mapdpglem6 - Mathbox for Norm Megill

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Theorem mapdpglem6 41723

Description: Lemma for mapdpg 41751. Baer p. 45, line 4: "If g were 0, then t would be in (Fy)*..." (Contributed by NM, 18-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpglem.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpglem.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpglem.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpglem.v	⊢ 𝑉 = (Base‘𝑈)
mapdpglem.s	⊢ − = (-_g‘𝑈)
mapdpglem.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpglem.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpglem.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpglem.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
mapdpglem.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
mapdpglem1.p	⊢ ⊕ = (LSSum‘𝐶)
mapdpglem2.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpglem3.f	⊢ 𝐹 = (Base‘𝐶)
mapdpglem3.te	⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
mapdpglem3.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem3.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem3.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem3.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpglem3.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpglem3.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpglem4.q	⊢ 𝑄 = (0_g‘𝑈)
mapdpglem.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpglem4.jt	⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
mapdpglem4.z	⊢ 0 = (0_g‘𝐴)
mapdpglem4.g4	⊢ (𝜑 → 𝑔 ∈ 𝐵)
mapdpglem4.z4	⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
mapdpglem4.t4	⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
mapdpglem4.xn	⊢ (𝜑 → 𝑋 ≠ 𝑄)
mapdpglem4.g0	⊢ (𝜑 → 𝑔 = 0 )

**Assertion**
Ref	Expression
mapdpglem6	⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝐵,𝑔 𝑧,𝑔,𝐶 𝑔,𝐹 𝑔,𝐺,𝑧 𝑔,𝐽,𝑧 𝑔,𝑀,𝑧 𝑔,𝑁,𝑧 𝑅,𝑔,𝑧 · ,𝑔,𝑧 𝑔,𝑌,𝑧,𝑡 Allowed substitution hints: 𝜑(𝑧,𝑡,𝑔) 𝐴(𝑧,𝑡,𝑔) 𝐵(𝑧,𝑡) ⊕ (𝑧,𝑡,𝑔) 𝑄(𝑧,𝑡,𝑔) 𝑅(𝑡) · (𝑡) 𝑈(𝑧,𝑡,𝑔) 𝐹(𝑧,𝑡) 𝐺(𝑡) 𝐻(𝑧,𝑡,𝑔) 𝐾(𝑧,𝑡,𝑔) − (𝑧,𝑔) 𝑉(𝑧,𝑡,𝑔) 𝑊(𝑧,𝑡,𝑔) 𝑋(𝑧,𝑔) 0 (𝑧,𝑡,𝑔)

**Proof of Theorem mapdpglem6**
Step	Hyp	Ref	Expression
1		mapdpglem4.t4	. 2 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
2		mapdpglem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
3		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
4		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
5	2, 3, 4	lcdlmod 41637	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
6		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
7		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		eqid 2731	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9		eqid 2731	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
10	2, 7, 4	dvhlmod 41155	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		mapdpglem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
12		mapdpglem.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
13		mapdpglem.n	. . . . . 6 ⊢ 𝑁 = (LSpan‘𝑈)
14	12, 8, 13	lspsncl 20911	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
15	10, 11, 14	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
16	2, 6, 7, 8, 3, 9, 4, 15	mapdcl2 41701	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
17		mapdpglem4.g0	. . . . . 6 ⊢ (𝜑 → 𝑔 = 0 )
18	17	oveq1d 7361	. . . . 5 ⊢ (𝜑 → (𝑔 · 𝐺) = ( 0 · 𝐺))
19		mapdpglem3.a	. . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈)
20		mapdpglem4.z	. . . . . 6 ⊢ 0 = (0_g‘𝐴)
21		mapdpglem3.f	. . . . . 6 ⊢ 𝐹 = (Base‘𝐶)
22		mapdpglem3.t	. . . . . 6 ⊢ · = ( ·_𝑠 ‘𝐶)
23		eqid 2731	. . . . . 6 ⊢ (0_g‘𝐶) = (0_g‘𝐶)
24		mapdpglem3.g	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
25	2, 7, 19, 20, 3, 21, 22, 23, 4, 24	lcd0vs 41660	. . . . 5 ⊢ (𝜑 → ( 0 · 𝐺) = (0_g‘𝐶))
26	18, 25	eqtrd 2766	. . . 4 ⊢ (𝜑 → (𝑔 · 𝐺) = (0_g‘𝐶))
27	23, 9	lss0cl 20881	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) → (0_g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌})))
28	5, 16, 27	syl2anc 584	. . . 4 ⊢ (𝜑 → (0_g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌})))
29	26, 28	eqeltrd 2831	. . 3 ⊢ (𝜑 → (𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌})))
30		mapdpglem4.z4	. . 3 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
31		mapdpglem3.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
32	31, 9	lssvsubcl 20878	. . 3 ⊢ (((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) ∧ ((𝑔 · 𝐺) ∈ (𝑀‘(𝑁‘{𝑌})) ∧ 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))) → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌})))
33	5, 16, 29, 30, 32	syl22anc 838	. 2 ⊢ (𝜑 → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌})))
34	1, 33	eqeltrd 2831	1 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ≠ wne 2928 {csn 4576 ‘cfv 6481 (class class class)co 7346 Basecbs 17120 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 -_gcsg 18848 LSSumclsm 19547 LModclmod 20794 LSubSpclss 20865 LSpanclspn 20905 HLchlt 39395 LHypclh 40029 DVecHcdvh 41123 LCDualclcd 41631 mapdcmpd 41669

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 38998

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 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 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19230 df-oppg 19259 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-nzr 20429 df-rlreg 20610 df-domn 20611 df-drng 20647 df-lmod 20796 df-lss 20866 df-lsp 20906 df-lvec 21038 df-lsatoms 39021 df-lshyp 39022 df-lcv 39064 df-lfl 39103 df-lkr 39131 df-ldual 39169 df-oposet 39221 df-ol 39223 df-oml 39224 df-covers 39311 df-ats 39312 df-atl 39343 df-cvlat 39367 df-hlat 39396 df-llines 39543 df-lplanes 39544 df-lvols 39545 df-lines 39546 df-psubsp 39548 df-pmap 39549 df-padd 39841 df-lhyp 40033 df-laut 40034 df-ldil 40149 df-ltrn 40150 df-trl 40204 df-tgrp 40788 df-tendo 40800 df-edring 40802 df-dveca 41048 df-disoa 41074 df-dvech 41124 df-dib 41184 df-dic 41218 df-dih 41274 df-doch 41393 df-djh 41440 df-lcdual 41632 df-mapd 41670

This theorem is referenced by: mapdpglem8 41724

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