hgmaprnlem1N - Mathbox for Norm Megill

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Theorem hgmaprnlem1N 37970

Description: Lemma for hgmaprnN 37975. (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 6436	. . . 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 3809	. . . . 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 37965	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑘 · 𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)))
19	2, 3, 18	3eqtr3d 2868	. . 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 37665	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
25		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
26	4, 5, 8, 9, 10, 21, 22, 13, 14, 17	hgmapdcl 37964	. . . 4 ⊢ (𝜑 → (𝐺‘𝑘) ∈ 𝐴)
27	4, 5, 6, 10, 20, 12, 14, 16	hdmapcl 37904	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28		eldifsni 4539	. . . . . 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 37918	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑡) = 𝑄 ↔ 𝑡 = 0 ))
32	31	necon3bid 3042	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑡) ≠ 𝑄 ↔ 𝑡 ≠ 0 ))
33	29, 32	mpbird 249	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ≠ 𝑄)
34	20, 11, 21, 22, 23, 24, 25, 26, 27, 33	lvecvscan2 19470	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)) ↔ 𝑧 = (𝐺‘𝑘)))
35	19, 34	mpbid 224	. 2 ⊢ (𝜑 → 𝑧 = (𝐺‘𝑘))
36	4, 5, 8, 9, 13, 14	hgmapfnN 37962	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
37		fnfvelrn 6604	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵) → (𝐺‘𝑘) ∈ ran 𝐺)
38	36, 17, 37	syl2anc 581	. 2 ⊢ (𝜑 → (𝐺‘𝑘) ∈ ran 𝐺)
39	35, 38	eqeltrd 2905	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ≠ wne 2998 ∖ cdif 3794 {csn 4396 ran crn 5342 Fn wfn 6117 ‘cfv 6122 (class class class)co 6904 Basecbs 16221 Scalarcsca 16307 ·_𝑠 cvsca 16308 0_gc0g 16452 HLchlt 35424 LHypclh 36058 DVecHcdvh 37152 LCDualclcd 37660 HDMapchdma 37866 HGMapchg 37957

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-rep 4993 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-1cn 10309 ax-icn 10310 ax-addcl 10311 ax-addrcl 10312 ax-mulcl 10313 ax-mulrcl 10314 ax-mulcom 10315 ax-addass 10316 ax-mulass 10317 ax-distr 10318 ax-i2m1 10319 ax-1ne0 10320 ax-1rid 10321 ax-rnegex 10322 ax-rrecex 10323 ax-cnre 10324 ax-pre-lttri 10325 ax-pre-lttrn 10326 ax-pre-ltadd 10327 ax-pre-mulgt0 10328 ax-riotaBAD 35027

This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-reu 3123 df-rmo 3124 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-pss 3813 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-tp 4401 df-op 4403 df-ot 4405 df-uni 4658 df-int 4697 df-iun 4741 df-iin 4742 df-br 4873 df-opab 4935 df-mpt 4952 df-tr 4975 df-id 5249 df-eprel 5254 df-po 5262 df-so 5263 df-fr 5300 df-we 5302 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-pred 5919 df-ord 5965 df-on 5966 df-lim 5967 df-suc 5968 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-riota 6865 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-of 7156 df-om 7326 df-1st 7427 df-2nd 7428 df-tpos 7616 df-undef 7663 df-wrecs 7671 df-recs 7733 df-rdg 7771 df-1o 7825 df-oadd 7829 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-fin 8225 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-sub 10586 df-neg 10587 df-nn 11350 df-2 11413 df-3 11414 df-4 11415 df-5 11416 df-6 11417 df-n0 11618 df-z 11704 df-uz 11968 df-fz 12619 df-struct 16223 df-ndx 16224 df-slot 16225 df-base 16227 df-sets 16228 df-ress 16229 df-plusg 16317 df-mulr 16318 df-sca 16320 df-vsca 16321 df-0g 16454 df-mre 16598 df-mrc 16599 df-acs 16601 df-proset 17280 df-poset 17298 df-plt 17310 df-lub 17326 df-glb 17327 df-join 17328 df-meet 17329 df-p0 17391 df-p1 17392 df-lat 17398 df-clat 17460 df-mgm 17594 df-sgrp 17636 df-mnd 17647 df-submnd 17688 df-grp 17778 df-minusg 17779 df-sbg 17780 df-subg 17941 df-cntz 18099 df-oppg 18125 df-lsm 18401 df-cmn 18547 df-abl 18548 df-mgp 18843 df-ur 18855 df-ring 18902 df-oppr 18976 df-dvdsr 18994 df-unit 18995 df-invr 19025 df-dvr 19036 df-drng 19104 df-lmod 19220 df-lss 19288 df-lsp 19330 df-lvec 19461 df-lsatoms 35050 df-lshyp 35051 df-lcv 35093 df-lfl 35132 df-lkr 35160 df-ldual 35198 df-oposet 35250 df-ol 35252 df-oml 35253 df-covers 35340 df-ats 35341 df-atl 35372 df-cvlat 35396 df-hlat 35425 df-llines 35572 df-lplanes 35573 df-lvols 35574 df-lines 35575 df-psubsp 35577 df-pmap 35578 df-padd 35870 df-lhyp 36062 df-laut 36063 df-ldil 36178 df-ltrn 36179 df-trl 36233 df-tgrp 36817 df-tendo 36829 df-edring 36831 df-dveca 37077 df-disoa 37103 df-dvech 37153 df-dib 37213 df-dic 37247 df-dih 37303 df-doch 37422 df-djh 37469 df-lcdual 37661 df-mapd 37699 df-hvmap 37831 df-hdmap1 37867 df-hdmap 37868 df-hgmap 37958

This theorem is referenced by: hgmaprnlem3N 37972

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