lcfl8b - Mathbox for Norm Megill

	Mathbox for Norm Megill		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > lcfl8b		Structured version Visualization version GIF version

Theorem lcfl8b 41874

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 3915	. . . 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 41861	. . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
12	11	simplbi 496	. . . . . 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 41872	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
15	2, 14	mpbid 232	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))
16		fveq2 6842	. . . . . . . . . 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 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉)
21	3, 5, 4, 6, 18, 19, 20	dochocsn 41751	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥}))
22	17, 21	eqtrd 2772	. . . . . . . 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 4748	. . . . . . . . . . . . 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 41480	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	6, 7, 8, 27, 28, 29, 13	lkr0f2 39531	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌))
31	30	necon3bid 2977	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌))
32	26, 31	mpbird 257	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉)
33	24, 32	eqnetrd 3000	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
34	33	ad2antrr 727	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
35		eqid 2737	. . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
36	13	ad2antrr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹)
37	3, 4, 5, 6, 35, 7, 8, 19, 36	dochkrsat2 41826	. . . . . . . . 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 2838	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
40		lcfl8b.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
41	29	ad2antrr 727	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod)
42	6, 18, 40, 35, 41, 20	lsatspn0 39370	. . . . . . 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 3151	. . 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 4752	. 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 1542 ∈ wcel 2114 ≠ wne 2933 ∃wrex 3062 {crab 3401 ∖ cdif 3900 {csn 4582 ‘cfv 6500 Basecbs 17148 0_gc0g 17371 LModclmod 20823 LSpanclspn 20934 LSAtomsclsa 39344 LFnlclfn 39427 LKerclk 39455 LDualcld 39493 HLchlt 39720 LHypclh 40354 DVecHcdvh 41448 ocHcoch 41717

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-riotaBAD 39323

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-tpos 8178 df-undef 8225 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-sca 17205 df-vsca 17206 df-0g 17373 df-proset 18229 df-poset 18248 df-plt 18263 df-lub 18279 df-glb 18280 df-join 18281 df-meet 18282 df-p0 18358 df-p1 18359 df-lat 18367 df-clat 18434 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-subg 19065 df-cntz 19258 df-lsm 19577 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-oppr 20285 df-dvdsr 20305 df-unit 20306 df-invr 20336 df-dvr 20349 df-drng 20676 df-lmod 20825 df-lss 20895 df-lsp 20935 df-lvec 21067 df-lsatoms 39346 df-lshyp 39347 df-lfl 39428 df-lkr 39456 df-ldual 39494 df-oposet 39546 df-ol 39548 df-oml 39549 df-covers 39636 df-ats 39637 df-atl 39668 df-cvlat 39692 df-hlat 39721 df-llines 39868 df-lplanes 39869 df-lvols 39870 df-lines 39871 df-psubsp 39873 df-pmap 39874 df-padd 40166 df-lhyp 40358 df-laut 40359 df-ldil 40474 df-ltrn 40475 df-trl 40529 df-tgrp 41113 df-tendo 41125 df-edring 41127 df-dveca 41373 df-disoa 41399 df-dvech 41449 df-dib 41509 df-dic 41543 df-dih 41599 df-doch 41718 df-djh 41765

This theorem is referenced by: mapdrvallem2 42015

W3C validator