mapdpglem6 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem6		Structured version Visualization version GIF version

Theorem mapdpglem6 42337

Description: Lemma for mapdpg 42365. 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 42251	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
6		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
7		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		eqid 2769	. . . 4 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
9		eqid 2769	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
10	2, 7, 4	dvhlmod 41769	. . . . 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 21072	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
15	10, 11, 14	syl2anc 595	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
16	2, 6, 7, 8, 3, 9, 4, 15	mapdcl2 42315	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
17		mapdpglem4.g0	. . . . . 6 ⊢ (𝜑 → 𝑔 = 0 )
18	17	oveq1d 7423	. . . . 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 2769	. . . . . 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 42274	. . . . 5 ⊢ (𝜑 → ( 0 · 𝐺) = (0_g‘𝐶))
26	18, 25	eqtrd 2804	. . . 4 ⊢ (𝜑 → (𝑔 · 𝐺) = (0_g‘𝐶))
27	23, 9	lss0cl 21042	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶)) → (0_g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌})))
28	5, 16, 27	syl2anc 595	. . . 4 ⊢ (𝜑 → (0_g‘𝐶) ∈ (𝑀‘(𝑁‘{𝑌})))
29	26, 28	eqeltrd 2869	. . 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 21039	. . 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 851	. 2 ⊢ (𝜑 → ((𝑔 · 𝐺)𝑅𝑧) ∈ (𝑀‘(𝑁‘{𝑌})))
34	1, 33	eqeltrd 2869	1 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 400 = wceq 1567 ∈ wcel 2149 ≠ wne 2964 {csn 4591 ‘cfv 6533 (class class class)co 7408 Basecbs 17265 Scalarcsca 17309 ·_𝑠 cvsca 17310 0_gc0g 17488 -_gcsg 18998 LSSumclsm 19700 LModclmod 20955 LSubSpclss 21026 LSpanclspn 21066 HLchlt 40009 LHypclh 40643 DVecHcdvh 41737 LCDualclcd 42245 mapdcmpd 42283

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5239 ax-sep 5258 ax-nul 5268 ax-pow 5334 ax-pr 5402 ax-un 7730 ax-cnex 11152 ax-resscn 11153 ax-1cn 11154 ax-icn 11155 ax-addcl 11156 ax-addrcl 11157 ax-mulcl 11158 ax-mulrcl 11159 ax-mulcom 11160 ax-addass 11161 ax-mulass 11162 ax-distr 11163 ax-i2m1 11164 ax-1ne0 11165 ax-1rid 11166 ax-rnegex 11167 ax-rrecex 11168 ax-cnre 11169 ax-pre-lttri 11170 ax-pre-lttrn 11171 ax-pre-ltadd 11172 ax-pre-mulgt0 11173 ax-riotaBAD 39612

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7365 df-ov 7411 df-oprab 7412 df-mpo 7413 df-of 7672 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-undef 8265 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-2o 8450 df-er 8690 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-pnf 11241 df-mnf 11242 df-xr 11243 df-ltxr 11244 df-le 11245 df-sub 11439 df-neg 11440 df-nn 12230 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-n0 12501 df-z 12588 df-uz 12859 df-fz 13532 df-struct 17203 df-sets 17220 df-slot 17238 df-ndx 17250 df-base 17266 df-ress 17287 df-plusg 17319 df-mulr 17320 df-sca 17322 df-vsca 17323 df-0g 17490 df-mre 17634 df-mrc 17635 df-acs 17637 df-proset 18346 df-poset 18365 df-plt 18380 df-lub 18396 df-glb 18397 df-join 18398 df-meet 18399 df-p0 18475 df-p1 18476 df-lat 18484 df-clat 18551 df-mgm 18694 df-sgrp 18773 df-mnd 18789 df-submnd 18838 df-grp 18999 df-minusg 19000 df-sbg 19001 df-subg 19185 df-cntz 19383 df-oppg 19412 df-lsm 19702 df-cmn 19848 df-abl 19849 df-mgp 20213 df-rng 20227 df-ur 20260 df-ring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-dvr 20479 df-nzr 20592 df-rlreg 20775 df-domn 20776 df-drng 20811 df-lmod 20957 df-lss 21027 df-lsp 21067 df-lvec 21198 df-lsatoms 39635 df-lshyp 39636 df-lcv 39678 df-lfl 39717 df-lkr 39745 df-ldual 39783 df-oposet 39835 df-ol 39837 df-oml 39838 df-covers 39925 df-ats 39926 df-atl 39957 df-cvlat 39981 df-hlat 40010 df-llines 40157 df-lplanes 40158 df-lvols 40159 df-lines 40160 df-psubsp 40162 df-pmap 40163 df-padd 40455 df-lhyp 40647 df-laut 40648 df-ldil 40763 df-ltrn 40764 df-trl 40818 df-tgrp 41402 df-tendo 41414 df-edring 41416 df-dveca 41662 df-disoa 41688 df-dvech 41738 df-dib 41798 df-dic 41832 df-dih 41888 df-doch 42007 df-djh 42054 df-lcdual 42246 df-mapd 42284

This theorem is referenced by: mapdpglem8 42338

W3C validator