hlhilhillem - Mathbox for Norm Megill

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Theorem hlhilhillem 37754

Description: Lemma for hlhil 23414. (Contributed by NM, 23-Jun-2015.)

**Hypotheses**
Ref	Expression
hlhilphl.h	⊢ 𝐻 = (LHyp‘𝐾)
hlhilphllem.u	⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)
hlhilphl.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
hlhilphllem.f	⊢ 𝐹 = (Scalar‘𝑈)
hlhilphllem.l	⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)
hlhilphllem.v	⊢ 𝑉 = (Base‘𝐿)
hlhilphllem.a	⊢ + = (+_g‘𝐿)
hlhilphllem.s	⊢ · = ( ·_𝑠 ‘𝐿)
hlhilphllem.r	⊢ 𝑅 = (Scalar‘𝐿)
hlhilphllem.b	⊢ 𝐵 = (Base‘𝑅)
hlhilphllem.p	⊢ ⨣ = (+_g‘𝑅)
hlhilphllem.t	⊢ × = (._r‘𝑅)
hlhilphllem.q	⊢ 𝑄 = (0_g‘𝑅)
hlhilphllem.z	⊢ 0 = (0_g‘𝐿)
hlhilphllem.i	⊢ , = (·_𝑖‘𝑈)
hlhilphllem.j	⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
hlhilphllem.g	⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
hlhilphllem.e	⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥))
hlhilphllem.o	⊢ 𝑂 = (ocv‘𝑈)
hlhilphllem.c	⊢ 𝐶 = (CSubSp‘𝑈)

**Assertion**
Ref	Expression
hlhilhillem	⊢ (𝜑 → 𝑈 ∈ Hil)

Distinct variable groups: 𝑥,𝑦,𝐾 𝑥,𝑈 𝑥,𝑊,𝑦 𝜑,𝑥 𝑥,𝐽,𝑦 𝑥,𝑉,𝑦 𝑥,𝐶 Allowed substitution hints: 𝜑(𝑦) 𝐵(𝑥,𝑦) 𝐶(𝑦) + (𝑥,𝑦) ⨣ (𝑥,𝑦) 𝑄(𝑥,𝑦) 𝑅(𝑥,𝑦) · (𝑥,𝑦) × (𝑥,𝑦) 𝑈(𝑦) 𝐸(𝑥,𝑦) 𝐹(𝑥,𝑦) 𝐺(𝑥,𝑦) 𝐻(𝑥,𝑦) , (𝑥,𝑦) 𝐿(𝑥,𝑦) 𝑂(𝑥,𝑦) 0 (𝑥,𝑦)

