hgmaprnlem1N - Mathbox for Norm Megill

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Theorem hgmaprnlem1N 41875

Description: Lemma for hgmaprnN 41880. (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 6826	. . . 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 3915	. . . . 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 41870	. . . 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 41570	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
25		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
26	4, 5, 8, 9, 10, 21, 22, 13, 14, 17	hgmapdcl 41869	. . . 4 ⊢ (𝜑 → (𝐺‘𝑘) ∈ 𝐴)
27	4, 5, 6, 10, 20, 12, 14, 16	hdmapcl 41809	. . . 4 ⊢ (𝜑 → (𝑆‘𝑡) ∈ 𝐷)
28		eldifsni 4741	. . . . . 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 41823	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝑡) = 𝑄 ↔ 𝑡 = 0 ))
32	31	necon3bid 2969	. . . . 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 21019	. . 3 ⊢ (𝜑 → ((𝑧 ∙ (𝑆‘𝑡)) = ((𝐺‘𝑘) ∙ (𝑆‘𝑡)) ↔ 𝑧 = (𝐺‘𝑘)))
35	19, 34	mpbid 232	. 2 ⊢ (𝜑 → 𝑧 = (𝐺‘𝑘))
36	4, 5, 8, 9, 13, 14	hgmapfnN 41867	. . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵)
37		fnfvelrn 7014	. . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝑘 ∈ 𝐵) → (𝐺‘𝑘) ∈ ran 𝐺)
38	36, 17, 37	syl2anc 584	. 2 ⊢ (𝜑 → (𝐺‘𝑘) ∈ ran 𝐺)
39	35, 38	eqeltrd 2828	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∖ cdif 3900 {csn 4577 ran crn 5620 Fn wfn 6477 ‘cfv 6482 (class class class)co 7349 Basecbs 17120 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 HLchlt 39329 LHypclh 39963 DVecHcdvh 41057 LCDualclcd 41565 HDMapchdma 41771 HGMapchg 41862

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-riotaBAD 38932

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-ot 4586 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-om 7800 df-1st 7924 df-2nd 7925 df-tpos 8159 df-undef 8206 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-n0 12385 df-z 12472 df-uz 12736 df-fz 13411 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-grp 18815 df-minusg 18816 df-sbg 18817 df-subg 19002 df-cntz 19196 df-oppg 19225 df-lsm 19515 df-cmn 19661 df-abl 19662 df-mgp 20026 df-rng 20038 df-ur 20067 df-ring 20120 df-oppr 20222 df-dvdsr 20242 df-unit 20243 df-invr 20273 df-dvr 20286 df-nzr 20398 df-rlreg 20579 df-domn 20580 df-drng 20616 df-lmod 20765 df-lss 20835 df-lsp 20875 df-lvec 21007 df-lsatoms 38955 df-lshyp 38956 df-lcv 38998 df-lfl 39037 df-lkr 39065 df-ldual 39103 df-oposet 39155 df-ol 39157 df-oml 39158 df-covers 39245 df-ats 39246 df-atl 39277 df-cvlat 39301 df-hlat 39330 df-llines 39477 df-lplanes 39478 df-lvols 39479 df-lines 39480 df-psubsp 39482 df-pmap 39483 df-padd 39775 df-lhyp 39967 df-laut 39968 df-ldil 40083 df-ltrn 40084 df-trl 40138 df-tgrp 40722 df-tendo 40734 df-edring 40736 df-dveca 40982 df-disoa 41008 df-dvech 41058 df-dib 41118 df-dic 41152 df-dih 41208 df-doch 41327 df-djh 41374 df-lcdual 41566 df-mapd 41604 df-hvmap 41736 df-hdmap1 41772 df-hdmap 41773 df-hgmap 41863

This theorem is referenced by: hgmaprnlem3N 41877

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