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 41343

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 41060	. . 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 2728	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		hdmap14lem8.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3957	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 41298	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
13		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3957	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	1, 6, 7, 2, 8, 9, 3, 14	hdmapcl 41298	. . 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 20762	. . 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 1370	. 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 41311	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
24	23	oveq2d 7431	. . 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 40578	. . . . . . 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 20762	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
32	26, 27, 11, 14, 31	syl13anc 1370	. . . . . 6 ⊢ (𝜑 → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
33	32	fveq2d 6896	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))))
34	7, 28, 29, 30	lmodvscl 20755	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
35	26, 27, 11, 34	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
36	7, 28, 29, 30	lmodvscl 20755	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → (𝐹 · 𝑌) ∈ 𝑉)
37	26, 27, 14, 36	syl3anc 1369	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑌) ∈ 𝑉)
38	1, 6, 7, 22, 2, 16, 9, 3, 35, 37	hdmapadd 41311	. . . . 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 7433	. . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
42	33, 38, 41	3eqtrd 2772	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
43	25, 42	eqtr3d 2770	. . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
44	24, 43	eqtr3d 2770	. 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
45	21, 44	eqtr3d 2770	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 395 = wceq 1534 ∈ wcel 2099 ≠ wne 2936 ∖ cdif 3942 {csn 4625 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +_gcplusg 17227 Scalarcsca 17230 ·_𝑠 cvsca 17231 0_gc0g 17415 LModclmod 20737 LSpanclspn 20849 HLchlt 38817 LHypclh 39452 DVecHcdvh 40546 LCDualclcd 41054 HDMapchdma 41260

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 ax-riotaBAD 38420

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-ot 4634 df-uni 4905 df-int 4946 df-iun 4994 df-iin 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-of 7680 df-om 7866 df-1st 7988 df-2nd 7989 df-tpos 8226 df-undef 8273 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-1o 8481 df-er 8719 df-map 8841 df-en 8959 df-dom 8960 df-sdom 8961 df-fin 8962 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-3 12301 df-4 12302 df-5 12303 df-6 12304 df-n0 12498 df-z 12584 df-uz 12848 df-fz 13512 df-struct 17110 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-ress 17204 df-plusg 17240 df-mulr 17241 df-sca 17243 df-vsca 17244 df-0g 17417 df-mre 17560 df-mrc 17561 df-acs 17563 df-proset 18281 df-poset 18299 df-plt 18316 df-lub 18332 df-glb 18333 df-join 18334 df-meet 18335 df-p0 18411 df-p1 18412 df-lat 18418 df-clat 18485 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-submnd 18735 df-grp 18887 df-minusg 18888 df-sbg 18889 df-subg 19072 df-cntz 19262 df-oppg 19291 df-lsm 19585 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-oppr 20267 df-dvdsr 20290 df-unit 20291 df-invr 20321 df-dvr 20334 df-drng 20620 df-lmod 20739 df-lss 20810 df-lsp 20850 df-lvec 20982 df-lsatoms 38443 df-lshyp 38444 df-lcv 38486 df-lfl 38525 df-lkr 38553 df-ldual 38591 df-oposet 38643 df-ol 38645 df-oml 38646 df-covers 38733 df-ats 38734 df-atl 38765 df-cvlat 38789 df-hlat 38818 df-llines 38966 df-lplanes 38967 df-lvols 38968 df-lines 38969 df-psubsp 38971 df-pmap 38972 df-padd 39264 df-lhyp 39456 df-laut 39457 df-ldil 39572 df-ltrn 39573 df-trl 39627 df-tgrp 40211 df-tendo 40223 df-edring 40225 df-dveca 40471 df-disoa 40497 df-dvech 40547 df-dib 40607 df-dic 40641 df-dih 40697 df-doch 40816 df-djh 40863 df-lcdual 41055 df-mapd 41093 df-hvmap 41225 df-hdmap1 41261 df-hdmap 41262

This theorem is referenced by: hdmap14lem9 41344

W3C validator