lcfl8b - Mathbox for Norm Megill

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Theorem lcfl8b 42377

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 3911	. . . 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 42364	. . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
12	11	simplbi 502	. . . . . 6 ⊢ (𝐺 ∈ 𝐶 → 𝐺 ∈ 𝐹)
13	2, 12	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
14	3, 4, 5, 6, 7, 8, 9, 10, 13	lcfl8 42375	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
15	2, 14	mpbid 235	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))
16		fveq2 6878	. . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
17	16	adantl 487	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
18		lcfl8b.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
19	10	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simplr 781	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉)
21	3, 5, 4, 6, 18, 19, 20	dochocsn 42254	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥}))
22	17, 21	eqtrd 2795	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
23	2, 11	sylib 221	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
24	23	simprd 501	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
25		eldifsni 4753	. . . . . . . . . . . . 13 ⊢ (𝐺 ∈ (𝐶 ∖ {𝑌}) → 𝐺 ≠ 𝑌)
26	1, 25	syl 18	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐺 ≠ 𝑌)
27		lcfl8b.d	. . . . . . . . . . . . . 14 ⊢ 𝐷 = (LDual‘𝑈)
28		lcfl8b.y	. . . . . . . . . . . . . 14 ⊢ 𝑌 = (0_g‘𝐷)
29	3, 5, 10	dvhlmod 41983	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	6, 7, 8, 27, 28, 29, 13	lkr0f2 40034	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌))
31	30	necon3bid 2999	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌))
32	26, 31	mpbird 260	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉)
33	24, 32	eqnetrd 3022	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
34	33	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
35		eqid 2760	. . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
36	13	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹)
37	3, 4, 5, 6, 35, 7, 8, 19, 36	dochkrsat2 42329	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)))
38	34, 37	mpbid 235	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))
39	22, 38	eqeltrrd 2861	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
40		lcfl8b.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
41	29	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod)
42	6, 18, 40, 35, 41, 20	lsatspn0 39873	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 ))
43	39, 42	mpbid 235	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 )
44	43, 22	jca 521	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
45	44	ex 418	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
46	45	reximdva 3175	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
47	15, 46	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
48		rexdifsn 4757	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
49	47, 48	sylibr 237	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∃wrex 3086 {crab 3412 ∖ cdif 3896 {csn 4584 ‘cfv 6533 Basecbs 17301 0_gc0g 17524 LModclmod 21044 LSpanclspn 21155 LSAtomsclsa 39847 LFnlclfn 39930 LKerclk 39958 LDualcld 39996 HLchlt 40223 LHypclh 40857 DVecHcdvh 41951 ocHcoch 42220

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7736 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-riotaBAD 39826

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-riota 7370 df-ov 7416 df-oprab 7417 df-mpo 7418 df-of 7678 df-om 7863 df-1st 7986 df-2nd 7987 df-tpos 8224 df-undef 8271 df-frecs 8280 df-wrecs 8311 df-recs 8360 df-rdg 8399 df-1o 8455 df-er 8696 df-map 8828 df-en 8953 df-dom 8954 df-sdom 8955 df-fin 8956 df-pnf 11269 df-mnf 11270 df-xr 11271 df-ltxr 11272 df-le 11273 df-sub 11467 df-neg 11468 df-nn 12258 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-n0 12529 df-z 12616 df-uz 12888 df-fz 13562 df-struct 17239 df-sets 17256 df-slot 17274 df-ndx 17286 df-base 17302 df-ress 17323 df-plusg 17355 df-mulr 17356 df-sca 17358 df-vsca 17359 df-0g 17526 df-proset 18382 df-poset 18401 df-plt 18416 df-lub 18432 df-glb 18433 df-join 18434 df-meet 18435 df-p0 18511 df-p1 18512 df-lat 18520 df-clat 18587 df-mgm 18730 df-sgrp 18821 df-mnd 18837 df-submnd 18892 df-grp 19060 df-minusg 19061 df-sbg 19062 df-subg 19246 df-cntz 19444 df-lsm 19763 df-cmn 19909 df-abl 19910 df-mgp 20274 df-rng 20288 df-ur 20321 df-ring 20374 df-oppr 20478 df-dvdsr 20498 df-unit 20499 df-invr 20529 df-dvr 20542 df-drng 20892 df-lmod 21046 df-lss 21116 df-lsp 21156 df-lvec 21287 df-lsatoms 39849 df-lshyp 39850 df-lfl 39931 df-lkr 39959 df-ldual 39997 df-oposet 40049 df-ol 40051 df-oml 40052 df-covers 40139 df-ats 40140 df-atl 40171 df-cvlat 40195 df-hlat 40224 df-llines 40371 df-lplanes 40372 df-lvols 40373 df-lines 40374 df-psubsp 40376 df-pmap 40377 df-padd 40669 df-lhyp 40861 df-laut 40862 df-ldil 40977 df-ltrn 40978 df-trl 41032 df-tgrp 41616 df-tendo 41628 df-edring 41630 df-dveca 41876 df-disoa 41902 df-dvech 41952 df-dib 42012 df-dic 42046 df-dih 42102 df-doch 42221 df-djh 42268

This theorem is used by: mapdrvallem2 42518

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