hgmaprnlem2N - Mathbox for Norm Megill

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Theorem hgmaprnlem2N 39838

Description: Lemma for hgmaprnN 39842. Part 15 of [Baer] p. 50 line 20. We only require a subset relation, rather than equality, so that the case of zero 𝑧 is taken care of automatically. (Contributed by NM, 7-Jun-2015.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
hgmaprnlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
hgmaprnlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hgmaprnlem1.v	⊢ 𝑉 = (Base‘𝑈)
hgmaprnlem1.r	⊢ 𝑅 = (Scalar‘𝑈)
hgmaprnlem1.b	⊢ 𝐵 = (Base‘𝑅)
hgmaprnlem1.t	⊢ · = ( ·_𝑠 ‘𝑈)
hgmaprnlem1.o	⊢ 0 = (0_g‘𝑈)
hgmaprnlem1.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hgmaprnlem1.d	⊢ 𝐷 = (Base‘𝐶)
hgmaprnlem1.p	⊢ 𝑃 = (Scalar‘𝐶)
hgmaprnlem1.a	⊢ 𝐴 = (Base‘𝑃)
hgmaprnlem1.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hgmaprnlem1.q	⊢ 𝑄 = (0_g‘𝐶)
hgmaprnlem1.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hgmaprnlem1.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hgmaprnlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hgmaprnlem1.z	⊢ (𝜑 → 𝑧 ∈ 𝐴)
hgmaprnlem1.t2	⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
hgmaprnlem1.s2	⊢ (𝜑 → 𝑠 ∈ 𝑉)
hgmaprnlem1.sz	⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
hgmaprnlem1.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hgmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hgmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)

**Assertion**
Ref	Expression
hgmaprnlem2N	⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))

**Proof of Theorem hgmaprnlem2N**
Step	Hyp	Ref	Expression
1		hgmaprnlem1.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		hgmaprnlem1.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hgmaprnlem1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 39533	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
6		hgmaprnlem1.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		hgmaprnlem1.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
8		hgmaprnlem1.d	. . . . 5 ⊢ 𝐷 = (Base‘𝐶)
9		hgmaprnlem1.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hgmaprnlem1.t2	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3895	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 39771	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
13		hgmaprnlem1.p	. . . . 5 ⊢ 𝑃 = (Scalar‘𝐶)
14		hgmaprnlem1.a	. . . . 5 ⊢ 𝐴 = (Base‘𝑃)
15		hgmaprnlem1.e	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
16		hgmaprnlem1.l	. . . . 5 ⊢ 𝐿 = (LSpan‘𝐶)
17	13, 14, 8, 15, 16	lspsnvsi 20181	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}) ⊆ (𝐿‘{(𝑆‘𝑡)}))
18	4, 5, 12, 17	syl3anc 1369	. . 3 ⊢ (𝜑 → (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}) ⊆ (𝐿‘{(𝑆‘𝑡)}))
19		hgmaprnlem1.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
20		hgmaprnlem1.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
21		hgmaprnlem1.s2	. . . . 5 ⊢ (𝜑 → 𝑠 ∈ 𝑉)
22	1, 6, 7, 19, 2, 16, 20, 9, 3, 21	hdmap10 39781	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑆‘𝑠)}))
23		hgmaprnlem1.sz	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
24	23	sneqd 4570	. . . . 5 ⊢ (𝜑 → {(𝑆‘𝑠)} = {(𝑧 ∙ (𝑆‘𝑡))})
25	24	fveq2d 6760	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑠)}) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}))
26	22, 25	eqtrd 2778	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}))
27	1, 6, 7, 19, 2, 16, 20, 9, 3, 11	hdmap10 39781	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
28	18, 26, 27	3sstr4d 3964	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡})))
29		eqid 2738	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
30	1, 6, 3	dvhlmod 39051	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
31	7, 29, 19	lspsncl 20154	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑠 ∈ 𝑉) → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈))
32	30, 21, 31	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈))
33	7, 29, 19	lspsncl 20154	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
34	30, 11, 33	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
35	1, 6, 29, 20, 3, 32, 34	mapdord 39579	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡})) ↔ (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})))
36	28, 35	mpbid 231	1 ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 LModclmod 20038 LSubSpclss 20108 LSpanclspn 20148 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 LCDualclcd 39527 mapdcmpd 39565 HDMapchdma 39733 HGMapchg 39824

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-ot 4567 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lcv 36960 df-lfl 36999 df-lkr 37027 df-ldual 37065 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336 df-lcdual 39528 df-mapd 39566 df-hvmap 39698 df-hdmap1 39734 df-hdmap 39735

This theorem is referenced by: hgmaprnlem3N 39839

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