hdmapcl - Mathbox for Norm Megill

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Theorem hdmapcl 42200

Description: Closure of map from vectors to functionals with closed kernels. (Contributed by NM, 15-May-2015.)

**Hypotheses**
Ref	Expression
hdmapcl.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapcl.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapcl.v	⊢ 𝑉 = (Base‘𝑈)
hdmapcl.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmapcl.d	⊢ 𝐷 = (Base‘𝐶)
hdmapcl.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapcl.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapcl.t	⊢ (𝜑 → 𝑇 ∈ 𝑉)

**Assertion**
Ref	Expression
hdmapcl	⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷)

Proof of Theorem hdmapcl Dummy variables 𝑦 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapcl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		eqid 2737	. . 3 ⊢ ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
3		hdmapcl.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		hdmapcl.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		eqid 2737	. . 3 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
6		hdmapcl.c	. . 3 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		hdmapcl.d	. . 3 ⊢ 𝐷 = (Base‘𝐶)
8		eqid 2737	. . 3 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
9		eqid 2737	. . 3 ⊢ ((HDMap1‘𝐾)‘𝑊) = ((HDMap1‘𝐾)‘𝑊)
10		hdmapcl.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
11		hdmapcl.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12		hdmapcl.t	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑉)
13	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12	hdmapval 42198	. 2 ⊢ (𝜑 → (𝑆‘𝑇) = (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑦, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑦⟩), 𝑇⟩))))
14		eqid 2737	. . . 4 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
15		eqid 2737	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
16		eqid 2737	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
17		eqid 2737	. . . . . 6 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2737	. . . . . 6 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
19	1, 17, 18, 3, 4, 14, 2, 11	dvheveccl 41482	. . . . 5 ⊢ (𝜑 → ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩ ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	1, 3, 4, 14, 5, 6, 15, 16, 8, 11, 19	mapdhvmap 42139	. . . 4 ⊢ (𝜑 → (((mapd‘𝐾)‘𝑊)‘((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩})) = ((LSpan‘𝐶)‘{(((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩)}))
21		eqid 2737	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
22	1, 3, 4, 14, 6, 7, 21, 8, 11, 19	hvmapcl2 42136	. . . . 5 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ (𝐷 ∖ {(0_g‘𝐶)}))
23	22	eldifad 3915	. . . 4 ⊢ (𝜑 → (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩) ∈ 𝐷)
24	1, 3, 4, 14, 5, 6, 7, 15, 16, 9, 11, 20, 19, 23, 12	hdmap1eu 42194	. . 3 ⊢ (𝜑 → ∃!ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑦, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑦⟩), 𝑇⟩)))
25		riotacl 7342	. . 3 ⊢ (∃!ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑦, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑦⟩), 𝑇⟩)) → (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑦, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑦⟩), 𝑇⟩))) ∈ 𝐷)
26	24, 25	syl 17	. 2 ⊢ (𝜑 → (℩ℎ ∈ 𝐷 ∀𝑦 ∈ 𝑉 (¬ 𝑦 ∈ (((LSpan‘𝑈)‘{⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩}) ∪ ((LSpan‘𝑈)‘{𝑇})) → ℎ = (((HDMap1‘𝐾)‘𝑊)‘⟨𝑦, (((HDMap1‘𝐾)‘𝑊)‘⟨⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩, (((HVMap‘𝐾)‘𝑊)‘⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩), 𝑦⟩), 𝑇⟩))) ∈ 𝐷)
27	13, 26	eqeltrd 2837	1 ⊢ (𝜑 → (𝑆‘𝑇) ∈ 𝐷)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃!wreu 3350 ∪ cun 3901 {csn 4582 ⟨cop 4588 ⟨cotp 4590 I cid 5526 ↾ cres 5634 ‘cfv 6500 ℩crio 7324 Basecbs 17148 0_gc0g 17371 LSpanclspn 20934 HLchlt 39720 LHypclh 40354 LTrncltrn 40471 DVecHcdvh 41448 LCDualclcd 41956 mapdcmpd 41994 HVMapchvm 42126 HDMap1chdma1 42161 HDMapchdma 42162

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-nzr 20458 df-rlreg 20639 df-domn 20640 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lshyp 39347 df-lcv 39389 df-lfl 39428 df-lkr 39456 df-ldual 39494 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tgrp 41113 df-tendo 41125 df-edring 41127 df-dveca 41373 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718 df-djh 41765 df-lcdual 41957 df-mapd 41995 df-hvmap 42127 df-hdmap1 42163 df-hdmap 42164

This theorem is referenced by: hdmapval2 42202 hdmap10lem 42209 hdmapeq0 42214 hdmapnzcl 42215 hdmapneg 42216 hdmapsub 42217 hdmap11 42218 hdmaprnlem3N 42220 hdmaprnlem3uN 42221 hdmaprnlem7N 42225 hdmaprnlem8N 42226 hdmaprnlem9N 42227 hdmaprnlem3eN 42228 hdmaprnN 42234 hdmap14lem2a 42237 hdmap14lem2N 42239 hdmap14lem3 42240 hdmap14lem4a 42241 hdmap14lem6 42243 hdmap14lem8 42245 hgmapval0 42262 hgmapval1 42263 hgmapadd 42264 hgmapmul 42265 hgmaprnlem1N 42266 hgmaprnlem2N 42267 hgmaprnlem4N 42269 hdmapipcl 42275 hdmapln1 42276 hdmaplna1 42277 hdmaplns1 42278 hdmaplnm1 42279 hdmaplna2 42280 hdmapglnm2 42281 hdmaplkr 42283 hdmapellkr 42284 hdmapip0 42285 hdmapinvlem1 42288 hdmapinvlem3 42290

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