hlhilhillem - Mathbox for Norm Megill

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Theorem hlhilhillem 42336

Description: Lemma for hlhil 25411. (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 42335	. 2 ⊢ (𝜑 → 𝑈 ∈ PreHil)
20	3	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
21		eqid 2737	. . . . . . 7 ⊢ ((ocH‘𝐾)‘𝑊) = ((ocH‘𝐾)‘𝑊)
22		hlhilphllem.o	. . . . . . 7 ⊢ 𝑂 = (ocv‘𝑈)
23		eqid 2737	. . . . . . . . . . 11 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
24		hlhilphllem.c	. . . . . . . . . . 11 ⊢ 𝐶 = (ClSubSp‘𝑈)
25	1, 23, 2, 24, 3	hlhillcs 42334	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 = ran ((DIsoH‘𝐾)‘𝑊))
26	25	eleq2d 2823	. . . . . . . . 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 41654	. . . . . . . . 9 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
29	3, 28	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ⊆ 𝑉)
30	27, 29	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ⊆ 𝑉)
31	1, 5, 2, 20, 6, 21, 22, 30	hlhilocv 42333	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑂‘𝑥) = (((ocH‘𝐾)‘𝑊)‘𝑥))
32	31	oveq2d 7384	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)))
33		eqid 2737	. . . . . . . 8 ⊢ (LSSum‘𝐿) = (LSSum‘𝐿)
34	1, 5, 2, 3, 33	hlhillsm 42332	. . . . . . 7 ⊢ (𝜑 → (LSSum‘𝐿) = (LSSum‘𝑈))
35	34	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (LSSum‘𝐿) = (LSSum‘𝑈))
36	35	oveqd 7385	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(𝑂‘𝑥)) = (𝑥(LSSum‘𝑈)(𝑂‘𝑥)))
37		eqid 2737	. . . . . . 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 41653	. . . . . . . 8 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
40	3, 39	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → 𝑥 ∈ (LSubSp‘𝐿))
41	1, 23, 5, 6, 21, 38, 29	dochoccl 41745	. . . . . . . . . . 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 41844	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ran ((DIsoH‘𝐾)‘𝑊)) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
47	27, 46	syldan 592	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝐿)(((ocH‘𝐾)‘𝑊)‘𝑥)) = 𝑉)
48	32, 36, 47	3eqtr3d 2780	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = 𝑉)
49	1, 2, 3, 5, 6	hlhilbase 42312	. . . . 5 ⊢ (𝜑 → 𝑉 = (Base‘𝑈))
50	49	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑉 = (Base‘𝑈))
51	48, 50	eqtrd 2772	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
52	51	ralrimiva 3130	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈))
53		eqid 2737	. . 3 ⊢ (Base‘𝑈) = (Base‘𝑈)
54		eqid 2737	. . 3 ⊢ (LSSum‘𝑈) = (LSSum‘𝑈)
55	53, 54, 22, 24	ishil2 21686	. 2 ⊢ (𝑈 ∈ Hil ↔ (𝑈 ∈ PreHil ∧ ∀𝑥 ∈ 𝐶 (𝑥(LSSum‘𝑈)(𝑂‘𝑥)) = (Base‘𝑈)))
56	19, 52, 55	sylanbrc 584	1 ⊢ (𝜑 → 𝑈 ∈ Hil)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ⊆ wss 3903 ran crn 5633 ‘cfv 6500 (class class class)co 7368 ∈ cmpo 7370 Basecbs 17148 +_gcplusg 17189 ._rcmulr 17190 Scalarcsca 17192 ·_𝑠 cvsca 17193 ·_𝑖cip 17194 0_gc0g 17371 LSSumclsm 19575 LSubSpclss 20894 PreHilcphl 21591 ocvcocv 21627 ClSubSpccss 21628 Hilchil 21668 HLchlt 39726 LHypclh 40360 DVecHcdvh 41454 DIsoHcdih 41604 ocHcoch 41723 HDMapchdma 42168 HGMapchg 42259 HLHilchlh 42308

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39329

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-ot 4591 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-0g 17373 df-mre 17517 df-mrc 17518 df-acs 17520 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-ghm 19154 df-cntz 19258 df-oppg 19287 df-lsm 19577 df-pj1 19578 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-rhm 20420 df-nzr 20458 df-subrg 20515 df-rlreg 20639 df-domn 20640 df-drng 20676 df-staf 20784 df-srng 20785 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lmhm 20986 df-lvec 21067 df-sra 21137 df-rgmod 21138 df-phl 21593 df-ocv 21630 df-css 21631 df-pj 21670 df-hil 21671 df-lsatoms 39352 df-lshyp 39353 df-lcv 39395 df-lfl 39434 df-lkr 39462 df-ldual 39500 df-oposet 39552 df-ol 39554 df-oml 39555 df-covers 39642 df-ats 39643 df-atl 39674 df-cvlat 39698 df-hlat 39727 df-llines 39874 df-lplanes 39875 df-lvols 39876 df-lines 39877 df-psubsp 39879 df-pmap 39880 df-padd 40172 df-lhyp 40364 df-laut 40365 df-ldil 40480 df-ltrn 40481 df-trl 40535 df-tgrp 41119 df-tendo 41131 df-edring 41133 df-dveca 41379 df-disoa 41405 df-dvech 41455 df-dib 41515 df-dic 41549 df-dih 41605 df-doch 41724 df-djh 41771 df-lcdual 41963 df-mapd 42001 df-hvmap 42133 df-hdmap1 42169 df-hdmap 42170 df-hgmap 42260 df-hlhil 42309

This theorem is referenced by: hlathil 42337

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