lcfl8b - Mathbox for Norm Megill

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Theorem lcfl8b 41623

Description: Property of a nonzero functional with a closed kernel. (Contributed by NM, 4-Feb-2015.)

**Hypotheses**
Ref	Expression
lcfl8b.h	⊢ 𝐻 = (LHyp‘𝐾)
lcfl8b.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcfl8b.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfl8b.v	⊢ 𝑉 = (Base‘𝑈)
lcfl8b.n	⊢ 𝑁 = (LSpan‘𝑈)
lcfl8b.z	⊢ 0 = (0_g‘𝑈)
lcfl8b.f	⊢ 𝐹 = (LFnl‘𝑈)
lcfl8b.l	⊢ 𝐿 = (LKer‘𝑈)
lcfl8b.d	⊢ 𝐷 = (LDual‘𝑈)
lcfl8b.y	⊢ 𝑌 = (0_g‘𝐷)
lcfl8b.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcfl8b.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfl8b.g	⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌}))

**Assertion**
Ref	Expression
lcfl8b	⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))

Distinct variable groups: 𝑥,𝐶 𝑓,𝐹 𝑥,𝑓,𝐺 𝑓,𝐿,𝑥 ⊥ ,𝑓,𝑥 𝑥,𝑈 𝑥,𝑉 𝜑,𝑥 Allowed substitution hints: 𝜑(𝑓) 𝐶(𝑓) 𝐷(𝑥,𝑓) 𝑈(𝑓) 𝐹(𝑥) 𝐻(𝑥,𝑓) 𝐾(𝑥,𝑓) 𝑁(𝑥,𝑓) 𝑉(𝑓) 𝑊(𝑥,𝑓) 𝑌(𝑥,𝑓) 0 (𝑥,𝑓)

**Proof of Theorem lcfl8b**
Step	Hyp	Ref	Expression
1		lcfl8b.g	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ (𝐶 ∖ {𝑌}))
2	1	eldifad 3910	. . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝐶)
3		lcfl8b.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
4		lcfl8b.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
5		lcfl8b.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
6		lcfl8b.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
7		lcfl8b.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑈)
8		lcfl8b.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
9		lcfl8b.c	. . . . 5 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
10		lcfl8b.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
11	9	lcfl1lem 41610	. . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
12	11	simplbi 497	. . . . . 6 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐹)
13	2, 12	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
14	3, 4, 5, 6, 7, 8, 9, 10, 13	lcfl8 41621	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
15	2, 14	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))
16		fveq2 6828	. . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
17	16	adantl 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
18		lcfl8b.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
19	10	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simplr 768	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉)
21	3, 5, 4, 6, 18, 19, 20	dochocsn 41500	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥}))
22	17, 21	eqtrd 2768	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
23	2, 11	sylib 218	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
24	23	simprd 495	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
25		eldifsni 4741	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐶 ∖ {𝑌}) → 𝐺 ≠ 𝑌)
26	1, 25	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ≠ 𝑌)
27		lcfl8b.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (LDual‘𝑈)
28		lcfl8b.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (0_g‘𝐷)
29	3, 5, 10	dvhlmod 41229	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	6, 7, 8, 27, 28, 29, 13	lkr0f2 39280	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌))
31	30	necon3bid 2973	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌))
32	26, 31	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉)
33	24, 32	eqnetrd 2996	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
34	33	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
35		eqid 2733	. . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
36	13	ad2antrr 726	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹)
37	3, 4, 5, 6, 35, 7, 8, 19, 36	dochkrsat2 41575	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)))
38	34, 37	mpbid 232	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))
39	22, 38	eqeltrrd 2834	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
40		lcfl8b.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
41	29	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod)
42	6, 18, 40, 35, 41, 20	lsatspn0 39119	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 ))
43	39, 42	mpbid 232	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 )
44	43, 22	jca 511	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
45	44	ex 412	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
46	45	reximdva 3146	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
47	15, 46	mpd 15	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
48		rexdifsn 4745	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
49	47, 48	sylibr 234	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ≠ wne 2929 ∃wrex 3057 {crab 3396 ∖ cdif 3895 {csn 4575 ‘cfv 6486 Basecbs 17122 0_gc0g 17345 LModclmod 20795 LSpanclspn 20906 LSAtomsclsa 39093 LFnlclfn 39176 LKerclk 39204 LDualcld 39242 HLchlt 39469 LHypclh 40103 DVecHcdvh 41197 ocHcoch 41466

