hdmapglem7a - Mathbox for Norm Megill

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Theorem hdmapglem7a 40188

Description: Lemma for hdmapg 40191. (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)

**Assertion**
Ref	Expression
hdmapglem7a	⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))

Distinct variable groups: 𝑢,𝑘, + 𝐵,𝑘,𝑢 𝑘,𝐸,𝑢 𝑘,𝑁,𝑢 𝑘,𝑂,𝑢 · ,𝑘,𝑢 𝑅,𝑘 𝑈,𝑘,𝑢 𝑘,𝑉 𝑘,𝑋,𝑢 𝜑,𝑘,𝑢 Allowed substitution hints: ⊕ (𝑢,𝑘) 𝑅(𝑢) 𝐻(𝑢,𝑘) 𝐾(𝑢,𝑘) 𝑉(𝑢) 𝑊(𝑢,𝑘)

Proof of Theorem hdmapglem7a Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hdmapglem7.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
2		hdmapglem7.h	. . . . . 6 ⊢ 𝐻 = (LHyp‘𝐾)
3		hdmapglem7.o	. . . . . 6 ⊢ 𝑂 = ((ocH‘𝐾)‘𝑊)
4		hdmapglem7.u	. . . . . 6 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
5		hdmapglem7.v	. . . . . 6 ⊢ 𝑉 = (Base‘𝑈)
6		eqid 2736	. . . . . 6 ⊢ (LSubSp‘𝑈) = (LSubSp‘𝑈)
7		hdmapglem7.a	. . . . . 6 ⊢ ⊕ = (LSSum‘𝑈)
8		hdmapglem7.k	. . . . . 6 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	2, 4, 8	dvhlmod 39371	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		eqid 2736	. . . . . . . . 9 ⊢ (Base‘𝐾) = (Base‘𝐾)
11		eqid 2736	. . . . . . . . 9 ⊢ ((LTrn‘𝐾)‘𝑊) = ((LTrn‘𝐾)‘𝑊)
12		eqid 2736	. . . . . . . . 9 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
13		hdmapglem7.e	. . . . . . . . 9 ⊢ 𝐸 = ⟨( I ↾ (Base‘𝐾)), ( I ↾ ((LTrn‘𝐾)‘𝑊))⟩
14	2, 10, 11, 4, 5, 12, 13, 8	dvheveccl 39373	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
15	14	eldifad 3909	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
16		hdmapglem7.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
17	5, 6, 16	lspsncl 20337	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
18	9, 15, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
19	15	snssd 4755	. . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
20	2, 4, 3, 5, 16, 8, 19	dochocsp 39640	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸}))
21	20	fveq2d 6823	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸})))
22	2, 4, 3, 5, 16, 8, 15	dochocsn 39642	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘{𝐸})) = (𝑁‘{𝐸}))
23	21, 22	eqtrd 2776	. . . . . 6 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑁‘{𝐸}))
24	2, 3, 4, 5, 6, 7, 8, 18, 23	dochexmid 39729	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉)
25	20	oveq2d 7345	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
26	24, 25	eqtr3d 2778	. . . 4 ⊢ (𝜑 → 𝑉 = ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
27	1, 26	eleqtrd 2839	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
28	6	lsssssubg 20318	. . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
29	9, 28	syl 17	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
30	29, 18	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈))
31	2, 4, 5, 6, 3	dochlss 39615	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝐸} ⊆ 𝑉) → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈))
32	8, 19, 31	syl2anc 584	. . . . 5 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (LSubSp‘𝑈))
33	29, 32	sseldd 3932	. . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈))
34		hdmapglem7.p	. . . . 5 ⊢ + = (+_g‘𝑈)
35	34, 7	lsmelval 19342	. . . 4 ⊢ (((𝑁‘{𝐸}) ∈ (SubGrp‘𝑈) ∧ (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈)) → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)))
36	30, 33, 35	syl2anc 584	. . 3 ⊢ (𝜑 → (𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})) ↔ ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢)))
37	27, 36	mpbid 231	. 2 ⊢ (𝜑 → ∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢))
38		rexcom 3269	. . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢))
39		df-rex 3071	. . . . 5 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)))
40		hdmapglem7.r	. . . . . . . . . . 11 ⊢ 𝑅 = (Scalar‘𝑈)
41		hdmapglem7.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑅)
42		hdmapglem7.q	. . . . . . . . . . 11 ⊢ · = ( ·_𝑠 ‘𝑈)
43	40, 41, 5, 42, 16	lspsnel 20363	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸)))
44	9, 15, 43	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑎 ∈ (𝑁‘{𝐸}) ↔ ∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸)))
45	44	anbi1d 630	. . . . . . . 8 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
46		r19.41v 3181	. . . . . . . 8 ⊢ (∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ (∃𝑘 ∈ 𝐵 𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
47	45, 46	bitr4di 288	. . . . . . 7 ⊢ (𝜑 → ((𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
48	47	exbidv 1923	. . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
49		rexcom4 3267	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
50		ovex 7362	. . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V
51		oveq1 7336	. . . . . . . . . 10 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑎 + 𝑢) = ((𝑘 · 𝐸) + 𝑢))
52	51	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑎 = (𝑘 · 𝐸) → (𝑋 = (𝑎 + 𝑢) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
53	50, 52	ceqsexv 3488	. . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))
54	53	rexbii 3093	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
55	49, 54	bitr3i 276	. . . . . 6 ⊢ (∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))
56	48, 55	bitrdi 286	. . . . 5 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
57	39, 56	bitrid 282	. . . 4 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
58	57	rexbidv 3171	. . 3 ⊢ (𝜑 → (∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
59	38, 58	bitrid 282	. 2 ⊢ (𝜑 → (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢)))
60	37, 59	mpbid 231	1 ⊢ (𝜑 → ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑘 ∈ 𝐵 𝑋 = ((𝑘 · 𝐸) + 𝑢))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ∃wrex 3070 ⊆ wss 3897 {csn 4572 ⟨cop 4578 I cid 5511 ↾ cres 5616 ‘cfv 6473 (class class class)co 7329 Basecbs 17001 +_gcplusg 17051 Scalarcsca 17054 ·_𝑠 cvsca 17055 0_gc0g 17239 SubGrpcsubg 18837 LSSumclsm 19327 LModclmod 20221 LSubSpclss 20291 LSpanclspn 20331 HLchlt 37610 LHypclh 38245 LTrncltrn 38362 DVecHcdvh 39339 ocHcoch 39608

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-pred 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-mre 17384 df-mrc 17385 df-acs 17387 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-oppg 19038 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lsatoms 37236 df-lcv 37279 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tgrp 39004 df-tendo 39016 df-edring 39018 df-dveca 39264 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609 df-djh 39656

This theorem is referenced by: hdmapglem7 40190

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