lcfl8b - Mathbox for Norm Megill

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Theorem lcfl8b 42003

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 3902	. . . 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 41990	. . . . . . 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 42001	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
15	2, 14	mpbid 233	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))
16		fveq2 6834	. . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
17	16	adantl 482	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
18		lcfl8b.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
19	10	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simplr 774	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉)
21	3, 5, 4, 6, 18, 19, 20	dochocsn 41880	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥}))
22	17, 21	eqtrd 2775	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
23	2, 11	sylib 219	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
24	23	simprd 496	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
25		eldifsni 4730	. . . . . . . . . . . . 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 41609	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	6, 7, 8, 27, 28, 29, 13	lkr0f2 39660	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌))
31	30	necon3bid 2979	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌))
32	26, 31	mpbird 258	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉)
33	24, 32	eqnetrd 3002	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
34	33	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
35		eqid 2740	. . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
36	13	ad2antrr 732	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹)
37	3, 4, 5, 6, 35, 7, 8, 19, 36	dochkrsat2 41955	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)))
38	34, 37	mpbid 233	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))
39	22, 38	eqeltrrd 2841	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
40		lcfl8b.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
41	29	ad2antrr 732	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod)
42	6, 18, 40, 35, 41, 20	lsatspn0 39499	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 ))
43	39, 42	mpbid 233	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 )
44	43, 22	jca 516	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
45	44	ex 413	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
46	45	reximdva 3153	. . 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 4734	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
49	47, 48	sylibr 235	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ≠ wne 2935 ∃wrex 3064 {crab 3392 ∖ cdif 3887 {csn 4562 ‘cfv 6492 Basecbs 17177 0_gc0g 17400 LModclmod 20857 LSpanclspn 20968 LSAtomsclsa 39473 LFnlclfn 39556 LKerclk 39584 LDualcld 39622 HLchlt 39849 LHypclh 40483 DVecHcdvh 41577 ocHcoch 41846

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-rep 5206 ax-sep 5225 ax-nul 5235 ax-pow 5301 ax-pr 5369 ax-un 7685 ax-cnex 11092 ax-resscn 11093 ax-1cn 11094 ax-icn 11095 ax-addcl 11096 ax-addrcl 11097 ax-mulcl 11098 ax-mulrcl 11099 ax-mulcom 11100 ax-addass 11101 ax-mulass 11102 ax-distr 11103 ax-i2m1 11104 ax-1ne0 11105 ax-1rid 11106 ax-rnegex 11107 ax-rrecex 11108 ax-cnre 11109 ax-pre-lttri 11110 ax-pre-lttrn 11111 ax-pre-ltadd 11112 ax-pre-mulgt0 11113 ax-riotaBAD 39452

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-nel 3040 df-ral 3055 df-rex 3065 df-rmo 3345 df-reu 3346 df-rab 3393 df-v 3434 df-sbc 3731 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4269 df-if 4462 df-pw 4538 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4846 df-int 4885 df-iun 4930 df-iin 4931 df-br 5080 df-opab 5142 df-mpt 5161 df-tr 5187 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-riota 7320 df-ov 7366 df-oprab 7367 df-mpo 7368 df-of 7627 df-om 7814 df-1st 7938 df-2nd 7939 df-tpos 8173 df-undef 8220 df-frecs 8228 df-wrecs 8259 df-recs 8308 df-rdg 8346 df-1o 8402 df-er 8640 df-map 8772 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-pnf 11179 df-mnf 11180 df-xr 11181 df-ltxr 11182 df-le 11183 df-sub 11377 df-neg 11378 df-nn 12173 df-2 12242 df-3 12243 df-4 12244 df-5 12245 df-6 12246 df-n0 12436 df-z 12523 df-uz 12787 df-fz 13460 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17178 df-ress 17199 df-plusg 17231 df-mulr 17232 df-sca 17234 df-vsca 17235 df-0g 17402 df-proset 18258 df-poset 18277 df-plt 18292 df-lub 18308 df-glb 18309 df-join 18310 df-meet 18311 df-p0 18387 df-p1 18388 df-lat 18396 df-clat 18463 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-submnd 18750 df-grp 18910 df-minusg 18911 df-sbg 18912 df-subg 19097 df-cntz 19290 df-lsm 19609 df-cmn 19755 df-abl 19756 df-mgp 20120 df-rng 20132 df-ur 20161 df-ring 20214 df-oppr 20315 df-dvdsr 20335 df-unit 20336 df-invr 20366 df-dvr 20379 df-drng 20710 df-lmod 20859 df-lss 20929 df-lsp 20969 df-lvec 21100 df-lsatoms 39475 df-lshyp 39476 df-lfl 39557 df-lkr 39585 df-ldual 39623 df-oposet 39675 df-ol 39677 df-oml 39678 df-covers 39765 df-ats 39766 df-atl 39797 df-cvlat 39821 df-hlat 39850 df-llines 39997 df-lplanes 39998 df-lvols 39999 df-lines 40000 df-psubsp 40002 df-pmap 40003 df-padd 40295 df-lhyp 40487 df-laut 40488 df-ldil 40603 df-ltrn 40604 df-trl 40658 df-tgrp 41242 df-tendo 41254 df-edring 41256 df-dveca 41502 df-disoa 41528 df-dvech 41578 df-dib 41638 df-dic 41672 df-dih 41728 df-doch 41847 df-djh 41894

This theorem is referenced by: mapdrvallem2 42144

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