lclkrlem1 - Mathbox for Norm Megill

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Theorem lclkrlem1 41465

Description: The set of functionals having closed kernels is closed under scalar product. (Contributed by NM, 28-Dec-2014.)

**Hypotheses**
Ref	Expression
lclkrlem1.h	⊢ 𝐻 = (LHyp‘𝐾)
lclkrlem1.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lclkrlem1.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lclkrlem1.f	⊢ 𝐹 = (LFnl‘𝑈)
lclkrlem1.l	⊢ 𝐿 = (LKer‘𝑈)
lclkrlem1.d	⊢ 𝐷 = (LDual‘𝑈)
lclkrlem1.r	⊢ 𝑅 = (Scalar‘𝑈)
lclkrlem1.b	⊢ 𝐵 = (Base‘𝑅)
lclkrlem1.t	⊢ · = ( ·_𝑠 ‘𝐷)
lclkrlem1.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lclkrlem1.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lclkrlem1.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
lclkrlem1.g	⊢ (𝜑 → 𝐺 ∈ 𝐶)

**Assertion**
Ref	Expression
lclkrlem1	⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Distinct variable groups: 𝑓,𝐹 𝑓,𝐿 ⊥ ,𝑓 · ,𝑓 𝑓,𝐺 𝑓,𝑋 Allowed substitution hints: 𝜑(𝑓) 𝐵(𝑓) 𝐶(𝑓) 𝐷(𝑓) 𝑅(𝑓) 𝑈(𝑓) 𝐻(𝑓) 𝐾(𝑓) 𝑊(𝑓)

**Proof of Theorem lclkrlem1**
Step	Hyp	Ref	Expression
1		lclkrlem1.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
2		lclkrlem1.r	. . 3 ⊢ 𝑅 = (Scalar‘𝑈)
3		lclkrlem1.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
4		lclkrlem1.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
5		lclkrlem1.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝐷)
6		lclkrlem1.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
7		lclkrlem1.u	. . . 4 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
8		lclkrlem1.k	. . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
9	6, 7, 8	dvhlmod 41069	. . 3 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		lclkrlem1.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11		lclkrlem1.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
12		lclkrlem1.c	. . . . . 6 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
13	12	lcfl1lem 41450	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
14	11, 13	sylib 218	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
15	14	simpld 494	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
16	1, 2, 3, 4, 5, 9, 10, 15	ldualvscl 39097	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
17		lclkrlem1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18		eqid 2740	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6, 7, 17, 18, 8	dochoc1 41320	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		fvoveq1 7473	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑅) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘((0_g‘𝑅) · 𝐺)))
22	4, 9	lduallmod 39111	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ LMod)
23		eqid 2740	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
24	1, 4, 23, 9, 15	ldualelvbase 39085	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷))
25		eqid 2740	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
26		eqid 2740	. . . . . . . . . . . 12 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
27		eqid 2740	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
28	23, 25, 5, 26, 27	lmod0vs 20917	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
29	22, 24, 28	syl2anc 583	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
30		eqid 2740	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31	2, 30, 4, 25, 26, 9	ldual0 39105	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = (0_g‘𝑅))
32	31	oveq1d 7465	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = ((0_g‘𝑅) · 𝐺))
33	18, 2, 30, 4, 27, 9	ldual0v 39108	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {(0_g‘𝑅)}))
34	29, 32, 33	3eqtr3d 2788	. . . . . . . . 9 ⊢ (𝜑 → ((0_g‘𝑅) · 𝐺) = ((Base‘𝑈) × {(0_g‘𝑅)}))
35	34	fveq2d 6926	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})))
36		eqid 2740	. . . . . . . . 9 ⊢ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})
37	2, 30, 18, 1	lfl0f 39027	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹)
38		lclkrlem1.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
39	2, 30, 18, 1, 38	lkr0f 39052	. . . . . . . . . 10 ⊢ ((𝑈 ∈ LMod ∧ ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹) → ((𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})))
40	9, 37, 39	syl2anc2 584	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})))
41	36, 40	mpbiri 258	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈))
42	35, 41	eqtrd 2780	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (Base‘𝑈))
43	21, 42	sylan9eqr 2802	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (Base‘𝑈))
44	43	fveq2d 6926	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(Base‘𝑈)))
45	44	fveq2d 6926	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
46	20, 45, 43	3eqtr4d 2790	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
47	14	simprd 495	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
48	47	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
49	6, 7, 8	dvhlvec 41068	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑈 ∈ LVec)
51	15	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝐺 ∈ 𝐹)
52	10	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ∈ 𝐵)
53		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ≠ (0_g‘𝑅))
54	2, 3, 30, 1, 38, 4, 5, 50, 51, 52, 53	ldualkrsc 39125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
55	54	fveq2d 6926	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(𝐿‘𝐺)))
56	55	fveq2d 6926	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
57	48, 56, 54	3eqtr4d 2790	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
58	46, 57	pm2.61dane 3035	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
59	12	lcfl1lem 41450	. 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺))))
60	16, 58, 59	sylanbrc 582	1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 {crab 3443 {csn 4648 × cxp 5698 ‘cfv 6575 (class class class)co 7450 Basecbs 17260 Scalarcsca 17316 ·_𝑠 cvsca 17317 0_gc0g 17501 LModclmod 20882 LVecclvec 21126 LFnlclfn 39015 LKerclk 39043 LDualcld 39081 HLchlt 39308 LHypclh 39943 DVecHcdvh 41037 ocHcoch 41306

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-riotaBAD 38911

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-of 7716 df-om 7906 df-1st 8032 df-2nd 8033 df-tpos 8269 df-undef 8316 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-nn 12296 df-2 12358 df-3 12359 df-4 12360 df-5 12361 df-6 12362 df-n0 12556 df-z 12642 df-uz 12906 df-fz 13570 df-struct 17196 df-sets 17213 df-slot 17231 df-ndx 17243 df-base 17261 df-ress 17290 df-plusg 17326 df-mulr 17327 df-sca 17329 df-vsca 17330 df-0g 17503 df-proset 18367 df-poset 18385 df-plt 18402 df-lub 18418 df-glb 18419 df-join 18420 df-meet 18421 df-p0 18497 df-p1 18498 df-lat 18504 df-clat 18571 df-mgm 18680 df-sgrp 18759 df-mnd 18775 df-submnd 18821 df-grp 18978 df-minusg 18979 df-sbg 18980 df-subg 19165 df-cntz 19359 df-lsm 19680 df-cmn 19826 df-abl 19827 df-mgp 20164 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20362 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-nzr 20541 df-rlreg 20718 df-domn 20719 df-drng 20755 df-lmod 20884 df-lss 20955 df-lsp 20995 df-lvec 21127 df-lfl 39016 df-lkr 39044 df-ldual 39082 df-oposet 39134 df-ol 39136 df-oml 39137 df-covers 39224 df-ats 39225 df-atl 39256 df-cvlat 39280 df-hlat 39309 df-llines 39457 df-lplanes 39458 df-lvols 39459 df-lines 39460 df-psubsp 39462 df-pmap 39463 df-padd 39755 df-lhyp 39947 df-laut 39948 df-ldil 40063 df-ltrn 40064 df-trl 40118 df-tendo 40714 df-edring 40716 df-disoa 40988 df-dvech 41038 df-dib 41098 df-dic 41132 df-dih 41188 df-doch 41307

This theorem is referenced by: lclkr 41492 lclkrslem1 41496

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