lclkrlem1 - Mathbox for Norm Megill

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Theorem lclkrlem1 41495

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 41099	. . 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 41480	. . . . 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 39127	. 2 ⊢ (𝜑 → (𝑋 · 𝐺) ∈ 𝐹)
17		lclkrlem1.o	. . . . . 6 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
18		eqid 2730	. . . . . 6 ⊢ (Base‘𝑈) = (Base‘𝑈)
19	6, 7, 17, 18, 8	dochoc1 41350	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
20	19	adantr 480	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(Base‘𝑈))) = (Base‘𝑈))
21		fvoveq1 7412	. . . . . . 7 ⊢ (𝑋 = (0_g‘𝑅) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘((0_g‘𝑅) · 𝐺)))
22	4, 9	lduallmod 39141	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 ∈ LMod)
23		eqid 2730	. . . . . . . . . . . 12 ⊢ (Base‘𝐷) = (Base‘𝐷)
24	1, 4, 23, 9, 15	ldualelvbase 39115	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐺 ∈ (Base‘𝐷))
25		eqid 2730	. . . . . . . . . . . 12 ⊢ (Scalar‘𝐷) = (Scalar‘𝐷)
26		eqid 2730	. . . . . . . . . . . 12 ⊢ (0_g‘(Scalar‘𝐷)) = (0_g‘(Scalar‘𝐷))
27		eqid 2730	. . . . . . . . . . . 12 ⊢ (0_g‘𝐷) = (0_g‘𝐷)
28	23, 25, 5, 26, 27	lmod0vs 20807	. . . . . . . . . . 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 2730	. . . . . . . . . . . 12 ⊢ (0_g‘𝑅) = (0_g‘𝑅)
31	2, 30, 4, 25, 26, 9	ldual0 39135	. . . . . . . . . . 11 ⊢ (𝜑 → (0_g‘(Scalar‘𝐷)) = (0_g‘𝑅))
32	31	oveq1d 7404	. . . . . . . . . 10 ⊢ (𝜑 → ((0_g‘(Scalar‘𝐷)) · 𝐺) = ((0_g‘𝑅) · 𝐺))
33	18, 2, 30, 4, 27, 9	ldual0v 39138	. . . . . . . . . 10 ⊢ (𝜑 → (0_g‘𝐷) = ((Base‘𝑈) × {(0_g‘𝑅)}))
34	29, 32, 33	3eqtr3d 2773	. . . . . . . . 9 ⊢ (𝜑 → ((0_g‘𝑅) · 𝐺) = ((Base‘𝑈) × {(0_g‘𝑅)}))
35	34	fveq2d 6864	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})))
36		eqid 2730	. . . . . . . . 9 ⊢ ((Base‘𝑈) × {(0_g‘𝑅)}) = ((Base‘𝑈) × {(0_g‘𝑅)})
37	2, 30, 18, 1	lfl0f 39057	. . . . . . . . . 10 ⊢ (𝑈 ∈ LMod → ((Base‘𝑈) × {(0_g‘𝑅)}) ∈ 𝐹)
38		lclkrlem1.l	. . . . . . . . . . 11 ⊢ 𝐿 = (LKer‘𝑈)
39	2, 30, 18, 1, 38	lkr0f 39082	. . . . . . . . . 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 258	. . . . . . . 8 ⊢ (𝜑 → (𝐿‘((Base‘𝑈) × {(0_g‘𝑅)})) = (Base‘𝑈))
42	35, 41	eqtrd 2765	. . . . . . 7 ⊢ (𝜑 → (𝐿‘((0_g‘𝑅) · 𝐺)) = (Base‘𝑈))
43	21, 42	sylan9eqr 2787	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (Base‘𝑈))
44	43	fveq2d 6864	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(Base‘𝑈)))
45	44	fveq2d 6864	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(Base‘𝑈))))
46	20, 45, 43	3eqtr4d 2775	. . 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 41098	. . . . . . . 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 39155	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → (𝐿‘(𝑋 · 𝐺)) = (𝐿‘𝐺))
55	54	fveq2d 6864	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘(𝐿‘(𝑋 · 𝐺))) = ( ⊥ ‘(𝐿‘𝐺)))
56	55	fveq2d 6864	. . . 4 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
57	48, 56, 54	3eqtr4d 2775	. . 3 ⊢ ((𝜑 ∧ 𝑋 ≠ (0_g‘𝑅)) → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
58	46, 57	pm2.61dane 3013	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘(𝑋 · 𝐺)))) = (𝐿‘(𝑋 · 𝐺)))
59	12	lcfl1lem 41480	. 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 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ≠ wne 2926 {crab 3408 {csn 4591 × cxp 5638 ‘cfv 6513 (class class class)co 7389 Basecbs 17185 Scalarcsca 17229 ·_𝑠 cvsca 17230 0_gc0g 17408 LModclmod 20772 LVecclvec 21015 LFnlclfn 39045 LKerclk 39073 LDualcld 39111 HLchlt 39338 LHypclh 39973 DVecHcdvh 41067 ocHcoch 41336

