hdmapglem7b - Mathbox for Norm Megill

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Theorem hdmapglem7b 41911

Description: Lemma for hdmapg 41913. (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 41093	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
14		hdmapglem7b.l	. . . . 5 ⊢ (𝜑 → 𝑛 ∈ 𝐵)
15		eqid 2735	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2735	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
17		eqid 2735	. . . . . . 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 41095	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	19	eldifad 3975	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
21	3, 6, 5, 7	lmodvscl 20893	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝑛 · 𝐸) ∈ 𝑉)
22	13, 14, 20, 21	syl3anc 1370	. . . 4 ⊢ (𝜑 → (𝑛 · 𝐸) ∈ 𝑉)
23	20	snssd 4814	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
24		hdmapglem7.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
25	1, 2, 3, 24	dochssv 41338	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
26	12, 23, 25	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
27		hdmapglem7b.v	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))
28	26, 27	sseldd 3996	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
29	3, 4	lmodvacl 20890	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑛 · 𝐸) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
30	13, 22, 28, 29	syl3anc 1370	. . 3 ⊢ (𝜑 → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
31		hdmapglem7b.u	. . . 4 ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))
32	26, 31	sseldd 3996	. . 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 41895	. 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 41889	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)))
36		eqid 2735	. . . . . . . . 9 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
37		eqid 2735	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
38	1, 18, 36, 10, 12, 2, 6, 37	hdmapevec2 41819	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
39	38	oveq2d 7447	. . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1_r‘𝑅)))
40	6	lmodring 20883	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
41	13, 40	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	7, 9, 37	ringridm 20284	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × (1_r‘𝑅)) = 𝑛)
43	41, 14, 42	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝑛 × (1_r‘𝑅)) = 𝑛)
44	39, 43	eqtrd 2775	. . . . . 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 41901	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 )
47	44, 46	oveq12d 7449	. . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 ))
48		ringgrp 20256	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
49	41, 48	syl 17	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
50	7, 8, 45	grprid 18999	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵) → (𝑛 ✚ 0 ) = 𝑛)
51	49, 14, 50	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑛 ✚ 0 ) = 𝑛)
52	35, 47, 51	3eqtrd 2779	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = 𝑛)
53	52	oveq1d 7446	. . 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 41889	. . . 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 41902	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 )
56	55	oveq2d 7447	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 ))
57	7, 9, 45	ringrz 20308	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × 0 ) = 0 )
58	41, 14, 57	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑛 × 0 ) = 0 )
59	56, 58	eqtrd 2775	. . . . 5 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = 0 )
60	59	oveq1d 7446	. . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦)))
61	1, 2, 3, 6, 7, 10, 12, 28, 32	hdmapipcl 41888	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵)
62	7, 8, 45	grplid 18998	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
63	49, 61, 62	syl2anc 584	. . . 4 ⊢ (𝜑 → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
64	54, 60, 63	3eqtrd 2779	. . 3 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑆‘𝑥)‘𝑦))
65	53, 64	oveq12d 7449	. 2 ⊢ (𝜑 → ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
66	34, 65	eqtrd 2775	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 395 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 {csn 4631 ⟨cop 4637 I cid 5582 ↾ cres 5691 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +_gcplusg 17298 ._rcmulr 17299 Scalarcsca 17301 ·_𝑠 cvsca 17302 0_gc0g 17486 Grpcgrp 18964 LSSumclsm 19667 1_rcur 20199 Ringcrg 20251 LModclmod 20875 LSpanclspn 20987 HLchlt 39332 LHypclh 39967 LTrncltrn 40084 DVecHcdvh 41061 ocHcoch 41330 HVMapchvm 41739 HDMapchdma 41775 HGMapchg 41866

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-riotaBAD 38935

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-of 7697 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-undef 8297 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-0g 17488 df-mre 17631 df-mrc 17632 df-acs 17634 df-proset 18352 df-poset 18371 df-plt 18388 df-lub 18404 df-glb 18405 df-join 18406 df-meet 18407 df-p0 18483 df-p1 18484 df-lat 18490 df-clat 18557 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-cntz 19348 df-oppg 19377 df-lsm 19669 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-nzr 20530 df-rlreg 20711 df-domn 20712 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-lvec 21120 df-lsatoms 38958 df-lshyp 38959 df-lcv 39001 df-lfl 39040 df-lkr 39068 df-ldual 39106 df-oposet 39158 df-ol 39160 df-oml 39161 df-covers 39248 df-ats 39249 df-atl 39280 df-cvlat 39304 df-hlat 39333 df-llines 39481 df-lplanes 39482 df-lvols 39483 df-lines 39484 df-psubsp 39486 df-pmap 39487 df-padd 39779 df-lhyp 39971 df-laut 39972 df-ldil 40087 df-ltrn 40088 df-trl 40142 df-tgrp 40726 df-tendo 40738 df-edring 40740 df-dveca 40986 df-disoa 41012 df-dvech 41062 df-dib 41122 df-dic 41156 df-dih 41212 df-doch 41331 df-djh 41378 df-lcdual 41570 df-mapd 41608 df-hvmap 41740 df-hdmap1 41776 df-hdmap 41777 df-hgmap 41867

This theorem is referenced by: hdmapglem7 41912

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