lcfl9a - Mathbox for Norm Megill

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Theorem lcfl9a 39446

Description: Property implying that a functional has a closed kernel. (Contributed by NM, 16-Feb-2015.)

**Hypotheses**
Ref	Expression
lcfl9a.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfl9a.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfl9a.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfl9a.v	⊢ 𝑉 = (Base‘𝑈)
lcfl9a.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfl9a.l	⊢ 𝐿 = (LKer‘𝑈)
lcfl9a.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfl9a.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
lcfl9a.x	⊢ (𝜑 → 𝑋 ∈ 𝑉)
lcfl9a.s	⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))

**Assertion**
Ref	Expression
lcfl9a	⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

**Proof of Theorem lcfl9a**
Step	Hyp	Ref	Expression
1		lcfl9a.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcfl9a.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcfl9a.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
4		lcfl9a.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcfl9a.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
6	1, 2, 3, 4, 5	dochoc1 39302	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
7	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
8		lcfl9a.f	. . . . . . . 8 ⊢ 𝐹 = (LFnl‘𝑈)
9		lcfl9a.l	. . . . . . . 8 ⊢ 𝐿 = (LKer‘𝑈)
10	1, 2, 5	dvhlmod 39051	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		lcfl9a.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	4, 8, 9, 10, 11	lkrssv 37037	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉)
14		sneq 4568	. . . . . . . . 9 ⊢ (𝑋 = (0_g‘𝑈) → {𝑋} = {(0_g‘𝑈)})
15	14	fveq2d 6760	. . . . . . . 8 ⊢ (𝑋 = (0_g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0_g‘𝑈)}))
16		eqid 2738	. . . . . . . . . 10 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
17	1, 2, 3, 4, 16	doch0 39299	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
18	5, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
19	15, 18	sylan9eqr 2801	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) = 𝑉)
20		lcfl9a.s	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
22	19, 21	eqsstrrd 3956	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺))
23	13, 22	eqssd 3934	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) = 𝑉)
24	23	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
25	24	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
26	7, 25, 23	3eqtr4d 2788	. 2 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
27	6	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
28		simpr 484	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
29	28	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
30	29	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
31	27, 30, 28	3eqtr4d 2788	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
32		lcfl9a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
33	32	snssd 4739	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
34		eqid 2738	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
35	1, 34, 2, 4, 3	dochcl 39294	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
36	5, 33, 35	syl2anc 583	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
37	1, 34, 3	dochoc 39308	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
38	5, 36, 37	syl2anc 583	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
39	38	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
40	20	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
41		eqid 2738	. . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
42	1, 2, 5	dvhlvec 39050	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
43	42	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑈 ∈ LVec)
44	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	32	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ 𝑉)
46		simprl 767	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0_g‘𝑈))
47		eldifsn 4717	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑈)))
48	45, 46, 47	sylanbrc 582	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
49	1, 3, 2, 4, 16, 41, 44, 48	dochsnshp 39394	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
50		simprr 769	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉)
51	4, 41, 8, 9, 42, 11	lkrshp4 37049	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
52	51	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
53	50, 52	mpbid 231	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
54	41, 43, 49, 53	lshpcmp 36929	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)))
55	40, 54	mpbid 231	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))
56	55	fveq2d 6760	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺)))
57	56	fveq2d 6760	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
58	39, 57, 55	3eqtr3d 2786	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
59	26, 31, 58	pm2.61da2ne 3032	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 ∖ cdif 3880 ⊆ wss 3883 {csn 4558 ran crn 5581 ‘cfv 6418 Basecbs 16840 0_gc0g 17067 LVecclvec 20279 LSHypclsh 36916 LFnlclfn 36998 LKerclk 37026 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 DIsoHcdih 39169 ocHcoch 39288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lfl 36999 df-lkr 37027 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336

This theorem is referenced by: mapdsn 39582

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