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5236 ax-sep 5253 ax-nul 5263 ax-pow 5322 ax-pr 5389 ax-un 7713 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-riotaBAD 38941

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3756 df-csb 3865 df-dif 3919 df-un 3921 df-in 3923 df-ss 3933 df-pss 3936 df-nul 4299 df-if 4491 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4874 df-int 4913 df-iun 4959 df-iin 4960 df-br 5110 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5535 df-eprel 5540 df-po 5548 df-so 5549 df-fr 5593 df-we 5595 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-pred 6276 df-ord 6337 df-on 6338 df-lim 6339 df-suc 6340 df-iota 6466 df-fun 6515 df-fn 6516 df-f 6517 df-f1 6518 df-fo 6519 df-f1o 6520 df-fv 6521 df-riota 7346 df-ov 7392 df-oprab 7393 df-mpo 7394 df-of 7655 df-om 7845 df-1st 7970 df-2nd 7971 df-tpos 8207 df-undef 8254 df-frecs 8262 df-wrecs 8293 df-recs 8342 df-rdg 8380 df-1o 8436 df-er 8673 df-map 8803 df-en 8921 df-dom 8922 df-sdom 8923 df-fin 8924 df-pnf 11216 df-mnf 11217 df-xr 11218 df-ltxr 11219 df-le 11220 df-sub 11413 df-neg 11414 df-nn 12188 df-2 12250 df-3 12251 df-4 12252 df-5 12253 df-6 12254 df-n0 12449 df-z 12536 df-uz 12800 df-fz 13475 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17186 df-ress 17207 df-plusg 17239 df-mulr 17240 df-sca 17242 df-vsca 17243 df-0g 17410 df-proset 18261 df-poset 18280 df-plt 18295 df-lub 18311 df-glb 18312 df-join 18313 df-meet 18314 df-p0 18390 df-p1 18391 df-lat 18397 df-clat 18464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-submnd 18717 df-grp 18874 df-minusg 18875 df-sbg 18876 df-subg 19061 df-cntz 19255 df-lsm 19572 df-cmn 19718 df-abl 19719 df-mgp 20056 df-rng 20068 df-ur 20097 df-ring 20150 df-oppr 20252 df-dvdsr 20272 df-unit 20273 df-invr 20303 df-dvr 20316 df-nzr 20428 df-rlreg 20609 df-domn 20610 df-drng 20646 df-lmod 20774 df-lss 20844 df-lsp 20884 df-lvec 21016 df-lfl 39046 df-lkr 39074 df-ldual 39112 df-oposet 39164 df-ol 39166 df-oml 39167 df-covers 39254 df-ats 39255 df-atl 39286 df-cvlat 39310 df-hlat 39339 df-llines 39487 df-lplanes 39488 df-lvols 39489 df-lines 39490 df-psubsp 39492 df-pmap 39493 df-padd 39785 df-lhyp 39977 df-laut 39978 df-ldil 40093 df-ltrn 40094 df-trl 40148 df-tendo 40744 df-edring 40746 df-disoa 41018 df-dvech 41068 df-dib 41128 df-dic 41162 df-dih 41218 df-doch 41337

This theorem is referenced by: lclkr 41522 lclkrslem1 41526

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