hdmapglem7b - Mathbox for Norm Megill

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Theorem hdmapglem7b 42965

Description: Lemma for hdmapg 42967. (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 42147	. . . 4 ⊢ (𝜑 → 𝑈 ∈ LMod)
14		hdmapglem7b.l	. . . . 5 ⊢ (𝜑 → 𝑛 ∈ 𝐵)
15		eqid 2761	. . . . . . 7 ⊢ (Base‘𝐾) = (Base‘𝐾)
16		eqid 2761	. . . . . . 7 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
17		eqid 2761	. . . . . . 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 42149	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
20	19	eldifad 3911	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
21	3, 6, 5, 7	lmodvscl 21146	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝑛 ∈ 𝐵 ∧ 𝐸 ∈ 𝑉) → (𝑛 · 𝐸) ∈ 𝑉)
22	13, 14, 20, 21	syl3anc 1398	. . . 4 ⊢ (𝜑 → (𝑛 · 𝐸) ∈ 𝑉)
23	20	snssd 4747	. . . . . 6 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
24		hdmapglem7.o	. . . . . . 7 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
25	1, 2, 3, 24	dochssv 42392	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ⊆ 𝑉)
26	12, 23, 25	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ⊆ 𝑉)
27		hdmapglem7b.v	. . . . 5 ⊢ (𝜑 → 𝑦 ∈ (𝑂‘{𝐸}))
28	26, 27	sseldd 3932	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝑉)
29	3, 4	lmodvacl 21143	. . . 4 ⊢ ((𝑈 ∈ LMod ∧ (𝑛 · 𝐸) ∈ 𝑉 ∧ 𝑦 ∈ 𝑉) → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
30	13, 22, 28, 29	syl3anc 1398	. . 3 ⊢ (𝜑 → ((𝑛 · 𝐸) + 𝑦) ∈ 𝑉)
31		hdmapglem7b.u	. . . 4 ⊢ (𝜑 → 𝑥 ∈ (𝑂‘{𝐸}))
32	26, 31	sseldd 3932	. . 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 42949	. 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 42943	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)))
36		eqid 2761	. . . . . . . . 9 ⊢ ((HVMap‘𝐾)‘𝑊) = ((HVMap‘𝐾)‘𝑊)
37		eqid 2761	. . . . . . . . 9 ⊢ (1_r‘𝑅) = (1_r‘𝑅)
38	1, 18, 36, 10, 12, 2, 6, 37	hdmapevec2 42873	. . . . . . . 8 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝐸) = (1_r‘𝑅))
39	38	oveq2d 7434	. . . . . . 7 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝐸)‘𝐸)) = (𝑛 × (1_r‘𝑅)))
40	6	lmodring 21136	. . . . . . . . 9 ⊢ (𝑈 ∈ LMod → 𝑅 ∈ Ring)
41	13, 40	syl 18	. . . . . . . 8 ⊢ (𝜑 → 𝑅 ∈ Ring)
42	7, 9, 37	ringridm 20492	. . . . . . . 8 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × (1_r‘𝑅)) = 𝑛)
43	41, 14, 42	syl2anc 596	. . . . . . 7 ⊢ (𝜑 → (𝑛 × (1_r‘𝑅)) = 𝑛)
44	39, 43	eqtrd 2796	. . . . . 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 42955	. . . . . 6 ⊢ (𝜑 → ((𝑆‘𝐸)‘𝑦) = 0 )
47	44, 46	oveq12d 7436	. . . . 5 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝐸)‘𝐸)) ✚ ((𝑆‘𝐸)‘𝑦)) = (𝑛 ✚ 0 ))
48		ringgrp 20457	. . . . . . 7 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp)
49	41, 48	syl 18	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ Grp)
50	7, 8, 45	grprid 19172	. . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑛 ∈ 𝐵) → (𝑛 ✚ 0 ) = 𝑛)
51	49, 14, 50	syl2anc 596	. . . . 5 ⊢ (𝜑 → (𝑛 ✚ 0 ) = 𝑛)
52	35, 47, 51	3eqtrd 2800	. . . 4 ⊢ (𝜑 → ((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) = 𝑛)
53	52	oveq1d 7433	. . 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 42943	. . . 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 42956	. . . . . . 7 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝐸) = 0 )
56	55	oveq2d 7434	. . . . . 6 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = (𝑛 × 0 ))
57	7, 9, 45	ringrz 20518	. . . . . . 7 ⊢ ((𝑅 ∈ Ring ∧ 𝑛 ∈ 𝐵) → (𝑛 × 0 ) = 0 )
58	41, 14, 57	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑛 × 0 ) = 0 )
59	56, 58	eqtrd 2796	. . . . 5 ⊢ (𝜑 → (𝑛 × ((𝑆‘𝑥)‘𝐸)) = 0 )
60	59	oveq1d 7433	. . . 4 ⊢ (𝜑 → ((𝑛 × ((𝑆‘𝑥)‘𝐸)) ✚ ((𝑆‘𝑥)‘𝑦)) = ( 0 ✚ ((𝑆‘𝑥)‘𝑦)))
61	1, 2, 3, 6, 7, 10, 12, 28, 32	hdmapipcl 42942	. . . . 5 ⊢ (𝜑 → ((𝑆‘𝑥)‘𝑦) ∈ 𝐵)
62	7, 8, 45	grplid 19171	. . . . 5 ⊢ ((𝑅 ∈ Grp ∧ ((𝑆‘𝑥)‘𝑦) ∈ 𝐵) → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
63	49, 61, 62	syl2anc 596	. . . 4 ⊢ (𝜑 → ( 0 ✚ ((𝑆‘𝑥)‘𝑦)) = ((𝑆‘𝑥)‘𝑦))
64	54, 60, 63	3eqtrd 2800	. . 3 ⊢ (𝜑 → ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦)) = ((𝑆‘𝑥)‘𝑦))
65	53, 64	oveq12d 7436	. 2 ⊢ (𝜑 → ((((𝑆‘𝐸)‘((𝑛 · 𝐸) + 𝑦)) × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘((𝑛 · 𝐸) + 𝑦))) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))
66	34, 65	eqtrd 2796	1 ⊢ (𝜑 → ((𝑆‘((𝑚 · 𝐸) + 𝑥))‘((𝑛 · 𝐸) + 𝑦)) = ((𝑛 × (𝐺‘𝑚)) ✚ ((𝑆‘𝑥)‘𝑦)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ⊆ wss 3899 {csn 4584 ⟨cop 4590 I cid 5545 ↾ cres 5653 ‘cfv 6537 (class class class)co 7418 Basecbs 17380 +_gcplusg 17421 ._rcmulr 17422 Scalarcsca 17424 ·_𝑠 cvsca 17425 0_gc0g 17603 Grpcgrp 19137 LSSumclsm 19841 1_rcur 20400 Ringcrg 20452 LModclmod 21128 LSpanclspn 21239 HLchlt 40387 LHypclh 41021 LTrncltrn 41138 DVecHcdvh 42115 ocHcoch 42384 HVMapchvm 42793 HDMapchdma 42829 HGMapchg 42920

