hdmapglem7b - Mathbox for Norm Megill

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Theorem hdmapglem7b 37220

Description: Lemma for hdmapg 37222. (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 36399	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
14		hdmapglem7b.l	. . . . 5 ⊢ (𝜑 → 𝑛 ∈ 𝐵)
15		eqid 2622	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2622	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
17		eqid 2622	. . . . . . 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 36401	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	19	eldifad 3586	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
21	3, 6, 5, 7	lmodvscl 18880	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝑛 · 𝐸) ∈ 𝑉)
22	13, 14, 20, 21	syl3anc 1326	. . . 4 ⊢ (𝜑 → (𝑛 · 𝐸) ∈ 𝑉)
23	20	snssd 4340	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
24		hdmapglem7.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
25	1, 2, 3, 24	dochssv 36644	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
26	12, 23, 25	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
27		hdmapglem7b.v	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))
28	26, 27	sseldd 3604	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
29	3, 4	lmodvacl 18877	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑛 · 𝐸) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
30	13, 22, 28, 29	syl3anc 1326	. . 3 ⊢ (𝜑 → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
31		hdmapglem7b.u	. . . 4 ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))
32	26, 31	sseldd 3604	. . 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 37204	. 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 37198	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)))
36		eqid 2622	. . . . . . . . 9 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
37		eqid 2622	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
38	1, 18, 36, 10, 12, 2, 6, 37	hdmapevec2 37128	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
39	38	oveq2d 6666	. . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1_r‘𝑅)))
40	6	lmodring 18871	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
41	13, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	7, 9, 37	ringridm 18572	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × (1_r‘𝑅)) = 𝑛)
43	41, 14, 42	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝑛 × (1_r‘𝑅)) = 𝑛)
44	39, 43	eqtrd 2656	. . . . . 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 37210	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 )
47	44, 46	oveq12d 6668	. . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 ))
48		ringgrp 18552	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
49	41, 48	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
50	7, 8, 45	grprid 17453	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵) → (𝑛 ✚ 0 ) = 𝑛)
51	49, 14, 50	syl2anc 693	. . . . 5 ⊢ (𝜑 → (𝑛 ✚ 0 ) = 𝑛)
52	35, 47, 51	3eqtrd 2660	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = 𝑛)
53	52	oveq1d 6665	. . 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 37198	. . . 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 37211	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 )
56	55	oveq2d 6666	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 ))
57	7, 9, 45	ringrz 18588	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × 0 ) = 0 )
58	41, 14, 57	syl2anc 693	. . . . . 6 ⊢ (𝜑 → (𝑛 × 0 ) = 0 )
59	56, 58	eqtrd 2656	. . . . 5 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = 0 )
60	59	oveq1d 6665	. . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦)))
61	1, 2, 3, 6, 7, 10, 12, 28, 32	hdmapipcl 37197	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵)
62	7, 8, 45	grplid 17452	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
63	49, 61, 62	syl2anc 693	. . . 4 ⊢ (𝜑 → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
64	54, 60, 63	3eqtrd 2660	. . 3 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑆‘𝑥)‘𝑦))
65	53, 64	oveq12d 6668	. 2 ⊢ (𝜑 → ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
66	34, 65	eqtrd 2656	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 384 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 {csn 4177 ⟨cop 4183 I cid 5023 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +_gcplusg 15941 ._rcmulr 15942 Scalarcsca 15944 ·_𝑠 cvsca 15945 0_gc0g 16100 Grpcgrp 17422 LSSumclsm 18049 1_rcur 18501 Ringcrg 18547 LModclmod 18863 LSpanclspn 18971 HLchlt 34637 LHypclh 35270 LTrncltrn 35387 DVecHcdvh 36367 ocHcoch 36636 HVMapchvm 37045 HDMapchdma 37082 HGMapchg 37175

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-riotaBAD 34239

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-ot 4186 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-undef 7399 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-0g 16102 df-mre 16246 df-mrc 16247 df-acs 16249 df-preset 16928 df-poset 16946 df-plt 16958 df-lub 16974 df-glb 16975 df-join 16976 df-meet 16977 df-p0 17039 df-p1 17040 df-lat 17046 df-clat 17108 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-subg 17591 df-cntz 17750 df-oppg 17776 df-lsm 18051 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-lmod 18865 df-lss 18933 df-lsp 18972 df-lvec 19103 df-lsatoms 34263 df-lshyp 34264 df-lcv 34306 df-lfl 34345 df-lkr 34373 df-ldual 34411 df-oposet 34463 df-ol 34465 df-oml 34466 df-covers 34553 df-ats 34554 df-atl 34585 df-cvlat 34609 df-hlat 34638 df-llines 34784 df-lplanes 34785 df-lvols 34786 df-lines 34787 df-psubsp 34789 df-pmap 34790 df-padd 35082 df-lhyp 35274 df-laut 35275 df-ldil 35390 df-ltrn 35391 df-trl 35446 df-tgrp 36031 df-tendo 36043 df-edring 36045 df-dveca 36291 df-disoa 36318 df-dvech 36368 df-dib 36428 df-dic 36462 df-dih 36518 df-doch 36637 df-djh 36684 df-lcdual 36876 df-mapd 36914 df-hvmap 37046 df-hdmap1 37083 df-hdmap 37084 df-hgmap 37176

This theorem is referenced by: hdmapglem7 37221

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