mapdpglem28 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdpglem28		Structured version Visualization version GIF version

Theorem mapdpglem28 41678

Description: Lemma for mapdpg 41683. Baer p. 45 line 18: "vx'-vy'' = x'-uy''". (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)
mapdpglem28.ve	⊢ (𝜑 → 𝑣 ∈ 𝐵)
mapdpglem28.u1	⊢ (𝜑 → ℎ = (𝑢 · 𝑖))
mapdpglem28.u2	⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))

**Assertion**
Ref	Expression
mapdpglem28	⊢ (𝜑 → ((𝑣 · 𝐺)𝑅(𝑣 · 𝑖)) = (𝐺𝑅(𝑢 · 𝑖)))

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

**Proof of Theorem mapdpglem28**
Step	Hyp	Ref	Expression
1		mapdpglem28.u2	. 2 ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝑣 · (𝐺𝑅𝑖)))
2		mapdpglem28.u1	. . 3 ⊢ (𝜑 → ℎ = (𝑢 · 𝑖))
3	2	oveq2d 7429	. 2 ⊢ (𝜑 → (𝐺𝑅ℎ) = (𝐺𝑅(𝑢 · 𝑖)))
4		mapdpg.f	. . 3 ⊢ 𝐹 = (Base‘𝐶)
5		mapdpglem26.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐶)
6		eqid 2734	. . 3 ⊢ (Scalar‘𝐶) = (Scalar‘𝐶)
7		eqid 2734	. . 3 ⊢ (Base‘(Scalar‘𝐶)) = (Base‘(Scalar‘𝐶))
8		mapdpg.r	. . 3 ⊢ 𝑅 = (-_g‘𝐶)
9		mapdpg.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
10		mapdpg.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
11		mapdpg.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
12	9, 10, 11	lcdlmod 41569	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
13		mapdpglem28.ve	. . . 4 ⊢ (𝜑 → 𝑣 ∈ 𝐵)
14		mapdpg.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
15		mapdpglem26.a	. . . . 5 ⊢ 𝐴 = (Scalar‘𝑈)
16		mapdpglem26.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐴)
17	9, 14, 15, 16, 10, 6, 7, 11	lcdsbase 41577	. . . 4 ⊢ (𝜑 → (Base‘(Scalar‘𝐶)) = 𝐵)
18	13, 17	eleqtrrd 2836	. . 3 ⊢ (𝜑 → 𝑣 ∈ (Base‘(Scalar‘𝐶)))
19		mapdpg.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
20		mapdpgem25.i1	. . . 4 ⊢ (𝜑 → (𝑖 ∈ 𝐹 ∧ ((𝑀‘(𝑁‘{𝑌})) = (𝐽‘{𝑖}) ∧ (𝑀‘(𝑁‘{(𝑋 − 𝑌)})) = (𝐽‘{(𝐺𝑅𝑖)}))))
21	20	simpld 494	. . 3 ⊢ (𝜑 → 𝑖 ∈ 𝐹)
22	4, 5, 6, 7, 8, 12, 18, 19, 21	lmodsubdi 20886	. 2 ⊢ (𝜑 → (𝑣 · (𝐺𝑅𝑖)) = ((𝑣 · 𝐺)𝑅(𝑣 · 𝑖)))
23	1, 3, 22	3eqtr3rd 2778	1 ⊢ (𝜑 → ((𝑣 · 𝐺)𝑅(𝑣 · 𝑖)) = (𝐺𝑅(𝑢 · 𝑖)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 ∖ cdif 3928 {csn 4606 ‘cfv 6541 (class class class)co 7413 Basecbs 17230 Scalarcsca 17277 ·_𝑠 cvsca 17278 0_gc0g 17456 -_gcsg 18923 LSpanclspn 20938 HLchlt 39326 LHypclh 39961 DVecHcdvh 41055 LCDualclcd 41563 mapdcmpd 41601