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-riotaBAD 39990

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-ot 4593 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-tpos 8236 df-undef 8283 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-n0 12600 df-z 12687 df-uz 12959 df-fz 13633 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-sca 17437 df-vsca 17438 df-0g 17605 df-mre 17749 df-mrc 17750 df-acs 17752 df-proset 18461 df-poset 18480 df-plt 18495 df-lub 18511 df-glb 18512 df-join 18513 df-meet 18514 df-p0 18590 df-p1 18591 df-lat 18599 df-clat 18666 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-sbg 19142 df-subg 19326 df-cntz 19524 df-oppg 19553 df-lsm 19843 df-cmn 19989 df-abl 19990 df-mgp 20354 df-rng 20368 df-ur 20401 df-ring 20454 df-oppr 20560 df-dvdsr 20580 df-unit 20581 df-invr 20611 df-dvr 20624 df-nzr 20756 df-rlreg 20939 df-domn 20940 df-drng 20975 df-lmod 21130 df-lss 21200 df-lsp 21240 df-lvec 21371 df-lsatoms 40013 df-lshyp 40014 df-lcv 40056 df-lfl 40095 df-lkr 40123 df-ldual 40161 df-oposet 40213 df-ol 40215 df-oml 40216 df-covers 40303 df-ats 40304 df-atl 40335 df-cvlat 40359 df-hlat 40388 df-llines 40535 df-lplanes 40536 df-lvols 40537 df-lines 40538 df-psubsp 40540 df-pmap 40541 df-padd 40833 df-lhyp 41025 df-laut 41026 df-ldil 41141 df-ltrn 41142 df-trl 41196 df-tgrp 41780 df-tendo 41792 df-edring 41794 df-dveca 42040 df-disoa 42066 df-dvech 42116 df-dib 42176 df-dic 42210 df-dih 42266 df-doch 42385 df-djh 42432 df-lcdual 42624 df-mapd 42662 df-hvmap 42794 df-hdmap1 42830 df-hdmap 42831 df-hgmap 42921

This theorem is used by: hdmapglem7 42966

W3C validator