lcfl9a - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl9a		Structured version Visualization version GIF version

Theorem lcfl9a 39519

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 39375	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
7	6	adantr 481	. . 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 39124	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		lcfl9a.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	4, 8, 9, 10, 11	lkrssv 37110	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
13	12	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉)
14		sneq 4571	. . . . . . . . 9 ⊢ (𝑋 = (0_g‘𝑈) → {𝑋} = {(0_g‘𝑈)})
15	14	fveq2d 6778	. . . . . . . 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 39372	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
18	5, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
19	15, 18	sylan9eqr 2800	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) = 𝑉)
20		lcfl9a.s	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
21	20	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺))
22	19, 21	eqsstrrd 3960	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺))
23	13, 22	eqssd 3938	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) = 𝑉)
24	23	fveq2d 6778	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
25	24	fveq2d 6778	. . 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 481	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘𝑉)) = 𝑉)
28		simpr 485	. . . . 5 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → (𝐿‘𝐺) = 𝑉)
29	28	fveq2d 6778	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
30	29	fveq2d 6778	. . 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 4742	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
34		eqid 2738	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
35	1, 34, 2, 4, 3	dochcl 39367	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ {𝑋} ⊆ 𝑉) → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
36	5, 33, 35	syl2anc 584	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
37	1, 34, 3	dochoc 39381	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ ( ⊥ ‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
38	5, 36, 37	syl2anc 584	. . . 4 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
39	38	adantr 481	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘{𝑋}))
40	20	adantr 481	. . . . . 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 39123	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LVec)
43	42	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑈 ∈ LVec)
44	5	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
45	32	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ 𝑉)
46		simprl 768	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0_g‘𝑈))
47		eldifsn 4720	. . . . . . . . 9 ⊢ (𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}) ↔ (𝑋 ∈ 𝑉 ∧ 𝑋 ≠ (0_g‘𝑈)))
48	45, 46, 47	sylanbrc 583	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ∈ (𝑉 ∖ {(0_g‘𝑈)}))
49	1, 3, 2, 4, 16, 41, 44, 48	dochsnshp 39467	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
50		simprr 770	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉)
51	4, 41, 8, 9, 42, 11	lkrshp4 37122	. . . . . . . . 9 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ (𝐿‘𝐺) ∈ (LSHyp‘𝑈)))
52	51	adantr 481	. . . . . . . 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 37002	. . . . . 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 6778	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺)))
57	56	fveq2d 6778	. . 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 3033	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 ∖ cdif 3884 ⊆ wss 3887 {csn 4561 ran crn 5590 ‘cfv 6433 Basecbs 16912 0_gc0g 17150 LVecclvec 20364 LSHypclsh 36989 LFnlclfn 37071 LKerclk 37099 HLchlt 37364 LHypclh 37998 DVecHcdvh 39092 DIsoHcdih 39242 ocHcoch 39361

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-riotaBAD 36967

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-tpos 8042 df-undef 8089 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-sca 16978 df-vsca 16979 df-0g 17152 df-proset 18013 df-poset 18031 df-plt 18048 df-lub 18064 df-glb 18065 df-join 18066 df-meet 18067 df-p0 18143 df-p1 18144 df-lat 18150 df-clat 18217 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-grp 18580 df-minusg 18581 df-sbg 18582 df-subg 18752 df-cntz 18923 df-lsm 19241 df-cmn 19388 df-abl 19389 df-mgp 19721 df-ur 19738 df-ring 19785 df-oppr 19862 df-dvdsr 19883 df-unit 19884 df-invr 19914 df-dvr 19925 df-drng 19993 df-lmod 20125 df-lss 20194 df-lsp 20234 df-lvec 20365 df-lsatoms 36990 df-lshyp 36991 df-lfl 37072 df-lkr 37100 df-oposet 37190 df-ol 37192 df-oml 37193 df-covers 37280 df-ats 37281 df-atl 37312 df-cvlat 37336 df-hlat 37365 df-llines 37512 df-lplanes 37513 df-lvols 37514 df-lines 37515 df-psubsp 37517 df-pmap 37518 df-padd 37810 df-lhyp 38002 df-laut 38003 df-ldil 38118 df-ltrn 38119 df-trl 38173 df-tgrp 38757 df-tendo 38769 df-edring 38771 df-dveca 39017 df-disoa 39043 df-dvech 39093 df-dib 39153 df-dic 39187 df-dih 39243 df-doch 39362 df-djh 39409

This theorem is referenced by: mapdsn 39655

W3C validator