lclkrlem1 - Mathbox for Norm Megill

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Theorem lclkrlem1 42380

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 41984	. . 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 42365	. . . . 5 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
14	11, 13	sylib 221	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
15	14	simpld 500	. . 3 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
16	1, 2, 3, 4, 5, 9, 10, 15	ldualvscl 40013	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
17		lclkrlem1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18		eqid 2760	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6, 7, 17, 18, 8	dochoc1 42235	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
20	19	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		fvoveq1 7437	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑅) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘((0_g‘𝑅) · 𝐺)))
22	4, 9	lduallmod 40027	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ LMod)
23		eqid 2760	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
24	1, 4, 23, 9, 15	ldualelvbase 40001	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷))
25		eqid 2760	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
26		eqid 2760	. . . . . . . . . . . 12 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
27		eqid 2760	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
28	23, 25, 5, 26, 27	lmod0vs 21080	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ LMod ∧ 𝐺 ∈ (Base‘𝐷)) → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
29	22, 24, 28	syl2anc 596	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = (0_g‘𝐷))
30		eqid 2760	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31	2, 30, 4, 25, 26, 9	ldual0 40021	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = (0_g‘𝑅))
32	31	oveq1d 7429	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = ((0_g‘𝑅) · 𝐺))
33	18, 2, 30, 4, 27, 9	ldual0v 40024	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {(0_g‘𝑅)}))
34	29, 32, 33	3eqtr3d 2803	. . . . . . . . 9 ⊢ (𝜑 → ((0_g‘𝑅) · 𝐺) = ((Base‘𝑈) × {(0_g‘𝑅)}))
35	34	fveq2d 6883	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})))
36		eqid 2760	. . . . . . . . 9 ⊢ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})
37	2, 30, 18, 1	lfl0f 39943	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹)
38		lclkrlem1.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
39	2, 30, 18, 1, 38	lkr0f 39968	. . . . . . . . . 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 597	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈) ↔ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})))
41	36, 40	mpbiri 261	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈))
42	35, 41	eqtrd 2795	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (Base‘𝑈))
43	21, 42	sylan9eqr 2817	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (Base‘𝑈))
44	43	fveq2d 6883	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(Base‘𝑈)))
45	44	fveq2d 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
46	20, 45, 43	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
47	14	simprd 501	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
48	47	adantr 486	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
49	6, 7, 8	dvhlvec 41983	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
50	49	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑈 ∈ LVec)
51	15	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝐺 ∈ 𝐹)
52	10	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ∈ 𝐵)
53		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → 𝑋 ≠ (0_g‘𝑅))
54	2, 3, 30, 1, 38, 4, 5, 50, 51, 52, 53	ldualkrsc 40041	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
55	54	fveq2d 6883	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(𝐿‘𝐺)))
56	55	fveq2d 6883	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
57	48, 56, 54	3eqtr4d 2805	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
58	46, 57	pm2.61dane 3042	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
59	12	lcfl1lem 42365	. 2 ⊢ ((𝑋 · 𝐺) ∈ 𝐶 ↔ ((𝑋 · 𝐺) ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺))))
60	16, 58, 59	sylanbrc 595	1 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 {crab 3412 {csn 4584 × cxp 5653 ‘cfv 6533 (class class class)co 7414 Basecbs 17302 Scalarcsca 17346 ·_𝑠 cvsca 17347 0_gc0g 17525 LModclmod 21045 LVecclvec 21287 LFnlclfn 39931 LKerclk 39959 LDualcld 39997 HLchlt 40224 LHypclh 40858 DVecHcdvh 41952 ocHcoch 42221

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-riotaBAD 39827

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-tpos 8225 df-undef 8272 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-er 8697 df-map 8829 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-n0 12530 df-z 12617 df-uz 12889 df-fz 13563 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-sca 17359 df-vsca 17360 df-0g 17527 df-proset 18383 df-poset 18402 df-plt 18417 df-lub 18433 df-glb 18434 df-join 18435 df-meet 18436 df-p0 18512 df-p1 18513 df-lat 18521 df-clat 18588 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-grp 19061 df-minusg 19062 df-sbg 19063 df-subg 19247 df-cntz 19445 df-lsm 19764 df-cmn 19910 df-abl 19911 df-mgp 20275 df-rng 20289 df-ur 20322 df-ring 20375 df-oppr 20479 df-dvdsr 20499 df-unit 20500 df-invr 20530 df-dvr 20543 df-nzr 20674 df-rlreg 20857 df-domn 20858 df-drng 20893 df-lmod 21047 df-lss 21117 df-lsp 21157 df-lvec 21288 df-lfl 39932 df-lkr 39960 df-ldual 39998 df-oposet 40050 df-ol 40052 df-oml 40053 df-covers 40140 df-ats 40141 df-atl 40172 df-cvlat 40196 df-hlat 40225 df-llines 40372 df-lplanes 40373 df-lvols 40374 df-lines 40375 df-psubsp 40377 df-pmap 40378 df-padd 40670 df-lhyp 40862 df-laut 40863 df-ldil 40978 df-ltrn 40979 df-trl 41033 df-tendo 41629 df-edring 41631 df-disoa 41903 df-dvech 41953 df-dib 42013 df-dic 42047 df-dih 42103 df-doch 42222

This theorem is used by: lclkr 42407 lclkrslem1 42411

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