hgmaprnlem1N - Mathbox for Norm Megill

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Theorem hgmaprnlem1N 37682

Description: Lemma for hgmaprnN 37687. (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.k2	⊢ (𝜑 → 𝑘 ∈ 𝐵)
hgmaprnlem1.sk	⊢ (𝜑 → 𝑠 = (𝑘 · 𝑡))

**Assertion**
Ref	Expression
hgmaprnlem1N	⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

**Proof of Theorem hgmaprnlem1N**
Step	Hyp	Ref	Expression
1		hgmaprnlem1.sk	. . . . 5 ⊢ (𝜑 → 𝑠 = (𝑘 · 𝑡))
2	1	fveq2d 6348	. . . 4 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑆‘(𝑘 · 𝑡)))
3		hgmaprnlem1.sz	. . . 4 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
4		hgmaprnlem1.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		hgmaprnlem1.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		hgmaprnlem1.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
7		hgmaprnlem1.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑈)
8		hgmaprnlem1.r	. . . . 5 ⊢ 𝑅 = (Scalar‘𝑈)
9		hgmaprnlem1.b	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
10		hgmaprnlem1.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
11		hgmaprnlem1.e	. . . . 5 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
12		hgmaprnlem1.s	. . . . 5 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
13		hgmaprnlem1.g	. . . . 5 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
14		hgmaprnlem1.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
15		hgmaprnlem1.t2	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
16	15	eldifad 3719	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
17		hgmaprnlem1.k2	. . . . 5 ⊢ (𝜑 → 𝑘 ∈ 𝐵)
18	4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17	hgmapvs 37677	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑘 · 𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)))
19	2, 3, 18	3eqtr3d 2794	. . 3 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)))
20		hgmaprnlem1.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
21		hgmaprnlem1.p	. . . 4 ⊢ 𝑃 = (Scalar‘𝐶)
22		hgmaprnlem1.a	. . . 4 ⊢ 𝐴 = (Base‘𝑃)
23		hgmaprnlem1.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
24	4, 10, 14	lcdlvec 37374	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
25		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
26	4, 5, 8, 9, 10, 21, 22, 13, 14, 17	hgmapdcl 37676	. . . 4 ⊢ (𝜑 → (𝐺‘𝑘) ∈ 𝐴)
27	4, 5, 6, 10, 20, 12, 14, 16	hdmapcl 37616	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28		eldifsni 4458	. . . . . 6 ⊢ (𝑡 ∈ (𝑉 ∖ { 0 }) → 𝑡 ≠ 0 )
29	15, 28	syl 17	. . . . 5 ⊢ (𝜑 → 𝑡 ≠ 0 )
30		hgmaprnlem1.o	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
31	4, 5, 6, 30, 10, 23, 12, 14, 16	hdmapeq0 37630	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑡) = 𝑄 ↔ 𝑡 = 0 ))
32	31	necon3bid 2968	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑡) ≠ 𝑄 ↔ 𝑡 ≠ 0 ))
33	29, 32	mpbird 247	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ≠ 𝑄)
34	20, 11, 21, 22, 23, 24, 25, 26, 27, 33	lvecvscan2 19306	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)) ↔ 𝑧 = (𝐺‘𝑘)))
35	19, 34	mpbid 222	. 2 ⊢ (𝜑 → 𝑧 = (𝐺‘𝑘))
36	4, 5, 8, 9, 13, 14	hgmapfnN 37674	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
37		fnfvelrn 6511	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵) → (𝐺‘𝑘) ∈ ran 𝐺)
38	36, 17, 37	syl2anc 696	. 2 ⊢ (𝜑 → (𝐺‘𝑘) ∈ ran 𝐺)
39	35, 38	eqeltrd 2831	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ≠ wne 2924 ∖ cdif 3704 {csn 4313 ran crn 5259 Fn wfn 6036 ‘cfv 6041 (class class class)co 6805 Basecbs 16051 Scalarcsca 16138 ·_𝑠 cvsca 16139 0_gc0g 16294 HLchlt 35132 LHypclh 35765 DVecHcdvh 36861 LCDualclcd 37369 HDMapchdma 37576 HGMapchg 37669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-riotaBAD 34734

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-fal 1630 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-ot 4322 df-uni 4581 df-int 4620 df-iun 4666 df-iin 4667 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-of 7054 df-om 7223 df-1st 7325 df-2nd 7326 df-tpos 7513 df-undef 7560 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-oadd 7725 df-er 7903 df-map 8017 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-n0 11477 df-z 11562 df-uz 11872 df-fz 12512 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-sca 16151 df-vsca 16152 df-0g 16296 df-mre 16440 df-mrc 16441 df-acs 16443 df-preset 17121 df-poset 17139 df-plt 17151 df-lub 17167 df-glb 17168 df-join 17169 df-meet 17170 df-p0 17232 df-p1 17233 df-lat 17239 df-clat 17301 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-submnd 17529 df-grp 17618 df-minusg 17619 df-sbg 17620 df-subg 17784 df-cntz 17942 df-oppg 17968 df-lsm 18243 df-cmn 18387 df-abl 18388 df-mgp 18682 df-ur 18694 df-ring 18741 df-oppr 18815 df-dvdsr 18833 df-unit 18834 df-invr 18864 df-dvr 18875 df-drng 18943 df-lmod 19059 df-lss 19127 df-lsp 19166 df-lvec 19297 df-lsatoms 34758 df-lshyp 34759 df-lcv 34801 df-lfl 34840 df-lkr 34868 df-ldual 34906 df-oposet 34958 df-ol 34960 df-oml 34961 df-covers 35048 df-ats 35049 df-atl 35080 df-cvlat 35104 df-hlat 35133 df-llines 35279 df-lplanes 35280 df-lvols 35281 df-lines 35282 df-psubsp 35284 df-pmap 35285 df-padd 35577 df-lhyp 35769 df-laut 35770 df-ldil 35885 df-ltrn 35886 df-trl 35941 df-tgrp 36525 df-tendo 36537 df-edring 36539 df-dveca 36785 df-disoa 36812 df-dvech 36862 df-dib 36922 df-dic 36956 df-dih 37012 df-doch 37131 df-djh 37178 df-lcdual 37370 df-mapd 37408 df-hvmap 37540 df-hdmap1 37577 df-hdmap 37578 df-hgmap 37670

This theorem is referenced by: hgmaprnlem3N 37684

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