mapdpglem27 - Mathbox for Norm Megill

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Theorem mapdpglem27 37459

Description: Lemma for mapdpg 37466. 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 37457	. . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})))
21	20	simprd 482	. 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
22		eqid 2748	. . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
23		eqid 2748	. . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
24		eqid 2748	. . . 4 ⊢ (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘𝐶))
25		mapdpglem26.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
26	1, 8, 12	lcdlvec 37351	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
27	1, 8, 12	lcdlmod 37352	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	18	simpld 477	. . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹)
29	9, 10	lmodvsubcl 19081	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹)
30	27, 15, 28, 29	syl3anc 1463	. . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹)
31	19	simpld 477	. . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
32	9, 10	lmodvsubcl 19081	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹)
33	27, 15, 31, 32	syl3anc 1463	. . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹)
34	9, 22, 23, 24, 25, 11, 26, 30, 33	lspsneq 19295	. . 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 37360	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
38		mapdpglem26.o	. . . . . . 7 ⊢ 𝑂 = (0_g‘𝐴)
39	1, 3, 35, 38, 8, 22, 24, 12	lcd0 37368	. . . . . 6 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = 𝑂)
40	39	sneqd 4321	. . . . 5 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐶))} = {𝑂})
41	37, 40	difeq12d 3860	. . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂}))
42	41	rexeqdv 3272	. . 3 ⊢ (𝜑 → (∃𝑣 ∈ ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
43	34, 42	bitrd 268	. 2 ⊢ (𝜑 → ((𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}) ↔ ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖))))
44	21, 43	mpbid 222	1 ⊢ (𝜑 → ∃𝑣 ∈ (𝐵 ∖ {𝑂})(𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 ∃wrex 3039 ∖ cdif 3700 {csn 4309 ‘cfv 6037 (class class class)co 6801 Basecbs 16030 Scalarcsca 16117 ·_𝑠 cvsca 16118 0_gc0g 16273 -_gcsg 17596 LModclmod 19036 LSpanclspn 19144 HLchlt 35109 LHypclh 35742 DVecHcdvh 36838 LCDualclcd 37346 mapdcmpd 37384

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-rep 4911 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-1cn 10157 ax-icn 10158 ax-addcl 10159 ax-addrcl 10160 ax-mulcl 10161 ax-mulrcl 10162 ax-mulcom 10163 ax-addass 10164 ax-mulass 10165 ax-distr 10166 ax-i2m1 10167 ax-1ne0 10168 ax-1rid 10169 ax-rnegex 10170 ax-rrecex 10171 ax-cnre 10172 ax-pre-lttri 10173 ax-pre-lttrn 10174 ax-pre-ltadd 10175 ax-pre-mulgt0 10176 ax-riotaBAD 34711

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-fal 1626 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-reu 3045 df-rmo 3046 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-pss 3719 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-tp 4314 df-op 4316 df-uni 4577 df-int 4616 df-iun 4662 df-iin 4663 df-br 4793 df-opab 4853 df-mpt 4870 df-tr 4893 df-id 5162 df-eprel 5167 df-po 5175 df-so 5176 df-fr 5213 df-we 5215 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-pred 5829 df-ord 5875 df-on 5876 df-lim 5877 df-suc 5878 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-riota 6762 df-ov 6804 df-oprab 6805 df-mpt2 6806 df-of 7050 df-om 7219 df-1st 7321 df-2nd 7322 df-tpos 7509 df-undef 7556 df-wrecs 7564 df-recs 7625 df-rdg 7663 df-1o 7717 df-oadd 7721 df-er 7899 df-map 8013 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243 df-sub 10431 df-neg 10432 df-nn 11184 df-2 11242 df-3 11243 df-4 11244 df-5 11245 df-6 11246 df-n0 11456 df-z 11541 df-uz 11851 df-fz 12491 df-struct 16032 df-ndx 16033 df-slot 16034 df-base 16036 df-sets 16037 df-ress 16038 df-plusg 16127 df-mulr 16128 df-sca 16130 df-vsca 16131 df-0g 16275 df-mre 16419 df-mrc 16420 df-acs 16422 df-preset 17100 df-poset 17118 df-plt 17130 df-lub 17146 df-glb 17147 df-join 17148 df-meet 17149 df-p0 17211 df-p1 17212 df-lat 17218 df-clat 17280 df-mgm 17414 df-sgrp 17456 df-mnd 17467 df-submnd 17508 df-grp 17597 df-minusg 17598 df-sbg 17599 df-subg 17763 df-cntz 17921 df-oppg 17947 df-lsm 18222 df-cmn 18366 df-abl 18367 df-mgp 18661 df-ur 18673 df-ring 18720 df-oppr 18794 df-dvdsr 18812 df-unit 18813 df-invr 18843 df-dvr 18854 df-drng 18922 df-lmod 19038 df-lss 19106 df-lsp 19145 df-lvec 19276 df-lsatoms 34735 df-lshyp 34736 df-lcv 34778 df-lfl 34817 df-lkr 34845 df-ldual 34883 df-oposet 34935 df-ol 34937 df-oml 34938 df-covers 35025 df-ats 35026 df-atl 35057 df-cvlat 35081 df-hlat 35110 df-llines 35256 df-lplanes 35257 df-lvols 35258 df-lines 35259 df-psubsp 35261 df-pmap 35262 df-padd 35554 df-lhyp 35746 df-laut 35747 df-ldil 35862 df-ltrn 35863 df-trl 35918 df-tgrp 36502 df-tendo 36514 df-edring 36516 df-dveca 36762 df-disoa 36789 df-dvech 36839 df-dib 36899 df-dic 36933 df-dih 36989 df-doch 37108 df-djh 37155 df-lcdual 37347

This theorem is referenced by: mapdpglem32 37465

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