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-rep 5259 ax-sep 5276 ax-nul 5286 ax-pow 5345 ax-pr 5412 ax-un 7737 ax-cnex 11193 ax-resscn 11194 ax-1cn 11195 ax-icn 11196 ax-addcl 11197 ax-addrcl 11198 ax-mulcl 11199 ax-mulrcl 11200 ax-mulcom 11201 ax-addass 11202 ax-mulass 11203 ax-distr 11204 ax-i2m1 11205 ax-1ne0 11206 ax-1rid 11207 ax-rnegex 11208 ax-rrecex 11209 ax-cnre 11210 ax-pre-lttri 11211 ax-pre-lttrn 11212 ax-pre-ltadd 11213 ax-pre-mulgt0 11214 ax-riotaBAD 38929

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3880 df-dif 3934 df-un 3936 df-in 3938 df-ss 3948 df-pss 3951 df-nul 4314 df-if 4506 df-pw 4582 df-sn 4607 df-pr 4609 df-tp 4611 df-op 4613 df-uni 4888 df-int 4927 df-iun 4973 df-iin 4974 df-br 5124 df-opab 5186 df-mpt 5206 df-tr 5240 df-id 5558 df-eprel 5564 df-po 5572 df-so 5573 df-fr 5617 df-we 5619 df-xp 5671 df-rel 5672 df-cnv 5673 df-co 5674 df-dm 5675 df-rn 5676 df-res 5677 df-ima 5678 df-pred 6301 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7679 df-om 7870 df-1st 7996 df-2nd 7997 df-tpos 8233 df-undef 8280 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-2o 8489 df-er 8727 df-map 8850 df-en 8968 df-dom 8969 df-sdom 8970 df-fin 8971 df-pnf 11279 df-mnf 11280 df-xr 11281 df-ltxr 11282 df-le 11283 df-sub 11476 df-neg 11477 df-nn 12249 df-2 12311 df-3 12312 df-4 12313 df-5 12314 df-6 12315 df-n0 12510 df-z 12597 df-uz 12861 df-fz 13530 df-struct 17167 df-sets 17184 df-slot 17202 df-ndx 17214 df-base 17231 df-ress 17254 df-plusg 17287 df-mulr 17288 df-sca 17290 df-vsca 17291 df-0g 17458 df-mre 17601 df-mrc 17602 df-acs 17604 df-proset 18311 df-poset 18330 df-plt 18345 df-lub 18361 df-glb 18362 df-join 18363 df-meet 18364 df-p0 18440 df-p1 18441 df-lat 18447 df-clat 18514 df-mgm 18623 df-sgrp 18702 df-mnd 18718 df-submnd 18767 df-grp 18924 df-minusg 18925 df-sbg 18926 df-subg 19111 df-cntz 19305 df-oppg 19334 df-lsm 19623 df-cmn 19769 df-abl 19770 df-mgp 20107 df-rng 20119 df-ur 20148 df-ring 20201 df-oppr 20303 df-dvdsr 20326 df-unit 20327 df-invr 20357 df-dvr 20370 df-nzr 20482 df-rlreg 20663 df-domn 20664 df-drng 20700 df-lmod 20829 df-lss 20899 df-lsp 20939 df-lvec 21071 df-lsatoms 38952 df-lshyp 38953 df-lcv 38995 df-lfl 39034 df-lkr 39062 df-ldual 39100 df-oposet 39152 df-ol 39154 df-oml 39155 df-covers 39242 df-ats 39243 df-atl 39274 df-cvlat 39298 df-hlat 39327 df-llines 39475 df-lplanes 39476 df-lvols 39477 df-lines 39478 df-psubsp 39480 df-pmap 39481 df-padd 39773 df-lhyp 39965 df-laut 39966 df-ldil 40081 df-ltrn 40082 df-trl 40136 df-tgrp 40720 df-tendo 40732 df-edring 40734 df-dveca 40980 df-disoa 41006 df-dvech 41056 df-dib 41116 df-dic 41150 df-dih 41206 df-doch 41325 df-djh 41372 df-lcdual 41564

This theorem is referenced by: mapdpglem30 41679

W3C validator