lclkrlem1 - Mathbox for Norm Megill

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Theorem lclkrlem1 40365

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 39969	. . 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 40350	. . . . 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 37997	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
17		lclkrlem1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18		eqid 2732	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6, 7, 17, 18, 8	dochoc1 40220	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
20	19	adantr 481	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		fvoveq1 7428	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑅) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘((0_g‘𝑅) · 𝐺)))
22	4, 9	lduallmod 38011	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ LMod)
23		eqid 2732	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
24	1, 4, 23, 9, 15	ldualelvbase 37985	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷))
25		eqid 2732	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
26		eqid 2732	. . . . . . . . . . . 12 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
27		eqid 2732	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
28	23, 25, 5, 26, 27	lmod0vs 20497	. . . . . . . . . . 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 2732	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31	2, 30, 4, 25, 26, 9	ldual0 38005	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = (0_g‘𝑅))
32	31	oveq1d 7420	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = ((0_g‘𝑅) · 𝐺))
33	18, 2, 30, 4, 27, 9	ldual0v 38008	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {(0_g‘𝑅)}))
34	29, 32, 33	3eqtr3d 2780	. . . . . . . . 9 ⊢ (𝜑 → ((0_g‘𝑅) · 𝐺) = ((Base‘𝑈) × {(0_g‘𝑅)}))
35	34	fveq2d 6892	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})))
36		eqid 2732	. . . . . . . . 9 ⊢ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})
37	2, 30, 18, 1	lfl0f 37927	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹)
38		lclkrlem1.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
39	2, 30, 18, 1, 38	lkr0f 37952	. . . . . . . . . 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 2772	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (Base‘𝑈))
43	21, 42	sylan9eqr 2794	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (Base‘𝑈))
44	43	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(Base‘𝑈)))
45	44	fveq2d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
46	20, 45, 43	3eqtr4d 2782	. . 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 39968	. . . . . . . 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 38025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
55	54	fveq2d 6892	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(𝐿‘𝐺)))
56	55	fveq2d 6892	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
57	48, 56, 54	3eqtr4d 2782	. . 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 40350	. 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 1541 ∈ wcel 2106 ≠ wne 2940 {crab 3432 {csn 4627 × cxp 5673 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 Scalarcsca 17196 ·_𝑠 cvsca 17197 0_gc0g 17381 LModclmod 20463 LVecclvec 20705 LFnlclfn 37915 LKerclk 37943 LDualcld 37981 HLchlt 38208 LHypclh 38843 DVecHcdvh 39937 ocHcoch 40206

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-riotaBAD 37811

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-n0 12469 df-z 12555 df-uz 12819 df-fz 13481 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-0g 17383 df-proset 18244 df-poset 18262 df-plt 18279 df-lub 18295 df-glb 18296 df-join 18297 df-meet 18298 df-p0 18374 df-p1 18375 df-lat 18381 df-clat 18448 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-cntz 19175 df-lsm 19498 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-oppr 20142 df-dvdsr 20163 df-unit 20164 df-invr 20194 df-dvr 20207 df-drng 20309 df-lmod 20465 df-lss 20535 df-lsp 20575 df-lvec 20706 df-lfl 37916 df-lkr 37944 df-ldual 37982 df-oposet 38034 df-ol 38036 df-oml 38037 df-covers 38124 df-ats 38125 df-atl 38156 df-cvlat 38180 df-hlat 38209 df-llines 38357 df-lplanes 38358 df-lvols 38359 df-lines 38360 df-psubsp 38362 df-pmap 38363 df-padd 38655 df-lhyp 38847 df-laut 38848 df-ldil 38963 df-ltrn 38964 df-trl 39018 df-tendo 39614 df-edring 39616 df-disoa 39888 df-dvech 39938 df-dib 39998 df-dic 40032 df-dih 40088 df-doch 40207

This theorem is referenced by: lclkr 40392 lclkrslem1 40396

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