hdmapglem7b - Mathbox for Norm Megill

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Theorem hdmapglem7b 37996

Description: Lemma for hdmapg 37998. (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 37178	. . . 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 37180	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	19	eldifad 3810	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
21	3, 6, 5, 7	lmodvscl 19236	. . . . 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 4558	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
24		hdmapglem7.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
25	1, 2, 3, 24	dochssv 37423	. . . . . 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 19233	. . . 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 37980	. 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 37974	. . . . 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 37904	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
39	38	oveq2d 6921	. . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1_r‘𝑅)))
40	6	lmodring 19227	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
41	13, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	7, 9, 37	ringridm 18926	. . . . . . . 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 37986	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 )
47	44, 46	oveq12d 6923	. . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 ))
48		ringgrp 18906	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
49	41, 48	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
50	7, 8, 45	grprid 17807	. . . . . 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 6920	. . 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 37974	. . . 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 37987	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 )
56	55	oveq2d 6921	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 ))
57	7, 9, 45	ringrz 18942	. . . . . . 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 6920	. . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦)))
61	1, 2, 3, 6, 7, 10, 12, 28, 32	hdmapipcl 37973	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵)
62	7, 8, 45	grplid 17806	. . . . 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 6923	. 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 4397 ⟨cop 4403 I cid 5249 ↾ cres 5344 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +_gcplusg 16305 ._rcmulr 16306 Scalarcsca 16308 ·_𝑠 cvsca 16309 0_gc0g 16453 Grpcgrp 17776 LSSumclsm 18400 1_rcur 18855 Ringcrg 18901 LModclmod 19219 LSpanclspn 19330 HLchlt 35418 LHypclh 36052 LTrncltrn 36169 DVecHcdvh 37146 ocHcoch 37415 HVMapchvm 37824 HDMapchdma 37860 HGMapchg 37951

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 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 ax-riotaBAD 35021

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 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-ot 4406 df-uni 4659 df-int 4698 df-iun 4742 df-iin 4743 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-of 7157 df-om 7327 df-1st 7428 df-2nd 7429 df-tpos 7617 df-undef 7664 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-oadd 7830 df-er 8009 df-map 8124 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-2 11414 df-3 11415 df-4 11416 df-5 11417 df-6 11418 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-struct 16224 df-ndx 16225 df-slot 16226 df-base 16228 df-sets 16229 df-ress 16230 df-plusg 16318 df-mulr 16319 df-sca 16321 df-vsca 16322 df-0g 16455 df-mre 16599 df-mrc 16600 df-acs 16602 df-proset 17281 df-poset 17299 df-plt 17311 df-lub 17327 df-glb 17328 df-join 17329 df-meet 17330 df-p0 17392 df-p1 17393 df-lat 17399 df-clat 17461 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-submnd 17689 df-grp 17779 df-minusg 17780 df-sbg 17781 df-subg 17942 df-cntz 18100 df-oppg 18126 df-lsm 18402 df-cmn 18548 df-abl 18549 df-mgp 18844 df-ur 18856 df-ring 18903 df-oppr 18977 df-dvdsr 18995 df-unit 18996 df-invr 19026 df-dvr 19037 df-drng 19105 df-lmod 19221 df-lss 19289 df-lsp 19331 df-lvec 19462 df-lsatoms 35044 df-lshyp 35045 df-lcv 35087 df-lfl 35126 df-lkr 35154 df-ldual 35192 df-oposet 35244 df-ol 35246 df-oml 35247 df-covers 35334 df-ats 35335 df-atl 35366 df-cvlat 35390 df-hlat 35419 df-llines 35566 df-lplanes 35567 df-lvols 35568 df-lines 35569 df-psubsp 35571 df-pmap 35572 df-padd 35864 df-lhyp 36056 df-laut 36057 df-ldil 36172 df-ltrn 36173 df-trl 36227 df-tgrp 36811 df-tendo 36823 df-edring 36825 df-dveca 37071 df-disoa 37097 df-dvech 37147 df-dib 37207 df-dic 37241 df-dih 37297 df-doch 37416 df-djh 37463 df-lcdual 37655 df-mapd 37693 df-hvmap 37825 df-hdmap1 37861 df-hdmap 37862 df-hgmap 37952

This theorem is referenced by: hdmapglem7 37997

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