hdmap14lem8 - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hdmap14lem8		Structured version Visualization version GIF version

Theorem hdmap14lem8 39885

Description: Part of proof of part 14 in [Baer] p. 49 lines 33-35. (Contributed by NM, 1-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmap14lem8.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmap14lem8.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmap14lem8.v	⊢ 𝑉 = (Base‘𝑈)
hdmap14lem8.q	⊢ + = (+_g‘𝑈)
hdmap14lem8.t	⊢ · = ( ·_𝑠 ‘𝑈)
hdmap14lem8.o	⊢ 0 = (0_g‘𝑈)
hdmap14lem8.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmap14lem8.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmap14lem8.b	⊢ 𝐵 = (Base‘𝑅)
hdmap14lem8.c	⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
hdmap14lem8.d	⊢ ✚ = (+_g‘𝐶)
hdmap14lem8.e	⊢ ∙ = ( ·_𝑠 ‘𝐶)
hdmap14lem8.p	⊢ 𝑃 = (Scalar‘𝐶)
hdmap14lem8.a	⊢ 𝐴 = (Base‘𝑃)
hdmap14lem8.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmap14lem8.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmap14lem8.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
hdmap14lem8.y	⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
hdmap14lem8.f	⊢ (𝜑 → 𝐹 ∈ 𝐵)
hdmap14lem8.g	⊢ (𝜑 → 𝐺 ∈ 𝐴)
hdmap14lem8.i	⊢ (𝜑 → 𝐼 ∈ 𝐴)
hdmap14lem8.xx	⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
hdmap14lem8.yy	⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))
hdmap14lem8.ne	⊢ (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
hdmap14lem8.j	⊢ (𝜑 → 𝐽 ∈ 𝐴)
hdmap14lem8.xy	⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))))

**Assertion**
Ref	Expression
hdmap14lem8	⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))

**Proof of Theorem hdmap14lem8**
Step	Hyp	Ref	Expression
1		hdmap14lem8.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmap14lem8.c	. . . 4 ⊢ 𝐶 = ((LCDual‘𝐾)‘𝑊)
3		hdmap14lem8.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	lcdlmod 39602	. . 3 ⊢ (𝜑 → 𝐶 ∈ LMod)
5		hdmap14lem8.j	. . 3 ⊢ (𝜑 → 𝐽 ∈ 𝐴)
6		hdmap14lem8.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
7		hdmap14lem8.v	. . . 4 ⊢ 𝑉 = (Base‘𝑈)
8		eqid 2740	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		hdmap14lem8.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3904	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 39840	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
13		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3904	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	1, 6, 7, 2, 8, 9, 3, 14	hdmapcl 39840	. . 3 ⊢ (𝜑 → (𝑆‘𝑌) ∈ (Base‘𝐶))
16		hdmap14lem8.d	. . . 4 ⊢ ✚ = (+_g‘𝐶)
17		hdmap14lem8.p	. . . 4 ⊢ 𝑃 = (Scalar‘𝐶)
18		hdmap14lem8.e	. . . 4 ⊢ ∙ = ( ·_𝑠 ‘𝐶)
19		hdmap14lem8.a	. . . 4 ⊢ 𝐴 = (Base‘𝑃)
20	8, 16, 17, 18, 19	lmodvsdi 20144	. . 3 ⊢ ((𝐶 ∈ LMod ∧ (𝐽 ∈ 𝐴 ∧ (𝑆‘𝑋) ∈ (Base‘𝐶) ∧ (𝑆‘𝑌) ∈ (Base‘𝐶))) → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))))
21	4, 5, 12, 15, 20	syl13anc 1371	. 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))))
22		hdmap14lem8.q	. . . . 5 ⊢ + = (+_g‘𝑈)
23	1, 6, 7, 22, 2, 16, 9, 3, 11, 14	hdmapadd 39853	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
24	23	oveq2d 7287	. . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))))
25		hdmap14lem8.xy	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))))
26	1, 6, 3	dvhlmod 39120	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
27		hdmap14lem8.f	. . . . . . 7 ⊢ (𝜑 → 𝐹 ∈ 𝐵)
28		hdmap14lem8.r	. . . . . . . 8 ⊢ 𝑅 = (Scalar‘𝑈)
29		hdmap14lem8.t	. . . . . . . 8 ⊢ · = ( ·_𝑠 ‘𝑈)
30		hdmap14lem8.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝑅)
31	7, 22, 28, 29, 30	lmodvsdi 20144	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
32	26, 27, 11, 14, 31	syl13anc 1371	. . . . . 6 ⊢ (𝜑 → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
33	32	fveq2d 6775	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))))
34	7, 28, 29, 30	lmodvscl 20138	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
35	26, 27, 11, 34	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
36	7, 28, 29, 30	lmodvscl 20138	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → (𝐹 · 𝑌) ∈ 𝑉)
37	26, 27, 14, 36	syl3anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑌) ∈ 𝑉)
38	1, 6, 7, 22, 2, 16, 9, 3, 35, 37	hdmapadd 39853	. . . . 5 ⊢ (𝜑 → (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))) = ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))))
39		hdmap14lem8.xx	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑋)) = (𝐺 ∙ (𝑆‘𝑋)))
40		hdmap14lem8.yy	. . . . . 6 ⊢ (𝜑 → (𝑆‘(𝐹 · 𝑌)) = (𝐼 ∙ (𝑆‘𝑌)))
41	39, 40	oveq12d 7289	. . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
42	33, 38, 41	3eqtrd 2784	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
43	25, 42	eqtr3d 2782	. . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
44	24, 43	eqtr3d 2782	. 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
45	21, 44	eqtr3d 2782	1 ⊢ (𝜑 → ((𝐽 ∙ (𝑆‘𝑋)) ✚ (𝐽 ∙ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1542 ∈ wcel 2110 ≠ wne 2945 ∖ cdif 3889 {csn 4567 ‘cfv 6432 (class class class)co 7271 Basecbs 16910 +_gcplusg 16960 Scalarcsca 16963 ·_𝑠 cvsca 16964 0_gc0g 17148 LModclmod 20121 LSpanclspn 20231 HLchlt 37360 LHypclh 37994 DVecHcdvh 39088 LCDualclcd 39596 HDMapchdma 39802