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-rep 5219 ax-sep 5236 ax-nul 5246 ax-pow 5305 ax-pr 5372 ax-un 7674 ax-cnex 11069 ax-resscn 11070 ax-1cn 11071 ax-icn 11072 ax-addcl 11073 ax-addrcl 11074 ax-mulcl 11075 ax-mulrcl 11076 ax-mulcom 11077 ax-addass 11078 ax-mulass 11079 ax-distr 11080 ax-i2m1 11081 ax-1ne0 11082 ax-1rid 11083 ax-rnegex 11084 ax-rrecex 11085 ax-cnre 11086 ax-pre-lttri 11087 ax-pre-lttrn 11088 ax-pre-ltadd 11089 ax-pre-mulgt0 11090 ax-riotaBAD 39072

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4475 df-pw 4551 df-sn 4576 df-pr 4578 df-tp 4580 df-op 4582 df-uni 4859 df-int 4898 df-iun 4943 df-iin 4944 df-br 5094 df-opab 5156 df-mpt 5175 df-tr 5201 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6253 df-ord 6314 df-on 6315 df-lim 6316 df-suc 6317 df-iota 6442 df-fun 6488 df-fn 6489 df-f 6490 df-f1 6491 df-fo 6492 df-f1o 6493 df-fv 6494 df-riota 7309 df-ov 7355 df-oprab 7356 df-mpo 7357 df-of 7616 df-om 7803 df-1st 7927 df-2nd 7928 df-tpos 8162 df-undef 8209 df-frecs 8217 df-wrecs 8248 df-recs 8297 df-rdg 8335 df-1o 8391 df-er 8628 df-map 8758 df-en 8876 df-dom 8877 df-sdom 8878 df-fin 8879 df-pnf 11155 df-mnf 11156 df-xr 11157 df-ltxr 11158 df-le 11159 df-sub 11353 df-neg 11354 df-nn 12133 df-2 12195 df-3 12196 df-4 12197 df-5 12198 df-6 12199 df-n0 12389 df-z 12476 df-uz 12739 df-fz 13410 df-struct 17060 df-sets 17077 df-slot 17095 df-ndx 17107 df-base 17123 df-ress 17144 df-plusg 17176 df-mulr 17177 df-sca 17179 df-vsca 17180 df-0g 17347 df-proset 18202 df-poset 18221 df-plt 18236 df-lub 18252 df-glb 18253 df-join 18254 df-meet 18255 df-p0 18331 df-p1 18332 df-lat 18340 df-clat 18407 df-mgm 18550 df-sgrp 18629 df-mnd 18645 df-submnd 18694 df-grp 18851 df-minusg 18852 df-sbg 18853 df-subg 19038 df-cntz 19231 df-lsm 19550 df-cmn 19696 df-abl 19697 df-mgp 20061 df-rng 20073 df-ur 20102 df-ring 20155 df-oppr 20257 df-dvdsr 20277 df-unit 20278 df-invr 20308 df-dvr 20321 df-drng 20648 df-lmod 20797 df-lss 20867 df-lsp 20907 df-lvec 21039 df-lsatoms 39095 df-lshyp 39096 df-lfl 39177 df-lkr 39205 df-ldual 39243 df-oposet 39295 df-ol 39297 df-oml 39298 df-covers 39385 df-ats 39386 df-atl 39417 df-cvlat 39441 df-hlat 39470 df-llines 39617 df-lplanes 39618 df-lvols 39619 df-lines 39620 df-psubsp 39622 df-pmap 39623 df-padd 39915 df-lhyp 40107 df-laut 40108 df-ldil 40223 df-ltrn 40224 df-trl 40278 df-tgrp 40862 df-tendo 40874 df-edring 40876 df-dveca 41122 df-disoa 41148 df-dvech 41198 df-dib 41258 df-dic 41292 df-dih 41348 df-doch 41467 df-djh 41514

This theorem is referenced by: mapdrvallem2 41764

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