hgmaprnlem3N - Mathbox for Norm Megill

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Theorem hgmaprnlem3N 40757

Description: Lemma for hgmaprnN 40760. Eliminate 𝑘. (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.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
hgmaprnlem1.n	⊢ 𝑁 = (LSpan‘𝑈)
hgmaprnlem1.l	⊢ 𝐿 = (LSpan‘𝐶)

**Assertion**
Ref	Expression
hgmaprnlem3N	⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

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

Proof of Theorem hgmaprnlem3N Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hgmaprnlem1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hgmaprnlem1.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hgmaprnlem1.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
4		hgmaprnlem1.r	. . . 4 ⊢ 𝑅 = (Scalar‘𝑈)
5		hgmaprnlem1.b	. . . 4 ⊢ 𝐵 = (Base‘𝑅)
6		hgmaprnlem1.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑈)
7		hgmaprnlem1.o	. . . 4 ⊢ 0 = (0_g‘𝑈)
8		hgmaprnlem1.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		hgmaprnlem1.d	. . . 4 ⊢ 𝐷 = (Base‘𝐶)
10		hgmaprnlem1.p	. . . 4 ⊢ 𝑃 = (Scalar‘𝐶)
11		hgmaprnlem1.a	. . . 4 ⊢ 𝐴 = (Base‘𝑃)
12		hgmaprnlem1.e	. . . 4 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
13		hgmaprnlem1.q	. . . 4 ⊢ 𝑄 = (0_g‘𝐶)
14		hgmaprnlem1.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
15		hgmaprnlem1.g	. . . 4 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
16		hgmaprnlem1.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		hgmaprnlem1.z	. . . 4 ⊢ (𝜑 → 𝑧 ∈ 𝐴)
18		hgmaprnlem1.t2	. . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑉 ∖ { 0 }))
19		hgmaprnlem1.s2	. . . 4 ⊢ (𝜑 → 𝑠 ∈ 𝑉)
20		hgmaprnlem1.sz	. . . 4 ⊢ (𝜑 → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
21		hgmaprnlem1.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
22		hgmaprnlem1.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
23		hgmaprnlem1.l	. . . 4 ⊢ 𝐿 = (LSpan‘𝐶)
24	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23	hgmaprnlem2N 40756	. . 3 ⊢ (𝜑 → (𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}))
25	1, 2, 16	dvhlmod 39969	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
26	18	eldifad 3959	. . . 4 ⊢ (𝜑 → 𝑡 ∈ 𝑉)
27	3, 4, 5, 6, 22, 25, 19, 26	lspsnss2 20608	. . 3 ⊢ (𝜑 → ((𝑁‘{𝑠}) ⊆ (𝑁‘{𝑡}) ↔ ∃𝑘 ∈ 𝐵 𝑠 = (𝑘 · 𝑡)))
28	24, 27	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑘 ∈ 𝐵 𝑠 = (𝑘 · 𝑡))
29	16	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
30	17	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑧 ∈ 𝐴)
31	18	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑡 ∈ (𝑉 ∖ { 0 }))
32	19	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑠 ∈ 𝑉)
33	20	3ad2ant1 1133	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → (𝑆‘𝑠) = (𝑧 ∙ (𝑆‘𝑡)))
34		simp2 1137	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑘 ∈ 𝐵)
35		simp3 1138	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑠 = (𝑘 · 𝑡))
36	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 29, 30, 31, 32, 33, 34, 35	hgmaprnlem1N 40755	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐵 ∧ 𝑠 = (𝑘 · 𝑡)) → 𝑧 ∈ ran 𝐺)
37	36	rexlimdv3a 3159	. 2 ⊢ (𝜑 → (∃𝑘 ∈ 𝐵 𝑠 = (𝑘 · 𝑡) → 𝑧 ∈ ran 𝐺))
38	28, 37	mpd 15	1 ⊢ (𝜑 → 𝑧 ∈ ran 𝐺)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∃wrex 3070 ∖ cdif 3944 ⊆ wss 3947 {csn 4627 ran crn 5676 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 LSpanclspn 20574 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 LCDualclcd 40445 mapdcmpd 40483 HDMapchdma 40651 HGMapchg 40742

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-ot 4636 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 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 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-mre 17526 df-mrc 17527 df-acs 17529 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-oppg 19204 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lsatoms 37834 df-lshyp 37835 df-lcv 37877 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tgrp 39602 df-tendo 39614 df-edring 39616 df-dveca 39862 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207 df-djh 40254 df-lcdual 40446 df-mapd 40484 df-hvmap 40616 df-hdmap1 40652 df-hdmap 40653 df-hgmap 40743

This theorem is referenced by: hgmaprnlem4N 40758

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