mapdpglem8 - Mathbox for Norm Megill

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Theorem mapdpglem8 41622

Description: Lemma for mapdpg 41649. Baer p. 45, line 4: "...so that (F(x-y))* <= (Fy)*. This would imply that F(x-y) <= F(y)..." (Contributed by NM, 20-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
mapdpglem8	⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (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 mapdpglem8**
Step	Hyp	Ref	Expression
1		mapdpglem4.jt	. . 3 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
2		eqid 2734	. . . 4 ⊢ (LSubSp‘𝐶) = (LSubSp‘𝐶)
3		mapdpglem2.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
4		mapdpglem.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
5		mapdpglem.c	. . . . 5 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
6		mapdpglem.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7	4, 5, 6	lcdlmod 41535	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		mapdpglem.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
9		mapdpglem.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		eqid 2734	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
11	4, 9, 6	dvhlmod 41053	. . . . . 6 ⊢ (𝜑 → 𝑈 ∈ LMod)
12		mapdpglem.y	. . . . . 6 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
13		mapdpglem.v	. . . . . . 7 ⊢ 𝑉 = (Base‘𝑈)
14		mapdpglem.n	. . . . . . 7 ⊢ 𝑁 = (LSpan‘𝑈)
15	13, 10, 14	lspsncl 20948	. . . . . 6 ⊢ ((𝑈 ∈ LMod ∧ 𝑌 ∈ 𝑉) → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
16	11, 12, 15	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑌}) ∈ (LSubSp‘𝑈))
17	4, 8, 9, 10, 5, 2, 6, 16	mapdcl2 41599	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑌})) ∈ (LSubSp‘𝐶))
18		mapdpglem.s	. . . . 5 ⊢ − = (-_g‘𝑈)
19		mapdpglem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20		mapdpglem1.p	. . . . 5 ⊢ ⊕ = (LSSum‘𝐶)
21		mapdpglem3.f	. . . . 5 ⊢ 𝐹 = (Base‘𝐶)
22		mapdpglem3.te	. . . . 5 ⊢ (𝜑 → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
23		mapdpglem3.a	. . . . 5 ⊢ 𝐴 = (Scalar‘𝑈)
24		mapdpglem3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
25		mapdpglem3.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝐶)
26		mapdpglem3.r	. . . . 5 ⊢ 𝑅 = (-_g‘𝐶)
27		mapdpglem3.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
28		mapdpglem3.e	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
29		mapdpglem4.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝑈)
30		mapdpglem.ne	. . . . 5 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
31		mapdpglem4.z	. . . . 5 ⊢ 0 = (0_g‘𝐴)
32		mapdpglem4.g4	. . . . 5 ⊢ (𝜑 → 𝑔 ∈ 𝐵)
33		mapdpglem4.z4	. . . . 5 ⊢ (𝜑 → 𝑧 ∈ (𝑀‘(𝑁‘{𝑌})))
34		mapdpglem4.t4	. . . . 5 ⊢ (𝜑 → 𝑡 = ((𝑔 · 𝐺)𝑅𝑧))
35		mapdpglem4.xn	. . . . 5 ⊢ (𝜑 → 𝑋 ≠ 𝑄)
36		mapdpglem4.g0	. . . . 5 ⊢ (𝜑 → 𝑔 = 0 )
37	4, 8, 9, 13, 18, 14, 5, 6, 19, 12, 20, 3, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 1, 31, 32, 33, 34, 35, 36	mapdpglem6 41621	. . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))
38	2, 3, 7, 17, 37	ellspsn5 20967	. . 3 ⊢ (𝜑 → (𝐽‘{𝑡}) ⊆ (𝑀‘(𝑁‘{𝑌})))
39	1, 38	eqsstrd 4000	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌})))
40	13, 18	lmodvsubcl 20878	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
41	11, 19, 12, 40	syl3anc 1372	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
42	13, 10, 14	lspsncl 20948	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑋 − 𝑌) ∈ 𝑉) → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
43	11, 41, 42	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ∈ (LSubSp‘𝑈))
44	4, 9, 10, 8, 6, 43, 16	mapdord 41581	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})))
45	39, 44	mpbid 232	1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ⊆ wss 3933 {csn 4608 ‘cfv 6542 (class class class)co 7414 Basecbs 17230 Scalarcsca 17280 ·_𝑠 cvsca 17281 0_gc0g 17460 -_gcsg 18927 LSSumclsm 19625 LModclmod 20831 LSubSpclss 20902 LSpanclspn 20942 HLchlt 39292 LHypclh 39927 DVecHcdvh 41021 LCDualclcd 41529 mapdcmpd 41567

