lcfl9a - Mathbox for Norm Megill

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Theorem lcfl9a 41524

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 41380	. . . 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 41129	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ LMod)
11		lcfl9a.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
12	4, 8, 9, 10, 11	lkrssv 39114	. . . . . . 7 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
13	12	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) ⊆ 𝑉)
14		sneq 4611	. . . . . . . . 9 ⊢ (𝑋 = (0_g‘𝑈) → {𝑋} = {(0_g‘𝑈)})
15	14	fveq2d 6880	. . . . . . . 8 ⊢ (𝑋 = (0_g‘𝑈) → ( ⊥ ‘{𝑋}) = ( ⊥ ‘{(0_g‘𝑈)}))
16		eqid 2735	. . . . . . . . . 10 ⊢ (0_g‘𝑈) = (0_g‘𝑈)
17	1, 2, 3, 4, 16	doch0 41377	. . . . . . . . 9 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
18	5, 17	syl 17	. . . . . . . 8 ⊢ (𝜑 → ( ⊥ ‘{(0_g‘𝑈)}) = 𝑉)
19	15, 18	sylan9eqr 2792	. . . . . . 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 3994	. . . . . 6 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → 𝑉 ⊆ (𝐿‘𝐺))
23	13, 22	eqssd 3976	. . . . 5 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → (𝐿‘𝐺) = 𝑉)
24	23	fveq2d 6880	. . . 4 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
25	24	fveq2d 6880	. . 3 ⊢ ((𝜑 ∧ 𝑋 = (0_g‘𝑈)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
26	7, 25, 23	3eqtr4d 2780	. 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 6880	. . . 4 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘𝑉))
30	29	fveq2d 6880	. . 3 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = ( ⊥ ‘( ⊥ ‘𝑉)))
31	27, 30, 28	3eqtr4d 2780	. 2 ⊢ ((𝜑 ∧ (𝐿‘𝐺) = 𝑉) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
32		lcfl9a.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
33	32	snssd 4785	. . . . . 6 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
34		eqid 2735	. . . . . . 7 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
35	1, 34, 2, 4, 3	dochcl 41372	. . . . . 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 41386	. . . . 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 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 2735	. . . . . . 7 ⊢ (LSHyp‘𝑈) = (LSHyp‘𝑈)
42	1, 2, 5	dvhlvec 41128	. . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → 𝑋 ≠ (0_g‘𝑈))
47		eldifsn 4762	. . . . . . . . 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 41472	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) ∈ (LSHyp‘𝑈))
50		simprr 772	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ≠ 𝑉)
51	4, 41, 8, 9, 42, 11	lkrshp4 39126	. . . . . . . . 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 232	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (𝐿‘𝐺) ∈ (LSHyp‘𝑈))
54	41, 43, 49, 53	lshpcmp 39006	. . . . . 6 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → (( ⊥ ‘{𝑋}) ⊆ (𝐿‘𝐺) ↔ ( ⊥ ‘{𝑋}) = (𝐿‘𝐺)))
55	40, 54	mpbid 232	. . . . 5 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘{𝑋}) = (𝐿‘𝐺))
56	55	fveq2d 6880	. . . 4 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘{𝑋})) = ( ⊥ ‘(𝐿‘𝐺)))
57	56	fveq2d 6880	. . 3 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘( ⊥ ‘{𝑋}))) = ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))))
58	39, 57, 55	3eqtr3d 2778	. 2 ⊢ ((𝜑 ∧ (𝑋 ≠ (0_g‘𝑈) ∧ (𝐿‘𝐺) ≠ 𝑉)) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
59	26, 31, 58	pm2.61da2ne 3020	1 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 ∖ cdif 3923 ⊆ wss 3926 {csn 4601 ran crn 5655 ‘cfv 6531 Basecbs 17228 0_gc0g 17453 LVecclvec 21060 LSHypclsh 38993 LFnlclfn 39075 LKerclk 39103 HLchlt 39368 LHypclh 40003 DVecHcdvh 41097 DIsoHcdih 41247 ocHcoch 41366

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-riotaBAD 38971

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-tp 4606 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-iin 4970 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-tpos 8225 df-undef 8272 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8719 df-map 8842 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-n0 12502 df-z 12589 df-uz 12853 df-fz 13525 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17252 df-plusg 17284 df-mulr 17285 df-sca 17287 df-vsca 17288 df-0g 17455 df-proset 18306 df-poset 18325 df-plt 18340 df-lub 18356 df-glb 18357 df-join 18358 df-meet 18359 df-p0 18435 df-p1 18436 df-lat 18442 df-clat 18509 df-mgm 18618 df-sgrp 18697 df-mnd 18713 df-submnd 18762 df-grp 18919 df-minusg 18920 df-sbg 18921 df-subg 19106 df-cntz 19300 df-lsm 19617 df-cmn 19763 df-abl 19764 df-mgp 20101 df-rng 20113 df-ur 20142 df-ring 20195 df-oppr 20297 df-dvdsr 20317 df-unit 20318 df-invr 20348 df-dvr 20361 df-drng 20691 df-lmod 20819 df-lss 20889 df-lsp 20929 df-lvec 21061 df-lsatoms 38994 df-lshyp 38995 df-lfl 39076 df-lkr 39104 df-oposet 39194 df-ol 39196 df-oml 39197 df-covers 39284 df-ats 39285 df-atl 39316 df-cvlat 39340 df-hlat 39369 df-llines 39517 df-lplanes 39518 df-lvols 39519 df-lines 39520 df-psubsp 39522 df-pmap 39523 df-padd 39815 df-lhyp 40007 df-laut 40008 df-ldil 40123 df-ltrn 40124 df-trl 40178 df-tgrp 40762 df-tendo 40774 df-edring 40776 df-dveca 41022 df-disoa 41048 df-dvech 41098 df-dib 41158 df-dic 41192 df-dih 41248 df-doch 41367 df-djh 41414

This theorem is referenced by: mapdsn 41660

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