lclkrlem1 - Mathbox for Norm Megill

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Theorem lclkrlem1 39767

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 39371	. . 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 39752	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
14	11, 13	sylib 217	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
15	14	simpld 495	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
16	1, 2, 3, 4, 5, 9, 10, 15	ldualvscl 37399	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
17		lclkrlem1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18		eqid 2736	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6, 7, 17, 18, 8	dochoc1 39622	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
20	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		fvoveq1 7352	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑅) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘((0_g‘𝑅) · 𝐺)))
22	4, 9	lduallmod 37413	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ LMod)
23		eqid 2736	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
24	1, 4, 23, 9, 15	ldualelvbase 37387	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷))
25		eqid 2736	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
26		eqid 2736	. . . . . . . . . . . 12 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
27		eqid 2736	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
28	23, 25, 5, 26, 27	lmod0vs 20254	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
29	22, 24, 28	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
30		eqid 2736	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31	2, 30, 4, 25, 26, 9	ldual0 37407	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = (0_g‘𝑅))
32	31	oveq1d 7344	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = ((0_g‘𝑅) · 𝐺))
33	18, 2, 30, 4, 27, 9	ldual0v 37410	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {(0_g‘𝑅)}))
34	29, 32, 33	3eqtr3d 2784	. . . . . . . . 9 ⊢ (𝜑 → ((0_g‘𝑅) · 𝐺) = ((Base‘𝑈) × {(0_g‘𝑅)}))
35	34	fveq2d 6823	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})))
36		eqid 2736	. . . . . . . . 9 ⊢ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})
37	2, 30, 18, 1	lfl0f 37329	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹)
38		lclkrlem1.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
39	2, 30, 18, 1, 38	lkr0f 37354	. . . . . . . . . 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 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})))
41	36, 40	mpbiri 257	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈))
42	35, 41	eqtrd 2776	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (Base‘𝑈))
43	21, 42	sylan9eqr 2798	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (Base‘𝑈))
44	43	fveq2d 6823	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(Base‘𝑈)))
45	44	fveq2d 6823	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
46	20, 45, 43	3eqtr4d 2786	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
47	14	simprd 496	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
48	47	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
49	6, 7, 8	dvhlvec 39370	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑈 ∈ LVec)
51	15	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝐺 ∈ 𝐹)
52	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ∈ 𝐵)
53		simpr 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ≠ (0_g‘𝑅))
54	2, 3, 30, 1, 38, 4, 5, 50, 51, 52, 53	ldualkrsc 37427	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
55	54	fveq2d 6823	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(𝐿‘𝐺)))
56	55	fveq2d 6823	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
57	48, 56, 54	3eqtr4d 2786	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
58	46, 57	pm2.61dane 3029	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
59	12	lcfl1lem 39752	. 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺))))
60	16, 58, 59	sylanbrc 583	1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 {crab 3403 {csn 4572 × cxp 5612 ‘cfv 6473 (class class class)co 7329 Basecbs 17001 Scalarcsca 17054 ·_𝑠 cvsca 17055 0_gc0g 17239 LModclmod 20221 LVecclvec 20462 LFnlclfn 37317 LKerclk 37345 LDualcld 37383 HLchlt 37610 LHypclh 38245 DVecHcdvh 39339 ocHcoch 39608

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5226 ax-sep 5240 ax-nul 5247 ax-pow 5305 ax-pr 5369 ax-un 7642 ax-cnex 11020 ax-resscn 11021 ax-1cn 11022 ax-icn 11023 ax-addcl 11024 ax-addrcl 11025 ax-mulcl 11026 ax-mulrcl 11027 ax-mulcom 11028 ax-addass 11029 ax-mulass 11030 ax-distr 11031 ax-i2m1 11032 ax-1ne0 11033 ax-1rid 11034 ax-rnegex 11035 ax-rrecex 11036 ax-cnre 11037 ax-pre-lttri 11038 ax-pre-lttrn 11039 ax-pre-ltadd 11040 ax-pre-mulgt0 11041 ax-riotaBAD 37213

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3727 df-csb 3843 df-dif 3900 df-un 3902 df-in 3904 df-ss 3914 df-pss 3916 df-nul 4269 df-if 4473 df-pw 4548 df-sn 4573 df-pr 4575 df-tp 4577 df-op 4579 df-uni 4852 df-int 4894 df-iun 4940 df-iin 4941 df-br 5090 df-opab 5152 df-mpt 5173 df-tr 5207 df-id 5512 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5569 df-we 5571 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 6232 df-ord 6299 df-on 6300 df-lim 6301 df-suc 6302 df-iota 6425 df-fun 6475 df-fn 6476 df-f 6477 df-f1 6478 df-fo 6479 df-f1o 6480 df-fv 6481 df-riota 7286 df-ov 7332 df-oprab 7333 df-mpo 7334 df-of 7587 df-om 7773 df-1st 7891 df-2nd 7892 df-tpos 8104 df-undef 8151 df-frecs 8159 df-wrecs 8190 df-recs 8264 df-rdg 8303 df-1o 8359 df-er 8561 df-map 8680 df-en 8797 df-dom 8798 df-sdom 8799 df-fin 8800 df-pnf 11104 df-mnf 11105 df-xr 11106 df-ltxr 11107 df-le 11108 df-sub 11300 df-neg 11301 df-nn 12067 df-2 12129 df-3 12130 df-4 12131 df-5 12132 df-6 12133 df-n0 12327 df-z 12413 df-uz 12676 df-fz 13333 df-struct 16937 df-sets 16954 df-slot 16972 df-ndx 16984 df-base 17002 df-ress 17031 df-plusg 17064 df-mulr 17065 df-sca 17067 df-vsca 17068 df-0g 17241 df-proset 18102 df-poset 18120 df-plt 18137 df-lub 18153 df-glb 18154 df-join 18155 df-meet 18156 df-p0 18232 df-p1 18233 df-lat 18239 df-clat 18306 df-mgm 18415 df-sgrp 18464 df-mnd 18475 df-submnd 18520 df-grp 18668 df-minusg 18669 df-sbg 18670 df-subg 18840 df-cntz 19011 df-lsm 19329 df-cmn 19475 df-abl 19476 df-mgp 19808 df-ur 19825 df-ring 19872 df-oppr 19949 df-dvdsr 19970 df-unit 19971 df-invr 20001 df-dvr 20012 df-drng 20087 df-lmod 20223 df-lss 20292 df-lsp 20332 df-lvec 20463 df-lfl 37318 df-lkr 37346 df-ldual 37384 df-oposet 37436 df-ol 37438 df-oml 37439 df-covers 37526 df-ats 37527 df-atl 37558 df-cvlat 37582 df-hlat 37611 df-llines 37759 df-lplanes 37760 df-lvols 37761 df-lines 37762 df-psubsp 37764 df-pmap 37765 df-padd 38057 df-lhyp 38249 df-laut 38250 df-ldil 38365 df-ltrn 38366 df-trl 38420 df-tendo 39016 df-edring 39018 df-disoa 39290 df-dvech 39340 df-dib 39400 df-dic 39434 df-dih 39490 df-doch 39609

This theorem is referenced by: lclkr 39794 lclkrslem1 39798

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