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5261 ax-sep 5278 ax-nul 5288 ax-pow 5347 ax-pr 5414 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-riotaBAD 38895

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3364 df-reu 3365 df-rab 3421 df-v 3466 df-sbc 3773 df-csb 3882 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-pss 3953 df-nul 4316 df-if 4508 df-pw 4584 df-sn 4609 df-pr 4611 df-tp 4613 df-op 4615 df-uni 4890 df-int 4929 df-iun 4975 df-iin 4976 df-br 5126 df-opab 5188 df-mpt 5208 df-tr 5242 df-id 5560 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-pred 6303 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6495 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7680 df-om 7871 df-1st 7997 df-2nd 7998 df-tpos 8234 df-undef 8281 df-frecs 8289 df-wrecs 8320 df-recs 8394 df-rdg 8433 df-1o 8489 df-2o 8490 df-er 8728 df-map 8851 df-en 8969 df-dom 8970 df-sdom 8971 df-fin 8972 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11477 df-neg 11478 df-nn 12250 df-2 12312 df-3 12313 df-4 12314 df-5 12315 df-6 12316 df-n0 12511 df-z 12598 df-uz 12862 df-fz 13531 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17257 df-plusg 17290 df-mulr 17291 df-sca 17293 df-vsca 17294 df-0g 17462 df-mre 17605 df-mrc 17606 df-acs 17608 df-proset 18315 df-poset 18334 df-plt 18349 df-lub 18365 df-glb 18366 df-join 18367 df-meet 18368 df-p0 18444 df-p1 18445 df-lat 18451 df-clat 18518 df-mgm 18627 df-sgrp 18706 df-mnd 18722 df-submnd 18771 df-grp 18928 df-minusg 18929 df-sbg 18930 df-subg 19115 df-cntz 19309 df-oppg 19338 df-lsm 19627 df-cmn 19773 df-abl 19774 df-mgp 20111 df-rng 20123 df-ur 20152 df-ring 20205 df-oppr 20307 df-dvdsr 20330 df-unit 20331 df-invr 20361 df-dvr 20374 df-nzr 20486 df-rlreg 20667 df-domn 20668 df-drng 20704 df-lmod 20833 df-lss 20903 df-lsp 20943 df-lvec 21075 df-lsatoms 38918 df-lshyp 38919 df-lcv 38961 df-lfl 39000 df-lkr 39028 df-ldual 39066 df-oposet 39118 df-ol 39120 df-oml 39121 df-covers 39208 df-ats 39209 df-atl 39240 df-cvlat 39264 df-hlat 39293 df-llines 39441 df-lplanes 39442 df-lvols 39443 df-lines 39444 df-psubsp 39446 df-pmap 39447 df-padd 39739 df-lhyp 39931 df-laut 39932 df-ldil 40047 df-ltrn 40048 df-trl 40102 df-tgrp 40686 df-tendo 40698 df-edring 40700 df-dveca 40946 df-disoa 40972 df-dvech 41022 df-dib 41082 df-dic 41116 df-dih 41172 df-doch 41291 df-djh 41338 df-lcdual 41530 df-mapd 41568

This theorem is referenced by: mapdpglem9 41623

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