mapdpglem27 - Mathbox for Norm Megill

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Theorem mapdpglem27 37509

Description: Lemma for mapdpg 37516. 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 37507	. . 3 ⊢ (𝜑 → ((𝐽‘{ℎ}) = (𝐽‘{𝑖}) ∧ (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)})))
21	20	simprd 483	. 2 ⊢ (𝜑 → (𝐽‘{(𝐺𝑅ℎ)}) = (𝐽‘{(𝐺𝑅𝑖)}))
22		eqid 2771	. . . 4 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
23		eqid 2771	. . . 4 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
24		eqid 2771	. . . 4 ⊢ (0_g‘(Scalar‘𝐶)) = (0_g‘(Scalar‘𝐶))
25		mapdpglem26.t	. . . 4 ⊢ · = ( ·_𝑠 ‘𝐶)
26	1, 8, 12	lcdlvec 37401	. . . 4 ⊢ (𝜑 → 𝐶 ∈ LVec)
27	1, 8, 12	lcdlmod 37402	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ LMod)
28	18	simpld 482	. . . . 5 ⊢ (𝜑 → ℎ ∈ 𝐹)
29	9, 10	lmodvsubcl 19118	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ ℎ ∈ 𝐹) → (𝐺𝑅ℎ) ∈ 𝐹)
30	27, 15, 28, 29	syl3anc 1476	. . . 4 ⊢ (𝜑 → (𝐺𝑅ℎ) ∈ 𝐹)
31	19	simpld 482	. . . . 5 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
32	9, 10	lmodvsubcl 19118	. . . . 5 ⊢ ((𝐶 ∈ LMod ∧ 𝐺 ∈ 𝐹 ∧ 𝑖 ∈ 𝐹) → (𝐺𝑅𝑖) ∈ 𝐹)
33	27, 15, 31, 32	syl3anc 1476	. . . 4 ⊢ (𝜑 → (𝐺𝑅𝑖) ∈ 𝐹)
34	9, 22, 23, 24, 25, 11, 26, 30, 33	lspsneq 19335	. . 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 37410	. . . . 5 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
38		mapdpglem26.o	. . . . . . 7 ⊢ 𝑂 = (0_g‘𝐴)
39	1, 3, 35, 38, 8, 22, 24, 12	lcd0 37418	. . . . . 6 ⊢ (𝜑 → (0_g‘(Scalar‘𝐶)) = 𝑂)
40	39	sneqd 4328	. . . . 5 ⊢ (𝜑 → {(0_g‘(Scalar‘𝐶))} = {𝑂})
41	37, 40	difeq12d 3880	. . . 4 ⊢ (𝜑 → ((Base‘(Scalar‘𝐶)) ∖ {(0_g‘(Scalar‘𝐶))}) = (𝐵 ∖ {𝑂}))
42	41	rexeqdv 3294	. . 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 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 ∖ cdif 3720 {csn 4316 ‘cfv 6031 (class class class)co 6793 Basecbs 16064 Scalarcsca 16152 ·_𝑠 cvsca 16153 0_gc0g 16308 -_gcsg 17632 LModclmod 19073 LSpanclspn 19184 HLchlt 35159 LHypclh 35792 DVecHcdvh 36888 LCDualclcd 37396 mapdcmpd 37434

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-riotaBAD 34761

This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-of 7044 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-undef 7551 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-0g 16310 df-mre 16454 df-mrc 16455 df-acs 16457 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-oppg 17983 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lsatoms 34785 df-lshyp 34786 df-lcv 34828 df-lfl 34867 df-lkr 34895 df-ldual 34933 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 df-laut 35797 df-ldil 35912 df-ltrn 35913 df-trl 35968 df-tgrp 36552 df-tendo 36564 df-edring 36566 df-dveca 36812 df-disoa 36839 df-dvech 36889 df-dib 36949 df-dic 36983 df-dih 37039 df-doch 37158 df-djh 37205 df-lcdual 37397

This theorem is referenced by: mapdpglem32 37515

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