hdmapglem7b - Mathbox for Norm Megill

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Theorem hdmapglem7b 37998

Description: Lemma for hdmapg 38000. (Contributed by NM, 14-Jun-2015.)

**Hypotheses**
Ref	Expression
hdmapglem7.h	⊢ 𝐻 = (LHyp‘𝐾)
hdmapglem7.e	⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
hdmapglem7.o	⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
hdmapglem7.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
hdmapglem7.v	⊢ 𝑉 = (Base‘𝑈)
hdmapglem7.p	⊢ + = (+_g‘𝑈)
hdmapglem7.q	⊢ · = ( ·_𝑠 ‘𝑈)
hdmapglem7.r	⊢ 𝑅 = (Scalar‘𝑈)
hdmapglem7.b	⊢ 𝐵 = (Base‘𝑅)
hdmapglem7.a	⊢ ⊕ = (LSSum‘𝑈)
hdmapglem7.n	⊢ 𝑁 = (LSpan‘𝑈)
hdmapglem7.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hdmapglem7.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
hdmapglem7.t	⊢ × = (._r‘𝑅)
hdmapglem7.z	⊢ 0 = (0_g‘𝑅)
hdmapglem7.c	⊢ ✚ = (+_g‘𝑅)
hdmapglem7.s	⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
hdmapglem7.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hdmapglem7b.u	⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))
hdmapglem7b.v	⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))
hdmapglem7b.k	⊢ (𝜑 → 𝑚 ∈ 𝐵)
hdmapglem7b.l	⊢ (𝜑 → 𝑛 ∈ 𝐵)

**Assertion**
Ref	Expression
hdmapglem7b	⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))

**Proof of Theorem hdmapglem7b**
Step	Hyp	Ref	Expression
1		hdmapglem7.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hdmapglem7.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		hdmapglem7.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
4		hdmapglem7.p	. . 3 ⊢ + = (+_g‘𝑈)
5		hdmapglem7.q	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
6		hdmapglem7.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
7		hdmapglem7.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
8		hdmapglem7.c	. . 3 ⊢ ✚ = (+_g‘𝑅)
9		hdmapglem7.t	. . 3 ⊢ × = (._r‘𝑅)
10		hdmapglem7.s	. . 3 ⊢ 𝑆 = ((HDMap‘𝐾)‘𝑊)
11		hdmapglem7.g	. . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
12		hdmapglem7.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
13	1, 2, 12	dvhlmod 37180	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
14		hdmapglem7b.l	. . . . 5 ⊢ (𝜑 → 𝑛 ∈ 𝐵)
15		eqid 2825	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2825	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
17		eqid 2825	. . . . . . 7 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
18		hdmapglem7.e	. . . . . . 7 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
19	1, 15, 16, 2, 3, 17, 18, 12	dvheveccl 37182	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	19	eldifad 3810	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
21	3, 6, 5, 7	lmodvscl 19243	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝑛 · 𝐸) ∈ 𝑉)
22	13, 14, 20, 21	syl3anc 1494	. . . 4 ⊢ (𝜑 → (𝑛 · 𝐸) ∈ 𝑉)
23	20	snssd 4560	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
24		hdmapglem7.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
25	1, 2, 3, 24	dochssv 37425	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
26	12, 23, 25	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
27		hdmapglem7b.v	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))
28	26, 27	sseldd 3828	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
29	3, 4	lmodvacl 19240	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑛 · 𝐸) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
30	13, 22, 28, 29	syl3anc 1494	. . 3 ⊢ (𝜑 → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
31		hdmapglem7b.u	. . . 4 ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))
32	26, 31	sseldd 3828	. . 3 ⊢ (𝜑 → 𝑥 ∈ 𝑉)
33		hdmapglem7b.k	. . 3 ⊢ (𝜑 → 𝑚 ∈ 𝐵)
34	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 30, 20, 32, 33	hdmapgln2 37982	. 2 ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))))
35	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 28, 20, 14	hdmapln1 37976	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)))
36		eqid 2825	. . . . . . . . 9 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
37		eqid 2825	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
38	1, 18, 36, 10, 12, 2, 6, 37	hdmapevec2 37906	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
39	38	oveq2d 6926	. . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1_r‘𝑅)))
40	6	lmodring 19234	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
41	13, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	7, 9, 37	ringridm 18933	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × (1_r‘𝑅)) = 𝑛)
43	41, 14, 42	syl2anc 579	. . . . . . 7 ⊢ (𝜑 → (𝑛 × (1_r‘𝑅)) = 𝑛)
44	39, 43	eqtrd 2861	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = 𝑛)
45		hdmapglem7.z	. . . . . . 7 ⊢ 0 = (0_g‘𝑅)
46	1, 18, 24, 2, 3, 6, 7, 9, 45, 10, 12, 27	hdmapinvlem1 37988	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 )
47	44, 46	oveq12d 6928	. . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 ))
48		ringgrp 18913	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
49	41, 48	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
50	7, 8, 45	grprid 17814	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵) → (𝑛 ✚ 0 ) = 𝑛)
51	49, 14, 50	syl2anc 579	. . . . 5 ⊢ (𝜑 → (𝑛 ✚ 0 ) = 𝑛)
52	35, 47, 51	3eqtrd 2865	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = 𝑛)
53	52	oveq1d 6925	. . 3 ⊢ (𝜑 → (((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) = (𝑛 × (𝐺‘𝑚)))
54	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 20, 28, 32, 14	hdmapln1 37976	. . . 4 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)))
55	1, 18, 24, 2, 3, 6, 7, 9, 45, 10, 12, 31	hdmapinvlem2 37989	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 )
56	55	oveq2d 6926	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 ))
57	7, 9, 45	ringrz 18949	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × 0 ) = 0 )
58	41, 14, 57	syl2anc 579	. . . . . 6 ⊢ (𝜑 → (𝑛 × 0 ) = 0 )
59	56, 58	eqtrd 2861	. . . . 5 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = 0 )
60	59	oveq1d 6925	. . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦)))
61	1, 2, 3, 6, 7, 10, 12, 28, 32	hdmapipcl 37975	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵)
62	7, 8, 45	grplid 17813	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
63	49, 61, 62	syl2anc 579	. . . 4 ⊢ (𝜑 → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
64	54, 60, 63	3eqtrd 2865	. . 3 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑆‘𝑥)‘𝑦))
65	53, 64	oveq12d 6928	. 2 ⊢ (𝜑 → ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
66	34, 65	eqtrd 2861	1 ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 {csn 4399 ⟨cop 4405 I cid 5251 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +_gcplusg 16312 ._rcmulr 16313 Scalarcsca 16315 ·_𝑠 cvsca 16316 0_gc0g 16460 Grpcgrp 17783 LSSumclsm 18407 1_rcur 18862 Ringcrg 18908 LModclmod 19226 LSpanclspn 19337 HLchlt 35420 LHypclh 36054 LTrncltrn 36171 DVecHcdvh 37148 ocHcoch 37417 HVMapchvm 37826 HDMapchdma 37862 HGMapchg 37953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-riotaBAD 35023

