hgmaprnlem4N - Mathbox for Norm Megill

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Theorem hgmaprnlem4N 37920

Description: Lemma for hgmaprnN 37922. Eliminate 𝑠. (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 }))

**Assertion**
Ref	Expression
hgmaprnlem4N	⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Distinct variable group: 𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡) 𝐴(𝑧,𝑡) 𝐵(𝑧,𝑡) 𝐶(𝑧,𝑡) 𝐷(𝑧,𝑡) 𝑃(𝑧,𝑡) 𝑄(𝑧,𝑡) 𝑅(𝑧,𝑡) 𝑆(𝑧,𝑡) ∙ (𝑧,𝑡) · (𝑧,𝑡) 𝑈(𝑧,𝑡) 𝐺(𝑧,𝑡) 𝐻(𝑧,𝑡) 𝐾(𝑧,𝑡) 𝑉(𝑧,𝑡) 𝑊(𝑧,𝑡) 0 (𝑧,𝑡)

Proof of Theorem hgmaprnlem4N Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgmaprnlem1.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
2		hgmaprnlem1.c	. . . . . 6 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hgmaprnlem1.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 37613	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hgmaprnlem1.z	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
6		hgmaprnlem1.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		hgmaprnlem1.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
8		hgmaprnlem1.d	. . . . . 6 ⊢ 𝐷 = (Base‘𝐶)
9		hgmaprnlem1.s	. . . . . 6 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hgmaprnlem1.t2	. . . . . . 7 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3781	. . . . . 6 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 37851	. . . . 5 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
13		hgmaprnlem1.p	. . . . . 6 ⊢ 𝑃 = (Scalar‘𝐶)
14		hgmaprnlem1.e	. . . . . 6 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
15		hgmaprnlem1.a	. . . . . 6 ⊢ 𝐴 = (Base‘𝑃)
16	8, 13, 14, 15	lmodvscl 19198	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝑧 ∈ 𝐴 ∧ (𝑆‘𝑡) ∈ 𝐷) → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
17	4, 5, 12, 16	syl3anc 1491	. . . 4 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ 𝐷)
18	1, 2, 8, 9, 3	hdmaprnN 37885	. . . 4 ⊢ (𝜑 → ran 𝑆 = 𝐷)
19	17, 18	eleqtrrd 2881	. . 3 ⊢ (𝜑 → (𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆)
20	1, 6, 7, 9, 3	hdmapfnN 37850	. . . 4 ⊢ (𝜑 → 𝑆 Fn 𝑉)
21		fvelrnb 6468	. . . 4 ⊢ (𝑆 Fn 𝑉 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
22	20, 21	syl 17	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) ∈ ran 𝑆 ↔ ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))))
23	19, 22	mpbid 224	. 2 ⊢ (𝜑 → ∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
24		hgmaprnlem1.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
25		hgmaprnlem1.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
26		hgmaprnlem1.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑈)
27		hgmaprnlem1.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
28		hgmaprnlem1.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
29		hgmaprnlem1.g	. . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
30	3	3ad2ant1 1164	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
31	5	3ad2ant1 1164	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ 𝐴)
32	10	3ad2ant1 1164	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑡 ∈ (𝑉 ∖ { 0 }))
33		simp2 1168	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑠 ∈ 𝑉)
34		simp3 1169	. . . 4 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
35		eqid 2799	. . . 4 ⊢ ((mapd‘𝐾)‘𝑊) = ((mapd‘𝐾)‘𝑊)
36		eqid 2799	. . . 4 ⊢ (LSpan‘𝑈) = (LSpan‘𝑈)
37		eqid 2799	. . . 4 ⊢ (LSpan‘𝐶) = (LSpan‘𝐶)
38	1, 6, 7, 24, 25, 26, 27, 2, 8, 13, 15, 14, 28, 9, 29, 30, 31, 32, 33, 34, 35, 36, 37	hgmaprnlem3N 37919	. . 3 ⊢ ((𝜑 ∧ 𝑠 ∈ 𝑉 ∧ (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡))) → 𝑧 ∈ ran 𝐺)
39	38	rexlimdv3a 3214	. 2 ⊢ (𝜑 → (∃𝑠 ∈ 𝑉 (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)) → 𝑧 ∈ ran 𝐺))
40	23, 39	mpd 15	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 385 ∧ w3a 1108 = wceq 1653 ∈ wcel 2157 ∃wrex 3090 ∖ cdif 3766 {csn 4368 ran crn 5313 Fn wfn 6096 ‘cfv 6101 (class class class)co 6878 Basecbs 16184 Scalarcsca 16270 ·_𝑠 cvsca 16271 0_gc0g 16415 LModclmod 19181 LSpanclspn 19292 HLchlt 35371 LHypclh 36005 DVecHcdvh 37099 LCDualclcd 37607 mapdcmpd 37645 HDMapchdma 37813 HGMapchg 37904

