝑌 ∈ 𝑆) → (𝑋 ∩ 𝑌) ∈ 𝑆)
11	7, 8, 9, 10	syl3anc 1373	. . . . 5 ⊢ (𝜑 → (𝑋 ∩ 𝑌) ∈ 𝑆)
12	2, 3, 4, 5, 6, 11, 8	mapdord 41617	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑋 ∩ 𝑌)) ⊆ (𝑀‘𝑋) ↔ (𝑋 ∩ 𝑌) ⊆ 𝑋))
13	1, 12	mpbiri 258	. . 3 ⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) ⊆ (𝑀‘𝑋))
14		inss2 4189	. . . 4 ⊢ (𝑋 ∩ 𝑌) ⊆ 𝑌
15	2, 3, 4, 5, 6, 11, 9	mapdord 41617	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑋 ∩ 𝑌)) ⊆ (𝑀‘𝑌) ↔ (𝑋 ∩ 𝑌) ⊆ 𝑌))
16	14, 15	mpbiri 258	. . 3 ⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) ⊆ (𝑀‘𝑌))
17	13, 16	ssind 4192	. 2 ⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) ⊆ ((𝑀‘𝑋) ∩ (𝑀‘𝑌)))
18		eqid 2729	. . . . 5 ⊢ ((LCDual‘𝐾)‘𝑊) = ((LCDual‘𝐾)‘𝑊)
19		eqid 2729	. . . . . . 7 ⊢ (LSubSp‘((LCDual‘𝐾)‘𝑊)) = (LSubSp‘((LCDual‘𝐾)‘𝑊))
20	2, 5, 3, 4, 18, 19, 6, 8	mapdcl2 41635	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑋) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)))
21	2, 5, 18, 19, 6	mapdrn2 41630	. . . . . 6 ⊢ (𝜑 → ran 𝑀 = (LSubSp‘((LCDual‘𝐾)‘𝑊)))
22	20, 21	eleqtrrd 2831	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑋) ∈ ran 𝑀)
23	2, 5, 3, 4, 18, 19, 6, 9	mapdcl2 41635	. . . . . 6 ⊢ (𝜑 → (𝑀‘𝑌) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)))
24	23, 21	eleqtrrd 2831	. . . . 5 ⊢ (𝜑 → (𝑀‘𝑌) ∈ ran 𝑀)
25	2, 5, 3, 18, 6, 22, 24	mapdincl 41640	. . . 4 ⊢ (𝜑 → ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ∈ ran 𝑀)
26	2, 5, 6, 25	mapdcnvid2 41636	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) = ((𝑀‘𝑋) ∩ (𝑀‘𝑌)))
27		inss1 4188	. . . . . . 7 ⊢ ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ⊆ (𝑀‘𝑋)
28	26, 27	eqsstrdi 3980	. . . . . 6 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘𝑋))
29	2, 18, 6	lcdlmod 41571	. . . . . . . . . 10 ⊢ (𝜑 → ((LCDual‘𝐾)‘𝑊) ∈ LMod)
30	19	lssincl 20868	. . . . . . . . . 10 ⊢ ((((LCDual‘𝐾)‘𝑊) ∈ LMod ∧ (𝑀‘𝑋) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)) ∧ (𝑀‘𝑌) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊))) → ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)))
31	29, 20, 23, 30	syl3anc 1373	. . . . . . . . 9 ⊢ (𝜑 → ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ∈ (LSubSp‘((LCDual‘𝐾)‘𝑊)))
32	31, 21	eleqtrrd 2831	. . . . . . . 8 ⊢ (𝜑 → ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ∈ ran 𝑀)
33	2, 5, 3, 4, 6, 32	mapdcnvcl 41631	. . . . . . 7 ⊢ (𝜑 → (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ∈ 𝑆)
34	2, 3, 4, 5, 6, 33, 8	mapdord 41617	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘𝑋) ↔ (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ 𝑋))
35	28, 34	mpbid 232	. . . . 5 ⊢ (𝜑 → (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ 𝑋)
36	2, 5, 6, 32	mapdcnvid2 41636	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) = ((𝑀‘𝑋) ∩ (𝑀‘𝑌)))
37		inss2 4189	. . . . . . 7 ⊢ ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ⊆ (𝑀‘𝑌)
38	36, 37	eqsstrdi 3980	. . . . . 6 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘𝑌))
39	2, 3, 4, 5, 6, 33, 9	mapdord 41617	. . . . . 6 ⊢ (𝜑 → ((𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘𝑌) ↔ (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ 𝑌))
40	38, 39	mpbid 232	. . . . 5 ⊢ (𝜑 → (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ 𝑌)
41	35, 40	ssind 4192	. . . 4 ⊢ (𝜑 → (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ (𝑋 ∩ 𝑌))
42	2, 3, 4, 5, 6, 33, 11	mapdord 41617	. . . 4 ⊢ (𝜑 → ((𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘(𝑋 ∩ 𝑌)) ↔ (^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌))) ⊆ (𝑋 ∩ 𝑌)))
43	41, 42	mpbird 257	. . 3 ⊢ (𝜑 → (𝑀‘(^◡𝑀‘((𝑀‘𝑋) ∩ (𝑀‘𝑌)))) ⊆ (𝑀‘(𝑋 ∩ 𝑌)))
44	26, 43	eqsstrrd 3971	. 2 ⊢ (𝜑 → ((𝑀‘𝑋) ∩ (𝑀‘𝑌)) ⊆ (𝑀‘(𝑋 ∩ 𝑌)))
45	17, 44	eqssd 3953	1 ⊢ (𝜑 → (𝑀‘(𝑋 ∩ 𝑌)) = ((𝑀‘𝑋) ∩ (𝑀‘𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ∩ cin 3902 ⊆ wss 3903 ^◡ccnv 5618 ran crn 5620 ‘cfv 6482 LModclmod 20763 LSubSpclss 20834 HLchlt 39329 LHypclh 39963 DVecHcdvh 41057 LCDualclcd 41565 mapdcmpd 41603

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 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 38932

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 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-undef 8206 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-cntz 19196 df-oppg 19225 df-lsm 19515 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-nzr 20398 df-rlreg 20579 df-domn 20580 df-drng 20616 df-lmod 20765 df-lss 20835 df-lsp 20875 df-lvec 21007 df-lsatoms 38955 df-lshyp 38956 df-lcv 38998 df-lfl 39037 df-lkr 39065 df-ldual 39103 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374 df-lcdual 41566 df-mapd 41604

This theorem is referenced by: mapdheq4lem 41710 mapdh6lem1N 41712 mapdh6lem2N 41713 hdmap1l6lem1 41786 hdmap1l6lem2 41787

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