hdmap14lem8 - Mathbox for Norm Megill

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Theorem hdmap14lem8 42338

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 42055	. . 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 2737	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		hdmap14lem8.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3902	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 42293	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
13		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3902	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	1, 6, 7, 2, 8, 9, 3, 14	hdmapcl 42293	. . 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 20874	. . 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 1375	. 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 42306	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
24	23	oveq2d 7377	. . 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 41573	. . . . . . 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 20874	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
32	26, 27, 11, 14, 31	syl13anc 1375	. . . . . 6 ⊢ (𝜑 → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
33	32	fveq2d 6839	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))))
34	7, 28, 29, 30	lmodvscl 20867	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
35	26, 27, 11, 34	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
36	7, 28, 29, 30	lmodvscl 20867	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → (𝐹 · 𝑌) ∈ 𝑉)
37	26, 27, 14, 36	syl3anc 1374	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑌) ∈ 𝑉)
38	1, 6, 7, 22, 2, 16, 9, 3, 35, 37	hdmapadd 42306	. . . . 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 7379	. . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
42	33, 38, 41	3eqtrd 2776	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
43	25, 42	eqtr3d 2774	. . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
44	24, 43	eqtr3d 2774	. 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
45	21, 44	eqtr3d 2774	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 1542 ∈ wcel 2114 ≠ wne 2933 ∖ cdif 3887 {csn 4568 ‘cfv 6493 (class class class)co 7361 Basecbs 17173 +_gcplusg 17214 Scalarcsca 17217 ·_𝑠 cvsca 17218 0_gc0g 17396 LModclmod 20849 LSpanclspn 20960 HLchlt 39813 LHypclh 40447 DVecHcdvh 41541 LCDualclcd 42049 HDMapchdma 42255

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5213 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109 ax-riotaBAD 39416

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-ot 4577 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7625 df-om 7812 df-1st 7936 df-2nd 7937 df-tpos 8170 df-undef 8217 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-map 8769 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-2 12238 df-3 12239 df-4 12240 df-5 12241 df-6 12242 df-n0 12432 df-z 12519 df-uz 12783 df-fz 13456 df-struct 17111 df-sets 17128 df-slot 17146 df-ndx 17158 df-base 17174 df-ress 17195 df-plusg 17227 df-mulr 17228 df-sca 17230 df-vsca 17231 df-0g 17398 df-mre 17542 df-mrc 17543 df-acs 17545 df-proset 18254 df-poset 18273 df-plt 18288 df-lub 18304 df-glb 18305 df-join 18306 df-meet 18307 df-p0 18383 df-p1 18384 df-lat 18392 df-clat 18459 df-mgm 18602 df-sgrp 18681 df-mnd 18697 df-submnd 18746 df-grp 18906 df-minusg 18907 df-sbg 18908 df-subg 19093 df-cntz 19286 df-oppg 19315 df-lsm 19605 df-cmn 19751 df-abl 19752 df-mgp 20116 df-rng 20128 df-ur 20157 df-ring 20210 df-oppr 20311 df-dvdsr 20331 df-unit 20332 df-invr 20362 df-dvr 20375 df-nzr 20484 df-rlreg 20665 df-domn 20666 df-drng 20702 df-lmod 20851 df-lss 20921 df-lsp 20961 df-lvec 21093 df-lsatoms 39439 df-lshyp 39440 df-lcv 39482 df-lfl 39521 df-lkr 39549 df-ldual 39587 df-oposet 39639 df-ol 39641 df-oml 39642 df-covers 39729 df-ats 39730 df-atl 39761 df-cvlat 39785 df-hlat 39814 df-llines 39961 df-lplanes 39962 df-lvols 39963 df-lines 39964 df-psubsp 39966 df-pmap 39967 df-padd 40259 df-lhyp 40451 df-laut 40452 df-ldil 40567 df-ltrn 40568 df-trl 40622 df-tgrp 41206 df-tendo 41218 df-edring 41220 df-dveca 41466 df-disoa 41492 df-dvech 41542 df-dib 41602 df-dic 41636 df-dih 41692 df-doch 41811 df-djh 41858 df-lcdual 42050 df-mapd 42088 df-hvmap 42220 df-hdmap1 42256 df-hdmap 42257

This theorem is referenced by: hdmap14lem9 42339

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