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 ax-cnex 10928 ax-resscn 10929 ax-1cn 10930 ax-icn 10931 ax-addcl 10932 ax-addrcl 10933 ax-mulcl 10934 ax-mulrcl 10935 ax-mulcom 10936 ax-addass 10937 ax-mulass 10938 ax-distr 10939 ax-i2m1 10940 ax-1ne0 10941 ax-1rid 10942 ax-rnegex 10943 ax-rrecex 10944 ax-cnre 10945 ax-pre-lttri 10946 ax-pre-lttrn 10947 ax-pre-ltadd 10948 ax-pre-mulgt0 10949 ax-riotaBAD 36963

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rmo 3074 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-ot 4576 df-uni 4846 df-int 4886 df-iun 4932 df-iin 4933 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6201 df-ord 6268 df-on 6269 df-lim 6270 df-suc 6271 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-riota 7228 df-ov 7274 df-oprab 7275 df-mpo 7276 df-of 7527 df-om 7707 df-1st 7824 df-2nd 7825 df-tpos 8033 df-undef 8080 df-frecs 8088 df-wrecs 8119 df-recs 8193 df-rdg 8232 df-1o 8288 df-er 8481 df-map 8600 df-en 8717 df-dom 8718 df-sdom 8719 df-fin 8720 df-pnf 11012 df-mnf 11013 df-xr 11014 df-ltxr 11015 df-le 11016 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12582 df-fz 13239 df-struct 16846 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-ress 16940 df-plusg 16973 df-mulr 16974 df-sca 16976 df-vsca 16977 df-0g 17150 df-mre 17293 df-mrc 17294 df-acs 17296 df-proset 18011 df-poset 18029 df-plt 18046 df-lub 18062 df-glb 18063 df-join 18064 df-meet 18065 df-p0 18141 df-p1 18142 df-lat 18148 df-clat 18215 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-submnd 18429 df-grp 18578 df-minusg 18579 df-sbg 18580 df-subg 18750 df-cntz 18921 df-oppg 18948 df-lsm 19239 df-cmn 19386 df-abl 19387 df-mgp 19719 df-ur 19736 df-ring 19783 df-oppr 19860 df-dvdsr 19881 df-unit 19882 df-invr 19912 df-dvr 19923 df-drng 19991 df-lmod 20123 df-lss 20192 df-lsp 20232 df-lvec 20363 df-lsatoms 36986 df-lshyp 36987 df-lcv 37029 df-lfl 37068 df-lkr 37096 df-ldual 37134 df-oposet 37186 df-ol 37188 df-oml 37189 df-covers 37276 df-ats 37277 df-atl 37308 df-cvlat 37332 df-hlat 37361 df-llines 37508 df-lplanes 37509 df-lvols 37510 df-lines 37511 df-psubsp 37513 df-pmap 37514 df-padd 37806 df-lhyp 37998 df-laut 37999 df-ldil 38114 df-ltrn 38115 df-trl 38169 df-tgrp 38753 df-tendo 38765 df-edring 38767 df-dveca 39013 df-disoa 39039 df-dvech 39089 df-dib 39149 df-dic 39183 df-dih 39239 df-doch 39358 df-djh 39405 df-lcdual 39597 df-mapd 39635 df-hvmap 39767 df-hdmap1 39803 df-hdmap 39804

This theorem is referenced by: hdmap14lem9 39886

W3C validator