hdmap14lem8 - Mathbox for Norm Megill

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Theorem hdmap14lem8 39171

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 38888	. . 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 2798	. . . 4 ⊢ (Base‘𝐶) = (Base‘𝐶)
9		hdmap14lem8.s	. . . 4 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
10		hdmap14lem8.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
11	10	eldifad 3893	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
12	1, 6, 7, 2, 8, 9, 3, 11	hdmapcl 39126	. . 3 ⊢ (𝜑 → (𝑆‘𝑋) ∈ (Base‘𝐶))
13		hdmap14lem8.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ (𝑉 ∖ { 0 }))
14	13	eldifad 3893	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
15	1, 6, 7, 2, 8, 9, 3, 14	hdmapcl 39126	. . 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 19650	. . 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 1369	. 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 39139	. . . 4 ⊢ (𝜑 → (𝑆‘(𝑋 + 𝑌)) = ((𝑆‘𝑋) ✚ (𝑆‘𝑌)))
24	23	oveq2d 7151	. . 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 38406	. . . . . . 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 19650	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ (𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
32	26, 27, 11, 14, 31	syl13anc 1369	. . . . . 6 ⊢ (𝜑 → (𝐹 · (𝑋 + 𝑌)) = ((𝐹 · 𝑋) + (𝐹 · 𝑌)))
33	32	fveq2d 6649	. . . . 5 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = (𝑆‘((𝐹 · 𝑋) + (𝐹 · 𝑌))))
34	7, 28, 29, 30	lmodvscl 19644	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉) → (𝐹 · 𝑋) ∈ 𝑉)
35	26, 27, 11, 34	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑋) ∈ 𝑉)
36	7, 28, 29, 30	lmodvscl 19644	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐹 ∈ 𝐵 ∧ 𝑌 ∈ 𝑉) → (𝐹 · 𝑌) ∈ 𝑉)
37	26, 27, 14, 36	syl3anc 1368	. . . . . 6 ⊢ (𝜑 → (𝐹 · 𝑌) ∈ 𝑉)
38	1, 6, 7, 22, 2, 16, 9, 3, 35, 37	hdmapadd 39139	. . . . 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 7153	. . . . 5 ⊢ (𝜑 → ((𝑆‘(𝐹 · 𝑋)) ✚ (𝑆‘(𝐹 · 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
42	33, 38, 41	3eqtrd 2837	. . . 4 ⊢ (𝜑 → (𝑆‘(𝐹 · (𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
43	25, 42	eqtr3d 2835	. . 3 ⊢ (𝜑 → (𝐽 ∙ (𝑆‘(𝑋 + 𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
44	24, 43	eqtr3d 2835	. 2 ⊢ (𝜑 → (𝐽 ∙ ((𝑆‘𝑋) ✚ (𝑆‘𝑌))) = ((𝐺 ∙ (𝑆‘𝑋)) ✚ (𝐼 ∙ (𝑆‘𝑌))))
45	21, 44	eqtr3d 2835	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 399 = wceq 1538 ∈ wcel 2111 ≠ wne 2987 ∖ cdif 3878 {csn 4525 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +_gcplusg 16557 Scalarcsca 16560 ·_𝑠 cvsca 16561 0_gc0g 16705 LModclmod 19627 LSpanclspn 19736 HLchlt 36646 LHypclh 37280 DVecHcdvh 38374 LCDualclcd 38882 HDMapchdma 39088

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-riotaBAD 36249

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-ot 4534 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-tpos 7875 df-undef 7922 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-n0 11886 df-z 11970 df-uz 12232 df-fz 12886 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-sca 16573 df-vsca 16574 df-0g 16707 df-mre 16849 df-mrc 16850 df-acs 16852 df-proset 17530 df-poset 17548 df-plt 17560 df-lub 17576 df-glb 17577 df-join 17578 df-meet 17579 df-p0 17641 df-p1 17642 df-lat 17648 df-clat 17710 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-subg 18268 df-cntz 18439 df-oppg 18466 df-lsm 18753 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-lmod 19629 df-lss 19697 df-lsp 19737 df-lvec 19868 df-lsatoms 36272 df-lshyp 36273 df-lcv 36315 df-lfl 36354 df-lkr 36382 df-ldual 36420 df-oposet 36472 df-ol 36474 df-oml 36475 df-covers 36562 df-ats 36563 df-atl 36594 df-cvlat 36618 df-hlat 36647 df-llines 36794 df-lplanes 36795 df-lvols 36796 df-lines 36797 df-psubsp 36799 df-pmap 36800 df-padd 37092 df-lhyp 37284 df-laut 37285 df-ldil 37400 df-ltrn 37401 df-trl 37455 df-tgrp 38039 df-tendo 38051 df-edring 38053 df-dveca 38299 df-disoa 38325 df-dvech 38375 df-dib 38435 df-dic 38469 df-dih 38525 df-doch 38644 df-djh 38691 df-lcdual 38883 df-mapd 38921 df-hvmap 39053 df-hdmap1 39089 df-hdmap 39090

This theorem is referenced by: hdmap14lem9 39172

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