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-ot 4408 df-uni 4661 df-int 4700 df-iun 4744 df-iin 4745 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-of 7162 df-om 7332 df-1st 7433 df-2nd 7434 df-tpos 7622 df-undef 7669 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-sca 16328 df-vsca 16329 df-0g 16462 df-mre 16606 df-mrc 16607 df-acs 16609 df-proset 17288 df-poset 17306 df-plt 17318 df-lub 17334 df-glb 17335 df-join 17336 df-meet 17337 df-p0 17399 df-p1 17400 df-lat 17406 df-clat 17468 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-submnd 17696 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-cntz 18107 df-oppg 18133 df-lsm 18409 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-oppr 18984 df-dvdsr 19002 df-unit 19003 df-invr 19033 df-dvr 19044 df-drng 19112 df-lmod 19228 df-lss 19296 df-lsp 19338 df-lvec 19469 df-lsatoms 35046 df-lshyp 35047 df-lcv 35089 df-lfl 35128 df-lkr 35156 df-ldual 35194 df-oposet 35246 df-ol 35248 df-oml 35249 df-covers 35336 df-ats 35337 df-atl 35368 df-cvlat 35392 df-hlat 35421 df-llines 35568 df-lplanes 35569 df-lvols 35570 df-lines 35571 df-psubsp 35573 df-pmap 35574 df-padd 35866 df-lhyp 36058 df-laut 36059 df-ldil 36174 df-ltrn 36175 df-trl 36229 df-tgrp 36813 df-tendo 36825 df-edring 36827 df-dveca 37073 df-disoa 37099 df-dvech 37149 df-dib 37209 df-dic 37243 df-dih 37299 df-doch 37418 df-djh 37465 df-lcdual 37657 df-mapd 37695 df-hvmap 37827 df-hdmap1 37863 df-hdmap 37864 df-hgmap 37954

This theorem is referenced by: hdmapglem7 37999

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