hgmaprnlem1N - Mathbox for Norm Megill

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Theorem hgmaprnlem1N 41223

Description: Lemma for hgmaprnN 41228. (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 6885	. . . 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 3952	. . . . 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 41218	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑘 · 𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)))
19	2, 3, 18	3eqtr3d 2772	. . 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 40918	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
25		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
26	4, 5, 8, 9, 10, 21, 22, 13, 14, 17	hgmapdcl 41217	. . . 4 ⊢ (𝜑 → (𝐺‘𝑘) ∈ 𝐴)
27	4, 5, 6, 10, 20, 12, 14, 16	hdmapcl 41157	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28		eldifsni 4785	. . . . . 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 41171	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑡) = 𝑄 ↔ 𝑡 = 0 ))
32	31	necon3bid 2977	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑡) ≠ 𝑄 ↔ 𝑡 ≠ 0 ))
33	29, 32	mpbird 257	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ≠ 𝑄)
34	20, 11, 21, 22, 23, 24, 25, 26, 27, 33	lvecvscan2 20952	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)) ↔ 𝑧 = (𝐺‘𝑘)))
35	19, 34	mpbid 231	. 2 ⊢ (𝜑 → 𝑧 = (𝐺‘𝑘))
36	4, 5, 8, 9, 13, 14	hgmapfnN 41215	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
37		fnfvelrn 7072	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵) → (𝐺‘𝑘) ∈ ran 𝐺)
38	36, 17, 37	syl2anc 583	. 2 ⊢ (𝜑 → (𝐺‘𝑘) ∈ ran 𝐺)
39	35, 38	eqeltrd 2825	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ≠ wne 2932 ∖ cdif 3937 {csn 4620 ran crn 5667 Fn wfn 6528 ‘cfv 6533 (class class class)co 7401 Basecbs 17142 Scalarcsca 17198 ·_𝑠 cvsca 17199 0_gc0g 17383 HLchlt 38676 LHypclh 39311 DVecHcdvh 40405 LCDualclcd 40913 HDMapchdma 41119 HGMapchg 41210

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 ax-pre-mulgt0 11182 ax-riotaBAD 38279

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-rmo 3368 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-tp 4625 df-op 4627 df-ot 4629 df-uni 4900 df-int 4941 df-iun 4989 df-iin 4990 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7357 df-ov 7404 df-oprab 7405 df-mpo 7406 df-of 7663 df-om 7849 df-1st 7968 df-2nd 7969 df-tpos 8206 df-undef 8253 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17078 df-sets 17095 df-slot 17113 df-ndx 17125 df-base 17143 df-ress 17172 df-plusg 17208 df-mulr 17209 df-sca 17211 df-vsca 17212 df-0g 17385 df-mre 17528 df-mrc 17529 df-acs 17531 df-proset 18249 df-poset 18267 df-plt 18284 df-lub 18300 df-glb 18301 df-join 18302 df-meet 18303 df-p0 18379 df-p1 18380 df-lat 18386 df-clat 18453 df-mgm 18562 df-sgrp 18641 df-mnd 18657 df-submnd 18703 df-grp 18855 df-minusg 18856 df-sbg 18857 df-subg 19039 df-cntz 19222 df-oppg 19251 df-lsm 19545 df-cmn 19691 df-abl 19692 df-mgp 20029 df-rng 20047 df-ur 20076 df-ring 20129 df-oppr 20225 df-dvdsr 20248 df-unit 20249 df-invr 20279 df-dvr 20292 df-drng 20578 df-lmod 20697 df-lss 20768 df-lsp 20808 df-lvec 20940 df-lsatoms 38302 df-lshyp 38303 df-lcv 38345 df-lfl 38384 df-lkr 38412 df-ldual 38450 df-oposet 38502 df-ol 38504 df-oml 38505 df-covers 38592 df-ats 38593 df-atl 38624 df-cvlat 38648 df-hlat 38677 df-llines 38825 df-lplanes 38826 df-lvols 38827 df-lines 38828 df-psubsp 38830 df-pmap 38831 df-padd 39123 df-lhyp 39315 df-laut 39316 df-ldil 39431 df-ltrn 39432 df-trl 39486 df-tgrp 40070 df-tendo 40082 df-edring 40084 df-dveca 40330 df-disoa 40356 df-dvech 40406 df-dib 40466 df-dic 40500 df-dih 40556 df-doch 40675 df-djh 40722 df-lcdual 40914 df-mapd 40952 df-hvmap 41084 df-hdmap1 41120 df-hdmap 41121 df-hgmap 41211

This theorem is referenced by: hgmaprnlem3N 41225

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