hgmaprnlem2N - Mathbox for Norm Megill

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Theorem hgmaprnlem2N 39556

Description: Lemma for hgmaprnN 39560. 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 39251	. . . 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 3855	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 39489	. . . 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 19897	. . . 4 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}) ⊆ (𝐿‘{(𝑆‘𝑡)}))
18	4, 5, 12, 17	syl3anc 1372	. . 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 39499	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑆‘𝑠)}))
23		hgmaprnlem1.sz	. . . . . 6 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
24	23	sneqd 4528	. . . . 5 ⊢ (𝜑 → {(𝑆‘𝑠)} = {(𝑧 ∙ (𝑆‘𝑡))})
25	24	fveq2d 6680	. . . 4 ⊢ (𝜑 → (𝐿‘{(𝑆‘𝑠)}) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}))
26	22, 25	eqtrd 2773	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) = (𝐿‘{(𝑧 ∙ (𝑆‘𝑡))}))
27	1, 6, 7, 19, 2, 16, 20, 9, 3, 11	hdmap10 39499	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑡})) = (𝐿‘{(𝑆‘𝑡)}))
28	18, 26, 27	3sstr4d 3924	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡})))
29		eqid 2738	. . 3 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
30	1, 6, 3	dvhlmod 38769	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
31	7, 29, 19	lspsncl 19870	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑠 ∈ 𝑉) → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈))
32	30, 21, 31	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑁‘{𝑠}) ∈ (LSubSp‘𝑈))
33	7, 29, 19	lspsncl 19870	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ 𝑡 ∈ 𝑉) → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
34	30, 11, 33	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑁‘{𝑡}) ∈ (LSubSp‘𝑈))
35	1, 6, 29, 20, 3, 32, 34	mapdord 39297	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{𝑠})) ⊆ (𝑀‘(𝑁‘{𝑡})) ↔ (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡})))
36	28, 35	mpbid 235	1 ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∖ cdif 3840 ⊆ wss 3843 {csn 4516 ‘cfv 6339 (class class class)co 7172 Basecbs 16588 Scalarcsca 16673 ·_𝑠 cvsca 16674 0_gc0g 16818 LModclmod 19755 LSubSpclss 19824 LSpanclspn 19864 HLchlt 37009 LHypclh 37643 DVecHcdvh 38737 LCDualclcd 39245 mapdcmpd 39283 HDMapchdma 39451 HGMapchg 39542

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7481 ax-cnex 10673 ax-resscn 10674 ax-1cn 10675 ax-icn 10676 ax-addcl 10677 ax-addrcl 10678 ax-mulcl 10679 ax-mulrcl 10680 ax-mulcom 10681 ax-addass 10682 ax-mulass 10683 ax-distr 10684 ax-i2m1 10685 ax-1ne0 10686 ax-1rid 10687 ax-rnegex 10688 ax-rrecex 10689 ax-cnre 10690 ax-pre-lttri 10691 ax-pre-lttrn 10692 ax-pre-ltadd 10693 ax-pre-mulgt0 10694 ax-riotaBAD 36612

This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-ot 4525 df-uni 4797 df-int 4837 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7129 df-ov 7175 df-oprab 7176 df-mpo 7177 df-of 7427 df-om 7602 df-1st 7716 df-2nd 7717 df-tpos 7923 df-undef 7970 df-wrecs 7978 df-recs 8039 df-rdg 8077 df-1o 8133 df-er 8322 df-map 8441 df-en 8558 df-dom 8559 df-sdom 8560 df-fin 8561 df-pnf 10757 df-mnf 10758 df-xr 10759 df-ltxr 10760 df-le 10761 df-sub 10952 df-neg 10953 df-nn 11719 df-2 11781 df-3 11782 df-4 11783 df-5 11784 df-6 11785 df-n0 11979 df-z 12065 df-uz 12327 df-fz 12984 df-struct 16590 df-ndx 16591 df-slot 16592 df-base 16594 df-sets 16595 df-ress 16596 df-plusg 16683 df-mulr 16684 df-sca 16686 df-vsca 16687 df-0g 16820 df-mre 16962 df-mrc 16963 df-acs 16965 df-proset 17656 df-poset 17674 df-plt 17686 df-lub 17702 df-glb 17703 df-join 17704 df-meet 17705 df-p0 17767 df-p1 17768 df-lat 17774 df-clat 17836 df-mgm 17970 df-sgrp 18019 df-mnd 18030 df-submnd 18075 df-grp 18224 df-minusg 18225 df-sbg 18226 df-subg 18396 df-cntz 18567 df-oppg 18594 df-lsm 18881 df-cmn 19028 df-abl 19029 df-mgp 19361 df-ur 19373 df-ring 19420 df-oppr 19497 df-dvdsr 19515 df-unit 19516 df-invr 19546 df-dvr 19557 df-drng 19625 df-lmod 19757 df-lss 19825 df-lsp 19865 df-lvec 19996 df-lsatoms 36635 df-lshyp 36636 df-lcv 36678 df-lfl 36717 df-lkr 36745 df-ldual 36783 df-oposet 36835 df-ol 36837 df-oml 36838 df-covers 36925 df-ats 36926 df-atl 36957 df-cvlat 36981 df-hlat 37010 df-llines 37157 df-lplanes 37158 df-lvols 37159 df-lines 37160 df-psubsp 37162 df-pmap 37163 df-padd 37455 df-lhyp 37647 df-laut 37648 df-ldil 37763 df-ltrn 37764 df-trl 37818 df-tgrp 38402 df-tendo 38414 df-edring 38416 df-dveca 38662 df-disoa 38688 df-dvech 38738 df-dib 38798 df-dic 38832 df-dih 38888 df-doch 39007 df-djh 39054 df-lcdual 39246 df-mapd 39284 df-hvmap 39416 df-hdmap1 39452 df-hdmap 39453

This theorem is referenced by: hgmaprnlem3N 39557

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