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2377 ax-ext 2777 ax-rep 4964 ax-sep 4975 ax-nul 4983 ax-pow 5035 ax-pr 5097 ax-un 7183 ax-cnex 10280 ax-resscn 10281 ax-1cn 10282 ax-icn 10283 ax-addcl 10284 ax-addrcl 10285 ax-mulcl 10286 ax-mulrcl 10287 ax-mulcom 10288 ax-addass 10289 ax-mulass 10290 ax-distr 10291 ax-i2m1 10292 ax-1ne0 10293 ax-1rid 10294 ax-rnegex 10295 ax-rrecex 10296 ax-cnre 10297 ax-pre-lttri 10298 ax-pre-lttrn 10299 ax-pre-ltadd 10300 ax-pre-mulgt0 10301 ax-riotaBAD 34974

This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-fal 1667 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2591 df-eu 2609 df-clab 2786 df-cleq 2792 df-clel 2795 df-nfc 2930 df-ne 2972 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3387 df-sbc 3634 df-csb 3729 df-dif 3772 df-un 3774 df-in 3776 df-ss 3783 df-pss 3785 df-nul 4116 df-if 4278 df-pw 4351 df-sn 4369 df-pr 4371 df-tp 4373 df-op 4375 df-ot 4377 df-uni 4629 df-int 4668 df-iun 4712 df-iin 4713 df-br 4844 df-opab 4906 df-mpt 4923 df-tr 4946 df-id 5220 df-eprel 5225 df-po 5233 df-so 5234 df-fr 5271 df-we 5273 df-xp 5318 df-rel 5319 df-cnv 5320 df-co 5321 df-dm 5322 df-rn 5323 df-res 5324 df-ima 5325 df-pred 5898 df-ord 5944 df-on 5945 df-lim 5946 df-suc 5947 df-iota 6064 df-fun 6103 df-fn 6104 df-f 6105 df-f1 6106 df-fo 6107 df-f1o 6108 df-fv 6109 df-riota 6839 df-ov 6881 df-oprab 6882 df-mpt2 6883 df-of 7131 df-om 7300 df-1st 7401 df-2nd 7402 df-tpos 7590 df-undef 7637 df-wrecs 7645 df-recs 7707 df-rdg 7745 df-1o 7799 df-oadd 7803 df-er 7982 df-map 8097 df-en 8196 df-dom 8197 df-sdom 8198 df-fin 8199 df-pnf 10365 df-mnf 10366 df-xr 10367 df-ltxr 10368 df-le 10369 df-sub 10558 df-neg 10559 df-nn 11313 df-2 11376 df-3 11377 df-4 11378 df-5 11379 df-6 11380 df-n0 11581 df-z 11667 df-uz 11931 df-fz 12581 df-struct 16186 df-ndx 16187 df-slot 16188 df-base 16190 df-sets 16191 df-ress 16192 df-plusg 16280 df-mulr 16281 df-sca 16283 df-vsca 16284 df-0g 16417 df-mre 16561 df-mrc 16562 df-acs 16564 df-proset 17243 df-poset 17261 df-plt 17273 df-lub 17289 df-glb 17290 df-join 17291 df-meet 17292 df-p0 17354 df-p1 17355 df-lat 17361 df-clat 17423 df-mgm 17557 df-sgrp 17599 df-mnd 17610 df-submnd 17651 df-grp 17741 df-minusg 17742 df-sbg 17743 df-subg 17904 df-cntz 18062 df-oppg 18088 df-lsm 18364 df-cmn 18510 df-abl 18511 df-mgp 18806 df-ur 18818 df-ring 18865 df-oppr 18939 df-dvdsr 18957 df-unit 18958 df-invr 18988 df-dvr 18999 df-drng 19067 df-lmod 19183 df-lss 19251 df-lsp 19293 df-lvec 19424 df-lsatoms 34997 df-lshyp 34998 df-lcv 35040 df-lfl 35079 df-lkr 35107 df-ldual 35145 df-oposet 35197 df-ol 35199 df-oml 35200 df-covers 35287 df-ats 35288 df-atl 35319 df-cvlat 35343 df-hlat 35372 df-llines 35519 df-lplanes 35520 df-lvols 35521 df-lines 35522 df-psubsp 35524 df-pmap 35525 df-padd 35817 df-lhyp 36009 df-laut 36010 df-ldil 36125 df-ltrn 36126 df-trl 36180 df-tgrp 36764 df-tendo 36776 df-edring 36778 df-dveca 37024 df-disoa 37050 df-dvech 37100 df-dib 37160 df-dic 37194 df-dih 37250 df-doch 37369 df-djh 37416 df-lcdual 37608 df-mapd 37646 df-hvmap 37778 df-hdmap1 37814 df-hdmap 37815 df-hgmap 37905

This theorem is referenced by: hgmaprnlem5N 37921

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