**Proof of Theorem hlhilhillem**
Step	Hyp	Ref	Expression
1		hlhilphl.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		hlhilphllem.u	. . 3 ⊢ 𝑈 = ((HLHil‘𝐾)‘𝑊)
3		hlhilphl.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4		hlhilphllem.f	. . 3 ⊢ 𝐹 = (Scalar‘𝑈)
5		hlhilphllem.l	. . 3 ⊢ 𝐿 = ((DVecH‘𝐾)‘𝑊)
6		hlhilphllem.v	. . 3 ⊢ 𝑉 = (Base‘𝐿)
7		hlhilphllem.a	. . 3 ⊢ + = (+_g‘𝐿)
8		hlhilphllem.s	. . 3 ⊢ · = ( ·_𝑠 ‘𝐿)
9		hlhilphllem.r	. . 3 ⊢ 𝑅 = (Scalar‘𝐿)
10		hlhilphllem.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
11		hlhilphllem.p	. . 3 ⊢ ⨣ = (+_g‘𝑅)
12		hlhilphllem.t	. . 3 ⊢ × = (._r‘𝑅)
13		hlhilphllem.q	. . 3 ⊢ 𝑄 = (0_g‘𝑅)
14		hlhilphllem.z	. . 3 ⊢ 0 = (0_g‘𝐿)
15		hlhilphllem.i	. . 3 ⊢ , = (·_𝑖‘𝑈)
16		hlhilphllem.j	. . 3 ⊢ 𝐽 = ((HDMap‘𝐾)‘𝑊)
17		hlhilphllem.g	. . 3 ⊢ 𝐺 = ((HGMap‘𝐾)‘𝑊)
18		hlhilphllem.e	. . 3 ⊢ 𝐸 = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ ((𝐽‘𝑦)‘𝑥))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18	hlhilphllem 37753	. 2 ⊢ (𝜑 → 𝑈 ∈ PreHil)
20	3	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		eqid 2760	. . . . . . 7 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
22		hlhilphllem.o	. . . . . . 7 ⊢ 𝑂 = (ocv‘𝑈)
23		eqid 2760	. . . . . . . . . . 11 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
24		hlhilphllem.c	. . . . . . . . . . 11 ⊢ 𝐶 = (CSubSp‘𝑈)
25	1, 23, 2, 24, 3	hlhillcs 37752	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ran ((DIsoH‘𝐾)‘𝑊))
26	25	eleq2d 2825	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)))
27	26	biimpa 502	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊))
28	1, 5, 23, 6	dihrnss 37069	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
29	3, 28	sylan 489	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
30	27, 29	syldan 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ⊆ 𝑉)
31	1, 5, 2, 20, 6, 21, 22, 30	hlhilocv 37751	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑂‘𝑥) = (((ocH‘𝐾)‘𝑊)‘𝑥))
32	31	oveq2d 6829	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)))
33		eqid 2760	. . . . . . . 8 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	1, 5, 2, 3, 33	hlhillsm 37750	. . . . . . 7 ⊢ (𝜑 → (LSSum‘𝐿) = (LSSum‘𝑈))
35	34	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (LSSum‘𝐿) = (LSSum‘𝑈))
36	35	oveqd 6830	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝑈)(𝑂‘𝑥)))
37		eqid 2760	. . . . . . 7 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
38	3	adantr 472	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
39	1, 5, 23, 37	dihrnlss 37068	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
40	3, 39	sylan 489	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
41	1, 23, 5, 6, 21, 38, 29	dochoccl 37160	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))
42	41	biimpd 219	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))
43	42	ex 449	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥)))
44	43	pm2.43d 53	. . . . . . . 8 ⊢ (𝜑 → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))
45	44	imp 444	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥)
46	1, 21, 5, 6, 37, 33, 38, 40, 45	dochexmid 37259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
47	27, 46	syldan 488	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
48	32, 36, 47	3eqtr3d 2802	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = 𝑉)
49	1, 2, 3, 5, 6	hlhilbase 37730	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
50	49	adantr 472	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑉 = (Base‘𝑈))
51	48, 50	eqtrd 2794	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
52	51	ralrimiva 3104	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
53		eqid 2760	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
54		eqid 2760	. . 3 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	53, 54, 22, 24	ishil2 20265	. 2 ⊢ (𝑈 ∈ Hil ↔ (𝑈 ∈ PreHil ∧ ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈)))
56	19, 52, 55	sylanbrc 701	1 ⊢ (𝜑 → 𝑈 ∈ Hil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∀wral 3050 ⊆ wss 3715 ran crn 5267 ‘cfv 6049 (class class class)co 6813 ↦ cmpt2 6815 Basecbs 16059 +_gcplusg 16143 ._rcmulr 16144 Scalarcsca 16146 ·_𝑠 cvsca 16147 ·_𝑖cip 16148 0_gc0g 16302 LSSumclsm 18249 LSubSpclss 19134 PreHilcphl 20171 ocvcocv 20206 CSubSpccss 20207 Hilchs 20247 HLchlt 35140 LHypclh 35773 DVecHcdvh 36869 DIsoHcdih 37019 ocHcoch 37138 HDMapchdma 37584 HGMapchg 37677 HLHilchlh 37726

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-riotaBAD 34742

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-ot 4330 df-uni 4589 df-int 4628 df-iun 4674 df-iin 4675 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-of 7062 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-undef 7568 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-n0 11485 df-z 11570 df-uz 11880 df-fz 12520 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-0g 16304 df-mre 16448 df-mrc 16449 df-acs 16451 df-preset 17129 df-poset 17147 df-plt 17159 df-lub 17175 df-glb 17176 df-join 17177 df-meet 17178 df-p0 17240 df-p1 17241 df-lat 17247 df-clat 17309 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 df-grp 17626 df-minusg 17627 df-sbg 17628 df-subg 17792 df-ghm 17859 df-cntz 17950 df-oppg 17976 df-lsm 18251 df-pj1 18252 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-rnghom 18917 df-drng 18951 df-subrg 18980 df-staf 19047 df-srng 19048 df-lmod 19067 df-lss 19135 df-lsp 19174 df-lmhm 19224 df-lvec 19305 df-sra 19374 df-rgmod 19375 df-phl 20173 df-ocv 20209 df-css 20210 df-pj 20249 df-hil 20250 df-lsatoms 34766 df-lshyp 34767 df-lcv 34809 df-lfl 34848 df-lkr 34876 df-ldual 34914 df-oposet 34966 df-ol 34968 df-oml 34969 df-covers 35056 df-ats 35057 df-atl 35088 df-cvlat 35112 df-hlat 35141 df-llines 35287 df-lplanes 35288 df-lvols 35289 df-lines 35290 df-psubsp 35292 df-pmap 35293 df-padd 35585 df-lhyp 35777 df-laut 35778 df-ldil 35893 df-ltrn 35894 df-trl 35949 df-tgrp 36533 df-tendo 36545 df-edring 36547 df-dveca 36793 df-disoa 36820 df-dvech 36870 df-dib 36930 df-dic 36964 df-dih 37020 df-doch 37139 df-djh 37186 df-lcdual 37378 df-mapd 37416 df-hvmap 37548 df-hdmap1 37585 df-hdmap 37586 df-hgmap 37678 df-hlhil 37727

This theorem is referenced by: hlathil 37755

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