mapdpglem27 - Mathbox for Norm Megill

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Theorem mapdpglem27 39640

Description: Lemma for mapdpg 39647. Baer p. 45 line 16: "v(x'-y'') = x'-y'" (with equality swapped). (Contributed by NM, 22-Mar-2015.)

**Hypotheses**
Ref	Expression
mapdpg.h	⊢ 𝐻 = (LHyp‘𝐾)
mapdpg.m	⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
mapdpg.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
mapdpg.v	⊢ 𝑉 = (Base‘𝑈)
mapdpg.s	⊢ − = (-_g‘𝑈)
mapdpg.z	⊢ 0 = (0_g‘𝑈)
mapdpg.n	⊢ 𝑁 = (LSpan‘𝑈)
mapdpg.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
mapdpg.f	⊢ 𝐹 = (Base‘𝐶)
mapdpg.r	⊢ 𝑅 = (-_g‘𝐶)
mapdpg.j	⊢ 𝐽 = (LSpan‘𝐶)
mapdpg.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
mapdpg.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
mapdpg.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
mapdpg.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
mapdpg.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
mapdpg.e	⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
mapdpgem25.h1	⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
mapdpgem25.i1	⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
mapdpglem26.a	⊢ 𝐴 = (Scalar‘𝑈)
mapdpglem26.b	⊢ 𝐵 = (Base‘𝐴)
mapdpglem26.t	⊢ · = ( ·_𝑠 ‘𝐶)
mapdpglem26.o	⊢ 𝑂 = (0_g‘𝐴)

**Assertion**
Ref	Expression
mapdpglem27	⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

Distinct variable groups: ℎ,𝑖,𝑣 𝑣,𝐵 𝑣,𝐶 𝑣,𝑂 𝑣, · 𝑣,𝐺 𝑣,𝑅 𝜑,𝑣 Allowed substitution hints: 𝜑(ℎ,𝑖) 𝐴(𝑣,ℎ,𝑖) 𝐵(ℎ,𝑖) 𝐶(ℎ,𝑖) 𝑅(ℎ,𝑖) · (ℎ,𝑖) 𝑈(𝑣,ℎ,𝑖) 𝐹(𝑣,ℎ,𝑖) 𝐺(ℎ,𝑖) 𝐻(𝑣,ℎ,𝑖) 𝐽(𝑣,ℎ,𝑖) 𝐾(𝑣,ℎ,𝑖) 𝑀(𝑣,ℎ,𝑖) − (𝑣,ℎ,𝑖) 𝑁(𝑣,ℎ,𝑖) 𝑂(ℎ,𝑖) 𝑉(𝑣,ℎ,𝑖) 𝑊(𝑣,ℎ,𝑖) 𝑋(𝑣,ℎ,𝑖) 𝑌(𝑣,ℎ,𝑖) 0 (𝑣,ℎ,𝑖)

**Proof of Theorem mapdpglem27**
Step	Hyp	Ref	Expression
1		mapdpg.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		mapdpg.m	. . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊)
3		mapdpg.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		mapdpg.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
5		mapdpg.s	. . . 4 ⊢ − = (-_g‘𝑈)
6		mapdpg.z	. . . 4 ⊢ 0 = (0_g‘𝑈)
7		mapdpg.n	. . . 4 ⊢ 𝑁 = (LSpan‘𝑈)
8		mapdpg.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
9		mapdpg.f	. . . 4 ⊢ 𝐹 = (Base‘𝐶)
10		mapdpg.r	. . . 4 ⊢ 𝑅 = (-_g‘𝐶)
11		mapdpg.j	. . . 4 ⊢ 𝐽 = (LSpan‘𝐶)
12		mapdpg.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13		mapdpg.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
14		mapdpg.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
15		mapdpg.g	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
16		mapdpg.ne	. . . 4 ⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
17		mapdpg.e	. . . 4 ⊢ (𝜑 → (𝑀‘(𝑁‘{𝑋})) = (𝐽‘{𝐺}))
18		mapdpgem25.h1	. . . 4 ⊢ (𝜑 → (ℎ ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{ℎ}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅ℎ)}))))
19		mapdpgem25.i1	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
20	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19	mapdpglem25 39638	. . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})))
21	20	simprd 495	. 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
22		eqid 2738	. . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
23		eqid 2738	. . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
24		eqid 2738	. . . 4 ⊢ (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘𝐶))
25		mapdpglem26.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
26	1, 8, 12	lcdlvec 39532	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
27	1, 8, 12	lcdlmod 39533	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	18	simpld 494	. . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹)
29	9, 10	lmodvsubcl 20083	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹)
30	27, 15, 28, 29	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹)
31	19	simpld 494	. . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
32	9, 10	lmodvsubcl 20083	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹)
33	27, 15, 31, 32	syl3anc 1369	. . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹)
34	9, 22, 23, 24, 25, 11, 26, 30, 33	lspsneq 20299	. . 3 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
35		mapdpglem26.a	. . . . . 6 ⊢ 𝐴 = (Scalar‘𝑈)
36		mapdpglem26.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐴)
37	1, 3, 35, 36, 8, 22, 23, 12	lcdsbase 39541	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
38		mapdpglem26.o	. . . . . . 7 ⊢ 𝑂 = (0_g‘𝐴)
39	1, 3, 35, 38, 8, 22, 24, 12	lcd0 39549	. . . . . 6 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = 𝑂)
40	39	sneqd 4570	. . . . 5 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐶))} = {𝑂})
41	37, 40	difeq12d 4054	. . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂}))
42	41	rexeqdv 3340	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
43	34, 42	bitrd 278	. 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
44	21, 43	mpbid 231	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∃wrex 3064 ∖ cdif 3880 {csn 4558 ‘cfv 6418 (class class class)co 7255 Basecbs 16840 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 -_gcsg 18494 LModclmod 20038 LSpanclspn 20148 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 LCDualclcd 39527 mapdcmpd 39565

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-of 7511 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-mre 17212 df-mrc 17213 df-acs 17215 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-oppg 18865 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lcv 36960 df-lfl 36999 df-lkr 37027 df-ldual 37065 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336 df-lcdual 39528

This theorem is referenced by: mapdpglem32 39646

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