hdmapglem7a - Mathbox for Norm Megill

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Theorem hdmapglem7a 40390

Description: Lemma for hdmapg 40393. (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 39573	. . . . . . 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 39575	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
15	14	eldifad 3922	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ 𝑉)
16		hdmapglem7.n	. . . . . . . 8 ⊢ 𝑁 = (LSpan‘𝑈)
17	5, 6, 16	lspsncl 20438	. . . . . . 7 ⊢ ((𝑈 ∈ LMod ∧ 𝐸 ∈ 𝑉) → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
18	9, 15, 17	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (LSubSp‘𝑈))
19	15	snssd 4769	. . . . . . . . 9 ⊢ (𝜑 → {𝐸} ⊆ 𝑉)
20	2, 4, 3, 5, 16, 8, 19	dochocsp 39842	. . . . . . . 8 ⊢ (𝜑 → (𝑂‘(𝑁‘{𝐸})) = (𝑂‘{𝐸}))
21	20	fveq2d 6846	. . . . . . 7 ⊢ (𝜑 → (𝑂‘(𝑂‘(𝑁‘{𝐸}))) = (𝑂‘(𝑂‘{𝐸})))
22	2, 4, 3, 5, 16, 8, 15	dochocsn 39844	. . . . . . 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 39931	. . . . 5 ⊢ (𝜑 → ((𝑁‘{𝐸}) ⊕ (𝑂‘(𝑁‘{𝐸}))) = 𝑉)
25	20	oveq2d 7373	. . . . 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 2840	. . 3 ⊢ (𝜑 → 𝑋 ∈ ((𝑁‘{𝐸}) ⊕ (𝑂‘{𝐸})))
28	6	lsssssubg 20419	. . . . . 6 ⊢ (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
29	9, 28	syl 17	. . . . 5 ⊢ (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
30	29, 18	sseldd 3945	. . . 4 ⊢ (𝜑 → (𝑁‘{𝐸}) ∈ (SubGrp‘𝑈))
31	2, 4, 5, 6, 3	dochlss 39817	. . . . . 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 3945	. . . 4 ⊢ (𝜑 → (𝑂‘{𝐸}) ∈ (SubGrp‘𝑈))
34		hdmapglem7.p	. . . . 5 ⊢ + = (+_g‘𝑈)
35	34, 7	lsmelval 19431	. . . 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 3273	. . 3 ⊢ (∃𝑎 ∈ (𝑁‘{𝐸})∃𝑢 ∈ (𝑂‘{𝐸})𝑋 = (𝑎 + 𝑢) ↔ ∃𝑢 ∈ (𝑂‘{𝐸})∃𝑎 ∈ (𝑁‘{𝐸})𝑋 = (𝑎 + 𝑢))
39		df-rex 3074	. . . . 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 20464	. . . . . . . . . 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 3185	. . . . . . . 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 1924	. . . . . 6 ⊢ (𝜑 → (∃𝑎(𝑎 ∈ (𝑁‘{𝐸}) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢))))
49		rexcom4 3271	. . . . . . 7 ⊢ (∃𝑘 ∈ 𝐵 ∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ ∃𝑎∃𝑘 ∈ 𝐵 (𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)))
50		ovex 7390	. . . . . . . . 9 ⊢ (𝑘 · 𝐸) ∈ V
51		oveq1 7364	. . . . . . . . . 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 3494	. . . . . . . 8 ⊢ (∃𝑎(𝑎 = (𝑘 · 𝐸) ∧ 𝑋 = (𝑎 + 𝑢)) ↔ 𝑋 = ((𝑘 · 𝐸) + 𝑢))
54	53	rexbii 3097	. . . . . . 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 3175	. . 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 1541 ∃wex 1781 ∈ wcel 2106 ∃wrex 3073 ⊆ wss 3910 {csn 4586 ⟨cop 4592 I cid 5530 ↾ cres 5635 ‘cfv 6496 (class class class)co 7357 Basecbs 17083 +_gcplusg 17133 Scalarcsca 17136 ·_𝑠 cvsca 17137 0_gc0g 17321 SubGrpcsubg 18922 LSSumclsm 19416 LModclmod 20322 LSubSpclss 20392 LSpanclspn 20432 HLchlt 37812 LHypclh 38447 LTrncltrn 38564 DVecHcdvh 39541 ocHcoch 39810

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-riotaBAD 37415

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-tpos 8157 df-undef 8204 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-0g 17323 df-mre 17466 df-mrc 17467 df-acs 17469 df-proset 18184 df-poset 18202 df-plt 18219 df-lub 18235 df-glb 18236 df-join 18237 df-meet 18238 df-p0 18314 df-p1 18315 df-lat 18321 df-clat 18388 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-grp 18751 df-minusg 18752 df-sbg 18753 df-subg 18925 df-cntz 19097 df-oppg 19124 df-lsm 19418 df-cmn 19564 df-abl 19565 df-mgp 19897 df-ur 19914 df-ring 19966 df-oppr 20049 df-dvdsr 20070 df-unit 20071 df-invr 20101 df-dvr 20112 df-drng 20187 df-lmod 20324 df-lss 20393 df-lsp 20433 df-lvec 20564 df-lsatoms 37438 df-lcv 37481 df-oposet 37638 df-ol 37640 df-oml 37641 df-covers 37728 df-ats 37729 df-atl 37760 df-cvlat 37784 df-hlat 37813 df-llines 37961 df-lplanes 37962 df-lvols 37963 df-lines 37964 df-psubsp 37966 df-pmap 37967 df-padd 38259 df-lhyp 38451 df-laut 38452 df-ldil 38567 df-ltrn 38568 df-trl 38622 df-tgrp 39206 df-tendo 39218 df-edring 39220 df-dveca 39466 df-disoa 39492 df-dvech 39542 df-dib 39602 df-dic 39636 df-dih 39692 df-doch 39811 df-djh 39858

This theorem is referenced by: hdmapglem7 40392

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