hgmaprnlem1N - Mathbox for Norm Megill

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Theorem hgmaprnlem1N 42728

Description: Lemma for hgmaprnN 42733. (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 6889	. . . 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 3918	. . . . 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 42723	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑘 · 𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)))
19	2, 3, 18	3eqtr3d 2808	. . 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 42423	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
25		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
26	4, 5, 8, 9, 10, 21, 22, 13, 14, 17	hgmapdcl 42722	. . . 4 ⊢ (𝜑 → (𝐺‘𝑘) ∈ 𝐴)
27	4, 5, 6, 10, 20, 12, 14, 16	hdmapcl 42662	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28		eldifsni 4760	. . . . . 6 ⊢ (𝑡 ∈ (𝑉 ∖ { 0 }) → 𝑡 ≠ 0 )
29	15, 28	syl 18	. . . . 5 ⊢ (𝜑 → 𝑡 ≠ 0 )
30		hgmaprnlem1.o	. . . . . . 7 ⊢ 0 = (0_g‘𝑈)
31	4, 5, 6, 30, 10, 23, 12, 14, 16	hdmapeq0 42676	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑡) = 𝑄 ↔ 𝑡 = 0 ))
32	31	necon3bid 3004	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑡) ≠ 𝑄 ↔ 𝑡 ≠ 0 ))
33	29, 32	mpbird 260	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ≠ 𝑄)
34	20, 11, 21, 22, 23, 24, 25, 26, 27, 33	lvecvscan2 21286	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)) ↔ 𝑧 = (𝐺‘𝑘)))
35	19, 34	mpbid 235	. 2 ⊢ (𝜑 → 𝑧 = (𝐺‘𝑘))
36	4, 5, 8, 9, 13, 14	hgmapfnN 42720	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
37		fnfvelrn 7079	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵) → (𝐺‘𝑘) ∈ ran 𝐺)
38	36, 17, 37	syl2anc 596	. 2 ⊢ (𝜑 → (𝐺‘𝑘) ∈ ran 𝐺)
39	35, 38	eqeltrd 2865	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 ∖ cdif 3903 {csn 4591 ran crn 5664 Fn wfn 6535 ‘cfv 6540 (class class class)co 7419 Basecbs 17291 Scalarcsca 17335 ·_𝑠 cvsca 17336 0_gc0g 17514 HLchlt 40182 LHypclh 40816 DVecHcdvh 41910 LCDualclcd 42418 HDMapchdma 42624 HGMapchg 42715

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-rep 5240 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11171 ax-resscn 11172 ax-1cn 11173 ax-icn 11174 ax-addcl 11175 ax-addrcl 11176 ax-mulcl 11177 ax-mulrcl 11178 ax-mulcom 11179 ax-addass 11180 ax-mulass 11181 ax-distr 11182 ax-i2m1 11183 ax-1ne0 11184 ax-1rid 11185 ax-rnegex 11186 ax-rrecex 11187 ax-cnre 11188 ax-pre-lttri 11189 ax-pre-lttrn 11190 ax-pre-ltadd 11191 ax-pre-mulgt0 11192 ax-riotaBAD 39785

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-ot 4600 df-uni 4875 df-int 4915 df-iun 4960 df-iin 4961 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7684 df-om 7869 df-1st 7992 df-2nd 7993 df-tpos 8228 df-undef 8275 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-2o 8460 df-er 8700 df-map 8832 df-en 8950 df-dom 8951 df-sdom 8952 df-fin 8953 df-pnf 11260 df-mnf 11261 df-xr 11262 df-ltxr 11263 df-le 11264 df-sub 11458 df-neg 11459 df-nn 12249 df-2 12318 df-3 12319 df-4 12320 df-5 12321 df-6 12322 df-n0 12520 df-z 12607 df-uz 12879 df-fz 13552 df-struct 17229 df-sets 17246 df-slot 17264 df-ndx 17276 df-base 17292 df-ress 17313 df-plusg 17345 df-mulr 17346 df-sca 17348 df-vsca 17349 df-0g 17516 df-mre 17660 df-mrc 17661 df-acs 17663 df-proset 18372 df-poset 18391 df-plt 18406 df-lub 18422 df-glb 18423 df-join 18424 df-meet 18425 df-p0 18501 df-p1 18502 df-lat 18510 df-clat 18577 df-mgm 18720 df-sgrp 18809 df-mnd 18825 df-submnd 18879 df-grp 19047 df-minusg 19048 df-sbg 19049 df-subg 19233 df-cntz 19431 df-oppg 19460 df-lsm 19750 df-cmn 19896 df-abl 19897 df-mgp 20261 df-rng 20275 df-ur 20308 df-ring 20361 df-oppr 20465 df-dvdsr 20485 df-unit 20486 df-invr 20516 df-dvr 20529 df-nzr 20660 df-rlreg 20843 df-domn 20844 df-drng 20879 df-lmod 21033 df-lss 21103 df-lsp 21143 df-lvec 21274 df-lsatoms 39808 df-lshyp 39809 df-lcv 39851 df-lfl 39890 df-lkr 39918 df-ldual 39956 df-oposet 40008 df-ol 40010 df-oml 40011 df-covers 40098 df-ats 40099 df-atl 40130 df-cvlat 40154 df-hlat 40183 df-llines 40330 df-lplanes 40331 df-lvols 40332 df-lines 40333 df-psubsp 40335 df-pmap 40336 df-padd 40628 df-lhyp 40820 df-laut 40821 df-ldil 40936 df-ltrn 40937 df-trl 40991 df-tgrp 41575 df-tendo 41587 df-edring 41589 df-dveca 41835 df-disoa 41861 df-dvech 41911 df-dib 41971 df-dic 42005 df-dih 42061 df-doch 42180 df-djh 42227 df-lcdual 42419 df-mapd 42457 df-hvmap 42589 df-hdmap1 42625 df-hdmap 42626 df-hgmap 42716

This theorem is used by: hgmaprnlem3N 42730

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