mapdpglem2 - Mathbox for Norm Megill

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Theorem mapdpglem2 36962

Description: Lemma for mapdpg 36995. Baer p. 45, lines 1 and 2: "we have (F(x-y))* = Gt where t necessarily belongs to (Fx)*+(Fy)*." (We scope $d 𝑡𝜑 locally to avoid clashes with later substitutions into 𝜑.) (Contributed by NM, 15-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)

**Assertion**
Ref	Expression
mapdpglem2	⊢ (𝜑 → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))

Distinct variable groups: 𝑡, − 𝑡,𝐶 𝑡,𝐽 𝑡,𝑀 𝑡,𝑁 𝑡,𝑋 𝑡,𝑌 𝜑,𝑡 Allowed substitution hints: ⊕ (𝑡) 𝑈(𝑡) 𝐻(𝑡) 𝐾(𝑡) 𝑉(𝑡) 𝑊(𝑡)

**Proof of Theorem mapdpglem2**
Step	Hyp	Ref	Expression
1		mapdpglem.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpglem.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdpglem.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdpglem.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		mapdpglem.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
6		mapdpglem.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
7		eqid 2622	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
8		mapdpglem2.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
9		mapdpglem.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
10	1, 3, 9	dvhlmod 36399	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		mapdpglem.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12		mapdpglem.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
13		mapdpglem.s	. . . . . 6 ⊢ − = (-_g‘𝑈)
14	4, 13	lmodvsubcl 18908	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝑋 − 𝑌) ∈ 𝑉)
15	10, 11, 12, 14	syl3anc 1326	. . . 4 ⊢ (𝜑 → (𝑋 − 𝑌) ∈ 𝑉)
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 15	mapdspex 36957	. . 3 ⊢ (𝜑 → ∃𝑡 ∈ (Base‘𝐶)(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
17	1, 6, 9	lcdlmod 36881	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ LMod)
18	7, 8	lspsnid 18993	. . . . . . . . 9 ⊢ ((𝐶 ∈ LMod ∧ 𝑡 ∈ (Base‘𝐶)) → 𝑡 ∈ (𝐽‘{𝑡}))
19	17, 18	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (Base‘𝐶)) → 𝑡 ∈ (𝐽‘{𝑡}))
20	19	adantrr 753	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐶) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))) → 𝑡 ∈ (𝐽‘{𝑡}))
21		simprr 796	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐶) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))) → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
22	20, 21	eleqtrrd 2704	. . . . . 6 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐶) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))) → 𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})))
23	22, 21	jca 554	. . . . 5 ⊢ ((𝜑 ∧ (𝑡 ∈ (Base‘𝐶) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))) → (𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})))
24	23	ex 450	. . . 4 ⊢ (𝜑 → ((𝑡 ∈ (Base‘𝐶) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) → (𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))))
25	24	reximdv2 3014	. . 3 ⊢ (𝜑 → (∃𝑡 ∈ (Base‘𝐶)(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}) → ∃𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})))
26	16, 25	mpd 15	. 2 ⊢ (𝜑 → ∃𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))
27		mapdpglem1.p	. . . . . 6 ⊢ ⊕ = (LSSum‘𝐶)
28	1, 2, 3, 4, 13, 5, 6, 9, 11, 12, 27	mapdpglem1 36961	. . . . 5 ⊢ (𝜑 → (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ⊆ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))))
29	28	sseld 3602	. . . 4 ⊢ (𝜑 → (𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) → 𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))))
30	29	anim1d 588	. . 3 ⊢ (𝜑 → ((𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})) → (𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌}))) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))))
31	30	reximdv2 3014	. 2 ⊢ (𝜑 → (∃𝑡 ∈ (𝑀‘(𝑁‘{(𝑋 − 𝑌)}))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}) → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡})))
32	26, 31	mpd 15	1 ⊢ (𝜑 → ∃𝑡 ∈ ((𝑀‘(𝑁‘{𝑋})) ⊕ (𝑀‘(𝑁‘{𝑌})))(𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{𝑡}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 {csn 4177 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 -_gcsg 17424 LSSumclsm 18049 LModclmod 18863 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 DVecHcdvh 36367 LCDualclcd 36875 mapdcmpd 36913

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914

This theorem is referenced by: mapdpglem24 36993

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