mapdindp - Mathbox for Norm Megill

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Theorem mapdindp 42445

Description: Transfer (part of) vector independence condition from domain to range of projectivity mapd. (Contributed by NM, 11-Apr-2015.)

**Hypotheses**
Ref	Expression
mapdindp.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdindp.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdindp.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdindp.v	⊢ 𝑉 = (Base‘𝑈)
mapdindp.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdindp.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdindp.d	⊢ 𝐷 = (Base‘𝐶)
mapdindp.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdindp.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdindp.f	⊢ (𝜑 → 𝐹 ∈ 𝐷)
mapdindp.mx	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
mapdindp.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdindp.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdindp.g	⊢ (𝜑 → 𝐺 ∈ 𝐷)
mapdindp.my	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
mapdindp.z	⊢ (𝜑 → 𝑍 ∈ 𝑉)
mapdindp.e	⊢ (𝜑 → 𝐸 ∈ 𝐷)
mapdindp.mg	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
mapdindp.xn	⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))

**Assertion**
Ref	Expression
mapdindp	⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))

**Proof of Theorem mapdindp**
Step	Hyp	Ref	Expression
1		mapdindp.xn	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ (𝑁‘{𝑌, 𝑍}))
2		mapdindp.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
3		eqid 2763	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
4		mapdindp.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
5		mapdindp.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
6		mapdindp.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		mapdindp.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
8	5, 6, 7	lcdlmod 42366	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
9		mapdindp.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐷)
10		mapdindp.e	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝐷)
11	2, 3, 4, 8, 9, 10	lspprcl 21099	. . . 4 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) ∈ (LSubSp‘𝐶))
12		mapdindp.f	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷)
13	2, 3, 4, 8, 11, 12	ellspsn5b 21116	. . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐽‘{𝐺, 𝐸}) ↔ (𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸})))
14		mapdindp.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
15		eqid 2763	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
16		mapdindp.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
17		mapdindp.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
18	5, 17, 7	dvhlmod 41884	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
19		mapdindp.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
20		mapdindp.z	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
21	14, 15, 16, 18, 19, 20	lspprcl 21099	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) ∈ (LSubSp‘𝑈))
22		mapdindp.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
23	14, 15, 16, 18, 21, 22	ellspsn5b 21116	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝑁‘{𝑌, 𝑍}) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
24		mapdindp.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
25	14, 15, 16	lspsncl 21098	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
26	18, 22, 25	syl2anc 595	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ (LSubSp‘𝑈))
27	5, 17, 15, 24, 7, 26, 21	mapdord 42412	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊆ (𝑀‘(𝑁‘{𝑌, 𝑍})) ↔ (𝑁‘{𝑋}) ⊆ (𝑁‘{𝑌, 𝑍})))
28		mapdindp.mx	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐹}))
29		eqid 2763	. . . . . . . . 9 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
30	14, 16, 29, 18, 19, 20	lsmpr 21210	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌, 𝑍}) = ((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍})))
31	30	fveq2d 6885	. . . . . . 7 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))))
32		eqid 2763	. . . . . . . 8 ⊢ (LSSum‘𝐶) = (LSSum‘𝐶)
33	14, 15, 16	lspsncl 21098	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
34	18, 19, 33	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
35	14, 15, 16	lspsncl 21098	. . . . . . . . 9 ⊢ ((𝑈 ∈ LMod ∧ 𝑍 ∈ 𝑉) → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
36	18, 20, 35	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (𝑁‘{𝑍}) ∈ (LSubSp‘𝑈))
37	5, 24, 17, 15, 29, 6, 32, 7, 34, 36	mapdlsm 42438	. . . . . . 7 ⊢ (𝜑 → (𝑀‘((𝑁‘{𝑌})(LSSum‘𝑈)(𝑁‘{𝑍}))) = ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))))
38		mapdindp.my	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝐺}))
39		mapdindp.mg	. . . . . . . 8 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑍})) = (𝐽‘{𝐸}))
40	38, 39	oveq12d 7428	. . . . . . 7 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑌}))(LSSum‘𝐶)(𝑀‘(𝑁‘{𝑍}))) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸})))
41	31, 37, 40	3eqtrd 2802	. . . . . 6 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸})))
42	2, 4, 32, 8, 9, 10	lsmpr 21210	. . . . . 6 ⊢ (𝜑 → (𝐽‘{𝐺, 𝐸}) = ((𝐽‘{𝐺})(LSSum‘𝐶)(𝐽‘{𝐸})))
43	41, 42	eqtr4d 2801	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌, 𝑍})) = (𝐽‘{𝐺, 𝐸}))
44	28, 43	sseq12d 3970	. . . 4 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑋})) ⊆ (𝑀‘(𝑁‘{𝑌, 𝑍})) ↔ (𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸})))
45	23, 27, 44	3bitr2rd 311	. . 3 ⊢ (𝜑 → ((𝐽‘{𝐹}) ⊆ (𝐽‘{𝐺, 𝐸}) ↔ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
46	13, 45	bitrd 282	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐽‘{𝐺, 𝐸}) ↔ 𝑋 ∈ (𝑁‘{𝑌, 𝑍})))
47	1, 46	mtbird 328	1 ⊢ (𝜑 → ¬ 𝐹 ∈ (𝐽‘{𝐺, 𝐸}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 400 = wceq 1570 ∈ wcel 2143 ⊆ wss 3905 {csn 4589 {cpr 4591 ‘cfv 6536 (class class class)co 7410 Basecbs 17264 LSSumclsm 19699 LModclmod 20981 LSubSpclss 21052 LSpanclspn 21092 HLchlt 40124 LHypclh 40758 DVecHcdvh 41852 LCDualclcd 42360 mapdcmpd 42398

This theorem was proved from 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 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-riotaBAD 39727

This theorem 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 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-n0 12500 df-z 12587 df-uz 12858 df-fz 13531 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-sca 17321 df-vsca 17322 df-0g 17489 df-mre 17633 df-mrc 17634 df-acs 17636 df-proset 18345 df-poset 18364 df-plt 18379 df-lub 18395 df-glb 18396 df-join 18397 df-meet 18398 df-p0 18474 df-p1 18475 df-lat 18483 df-clat 18550 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-submnd 18837 df-grp 18998 df-minusg 18999 df-sbg 19000 df-subg 19184 df-cntz 19382 df-oppg 19411 df-lsm 19701 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-nzr 20610 df-rlreg 20793 df-domn 20794 df-drng 20829 df-lmod 20983 df-lss 21053 df-lsp 21093 df-lvec 21224 df-lsatoms 39750 df-lshyp 39751 df-lcv 39793 df-lfl 39832 df-lkr 39860 df-ldual 39898 df-oposet 39950 df-ol 39952 df-oml 39953 df-covers 40040 df-ats 40041 df-atl 40072 df-cvlat 40096 df-hlat 40125 df-llines 40272 df-lplanes 40273 df-lvols 40274 df-lines 40275 df-psubsp 40277 df-pmap 40278 df-padd 40570 df-lhyp 40762 df-laut 40763 df-ldil 40878 df-ltrn 40879 df-trl 40933 df-tgrp 41517 df-tendo 41529 df-edring 41531 df-dveca 41777 df-disoa 41803 df-dvech 41853 df-dib 41913 df-dic 41947 df-dih 42003 df-doch 42122 df-djh 42169 df-lcdual 42361 df-mapd 42399

This theorem is referenced by: mapdheq4lem 42505 mapdh6lem1N 42507 mapdh6lem2N 42508 hdmap1l6lem1 42581 hdmap1l6lem2 42582

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