mapdpglem8 - Mathbox for Norm Megill

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Theorem mapdpglem8 39693

Description: Lemma for mapdpg 39720. 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 2738	. . . 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 39606	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LMod)
8		mapdpglem.m	. . . . 5 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
9		mapdpglem.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
10		eqid 2738	. . . . 5 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
11	4, 9, 6	dvhlmod 39124	. . . . . 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 20239	. . . . . 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 39670	. . . 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 39692	. . . 4 ⊢ (𝜑 → 𝑡 ∈ (𝑀‘(𝑁‘{𝑌})))
38	2, 3, 7, 17, 37	lspsnel5a 20258	. . 3 ⊢ (𝜑 → (𝐽‘{𝑡}) ⊆ (𝑀‘(𝑁‘{𝑌})))
39	1, 38	eqsstrd 3959	. 2 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌})))
40	13, 18	lmodvsubcl 20168	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
41	11, 19, 12, 40	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
42	13, 10, 14	lspsncl 20239	. . . 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 39652	. 2 ⊢ (𝜑 → ((𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ (𝑀‘(𝑁‘{𝑌})) ↔ (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌})))
45	39, 44	mpbid 231	1 ⊢ (𝜑 → (𝑁‘{(𝑋 − 𝑌)}) ⊆ (𝑁‘{𝑌}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ⊆ wss 3887 {csn 4561 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 Scalarcsca 16965 ·_𝑠 cvsca 16966 0_gc0g 17150 -_gcsg 18579 LSSumclsm 19239 LModclmod 20123 LSubSpclss 20193 LSpanclspn 20233 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 LCDualclcd 39600 mapdcmpd 39638

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-mre 17295 df-mrc 17296 df-acs 17298 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-oppg 18950 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lcv 37033 df-lfl 37072 df-lkr 37100 df-ldual 37138 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409 df-lcdual 39601 df-mapd 39639

This theorem is referenced by: mapdpglem9 39694

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