hlhilhillem - Mathbox for Norm Megill

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Theorem hlhilhillem 42007

Description: Lemma for hlhil 25370. (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 = (ClSubSp‘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 42006	. 2 ⊢ (𝜑 → 𝑈 ∈ PreHil)
20	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		eqid 2731	. . . . . . 7 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
22		hlhilphllem.o	. . . . . . 7 ⊢ 𝑂 = (ocv‘𝑈)
23		eqid 2731	. . . . . . . . . . 11 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
24		hlhilphllem.c	. . . . . . . . . . 11 ⊢ 𝐶 = (ClSubSp‘𝑈)
25	1, 23, 2, 24, 3	hlhillcs 42005	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ran ((DIsoH‘𝐾)‘𝑊))
26	25	eleq2d 2817	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐶 ↔ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)))
27	26	biimpa 476	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊))
28	1, 5, 23, 6	dihrnss 41325	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
29	3, 28	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
30	27, 29	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ⊆ 𝑉)
31	1, 5, 2, 20, 6, 21, 22, 30	hlhilocv 42004	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑂‘𝑥) = (((ocH‘𝐾)‘𝑊)‘𝑥))
32	31	oveq2d 7362	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)))
33		eqid 2731	. . . . . . . 8 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	1, 5, 2, 3, 33	hlhillsm 42003	. . . . . . 7 ⊢ (𝜑 → (LSSum‘𝐿) = (LSSum‘𝑈))
35	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (LSSum‘𝐿) = (LSSum‘𝑈))
36	35	oveqd 7363	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝑈)(𝑂‘𝑥)))
37		eqid 2731	. . . . . . 7 ⊢ (LSubSp‘𝐿) = (LSubSp‘𝐿)
38	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
39	1, 5, 23, 37	dihrnlss 41324	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
40	3, 39	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
41	1, 23, 5, 6, 21, 38, 29	dochoccl 41416	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) ↔ (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))
42	41	biimpd 229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊) → (((ocH‘𝐾)‘𝑊)‘(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑥))
43	42	ex 412	. . . . . . . . 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 406	. . . . . . 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 41515	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
47	27, 46	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
48	32, 36, 47	3eqtr3d 2774	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = 𝑉)
49	1, 2, 3, 5, 6	hlhilbase 41983	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
50	49	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑉 = (Base‘𝑈))
51	48, 50	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
52	51	ralrimiva 3124	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
53		eqid 2731	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
54		eqid 2731	. . 3 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	53, 54, 22, 24	ishil2 21656	. 2 ⊢ (𝑈 ∈ Hil ↔ (𝑈 ∈ PreHil ∧ ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈)))
56	19, 52, 55	sylanbrc 583	1 ⊢ (𝜑 → 𝑈 ∈ Hil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∀wral 3047 ⊆ wss 3897 ran crn 5615 ‘cfv 6481 (class class class)co 7346 ∈ cmpo 7348 Basecbs 17120 +_gcplusg 17161 ._rcmulr 17162 Scalarcsca 17164 ·_𝑠 cvsca 17165 ·_𝑖cip 17166 0_gc0g 17343 LSSumclsm 19546 LSubSpclss 20864 PreHilcphl 21561 ocvcocv 21597 ClSubSpccss 21598 Hilchil 21638 HLchlt 39397 LHypclh 40031 DVecHcdvh 41125 DIsoHcdih 41275 ocHcoch 41394 HDMapchdma 41839 HGMapchg 41930 HLHilchlh 41979

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 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 39000

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3737 df-csb 3846 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3917 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-ot 4582 df-uni 4857 df-int 4896 df-iun 4941 df-iin 4942 df-br 5090 df-opab 5152 df-mpt 5171 df-tr 5197 df-id 5509 df-eprel 5514 df-po 5522 df-so 5523 df-fr 5567 df-we 5569 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 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-of 7610 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-7 12193 df-8 12194 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-0g 17345 df-mre 17488 df-mrc 17489 df-acs 17491 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-mhm 18691 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-ghm 19125 df-cntz 19229 df-oppg 19258 df-lsm 19548 df-pj1 19549 df-cmn 19694 df-abl 19695 df-mgp 20059 df-rng 20071 df-ur 20100 df-ring 20153 df-oppr 20255 df-dvdsr 20275 df-unit 20276 df-invr 20306 df-dvr 20319 df-rhm 20390 df-nzr 20428 df-subrg 20485 df-rlreg 20609 df-domn 20610 df-drng 20646 df-staf 20754 df-srng 20755 df-lmod 20795 df-lss 20865 df-lsp 20905 df-lmhm 20956 df-lvec 21037 df-sra 21107 df-rgmod 21108 df-phl 21563 df-ocv 21600 df-css 21601 df-pj 21640 df-hil 21641 df-lsatoms 39023 df-lshyp 39024 df-lcv 39066 df-lfl 39105 df-lkr 39133 df-ldual 39171 df-oposet 39223 df-ol 39225 df-oml 39226 df-covers 39313 df-ats 39314 df-atl 39345 df-cvlat 39369 df-hlat 39398 df-llines 39545 df-lplanes 39546 df-lvols 39547 df-lines 39548 df-psubsp 39550 df-pmap 39551 df-padd 39843 df-lhyp 40035 df-laut 40036 df-ldil 40151 df-ltrn 40152 df-trl 40206 df-tgrp 40790 df-tendo 40802 df-edring 40804 df-dveca 41050 df-disoa 41076 df-dvech 41126 df-dib 41186 df-dic 41220 df-dih 41276 df-doch 41395 df-djh 41442 df-lcdual 41634 df-mapd 41672 df-hvmap 41804 df-hdmap1 41840 df-hdmap 41841 df-hgmap 41931 df-hlhil 41980

This theorem is referenced by: hlathil 42008

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