lcfl8b - Mathbox for Norm Megill

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Theorem lcfl8b 42092

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 3916	. . . 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 42079	. . . . . . 7 ⊢ (𝐺 ∈ 𝐶 ↔ (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
12	11	simplbi 500	. . . . . 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 42090	. . . 4 ⊢ (𝜑 → (𝐺 ∈ 𝐶 ↔ ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥})))
15	2, 14	mpbid 234	. . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝑉 (𝐿‘𝐺) = ( ⊥ ‘{𝑥}))
16		fveq2 6863	. . . . . . . . . 10 ⊢ ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
17	16	adantl 485	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = ( ⊥ ‘( ⊥ ‘{𝑥})))
18		lcfl8b.n	. . . . . . . . . 10 ⊢ 𝑁 = (LSpan‘𝑈)
19	10	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
20		simplr 778	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ∈ 𝑉)
21	3, 5, 4, 6, 18, 19, 20	dochocsn 41969	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘{𝑥})) = (𝑁‘{𝑥}))
22	17, 21	eqtrd 2796	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))
23	2, 11	sylib 220	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐺 ∈ 𝐹 ∧ ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺)))
24	23	simprd 499	. . . . . . . . . . 11 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) = (𝐿‘𝐺))
25		eldifsni 4749	. . . . . . . . . . . . 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 41698	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑈 ∈ LMod)
30	6, 7, 8, 27, 28, 29, 13	lkr0f2 39749	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝐿‘𝐺) = 𝑉 ↔ 𝐺 = 𝑌))
31	30	necon3bid 3000	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝐿‘𝐺) ≠ 𝑉 ↔ 𝐺 ≠ 𝑌))
32	26, 31	mpbird 259	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐿‘𝐺) ≠ 𝑉)
33	24, 32	eqnetrd 3023	. . . . . . . . . 10 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
34	33	ad2antrr 736	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉)
35		eqid 2761	. . . . . . . . . 10 ⊢ (LSAtoms‘𝑈) = (LSAtoms‘𝑈)
36	13	ad2antrr 736	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝐺 ∈ 𝐹)
37	3, 4, 5, 6, 35, 7, 8, 19, 36	dochkrsat2 42044	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (( ⊥ ‘( ⊥ ‘(𝐿‘𝐺))) ≠ 𝑉 ↔ ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈)))
38	34, 37	mpbid 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ( ⊥ ‘(𝐿‘𝐺)) ∈ (LSAtoms‘𝑈))
39	22, 38	eqeltrrd 2862	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈))
40		lcfl8b.z	. . . . . . . 8 ⊢ 0 = (0_g‘𝑈)
41	29	ad2antrr 736	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑈 ∈ LMod)
42	6, 18, 40, 35, 41, 20	lsatspn0 39588	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → ((𝑁‘{𝑥}) ∈ (LSAtoms‘𝑈) ↔ 𝑥 ≠ 0 ))
43	39, 42	mpbid 234	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → 𝑥 ≠ 0 )
44	43, 22	jca 519	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝑉) ∧ (𝐿‘𝐺) = ( ⊥ ‘{𝑥})) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
45	44	ex 416	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑉) → ((𝐿‘𝐺) = ( ⊥ ‘{𝑥}) → (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))))
46	45	reximdva 3174	. . 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 4753	. 2 ⊢ (∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}) ↔ ∃𝑥 ∈ 𝑉 (𝑥 ≠ 0 ∧ ( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥})))
49	47, 48	sylibr 236	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑉 ∖ { 0 })( ⊥ ‘(𝐿‘𝐺)) = (𝑁‘{𝑥}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ≠ wne 2956 ∃wrex 3085 {crab 3413 ∖ cdif 3901 {csn 4581 ‘cfv 6517 Basecbs 17228 0_gc0g 17451 LModclmod 20907 LSpanclspn 21018 LSAtomsclsa 39562 LFnlclfn 39645 LKerclk 39673 LDualcld 39711 HLchlt 39938 LHypclh 40572 DVecHcdvh 41666 ocHcoch 41935

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-riotaBAD 39541

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-of 7656 df-om 7843 df-1st 7966 df-2nd 7967 df-tpos 8201 df-undef 8248 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-er 8673 df-map 8805 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-nn 12208 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-n0 12479 df-z 12566 df-uz 12837 df-fz 13510 df-struct 17166 df-sets 17183 df-slot 17201 df-ndx 17213 df-base 17229 df-ress 17250 df-plusg 17282 df-mulr 17283 df-sca 17285 df-vsca 17286 df-0g 17453 df-proset 18309 df-poset 18328 df-plt 18343 df-lub 18359 df-glb 18360 df-join 18361 df-meet 18362 df-p0 18438 df-p1 18439 df-lat 18447 df-clat 18514 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-submnd 18801 df-grp 18961 df-minusg 18962 df-sbg 18963 df-subg 19148 df-cntz 19340 df-lsm 19659 df-cmn 19805 df-abl 19806 df-mgp 20170 df-rng 20182 df-ur 20211 df-ring 20264 df-oppr 20365 df-dvdsr 20385 df-unit 20386 df-invr 20416 df-dvr 20429 df-drng 20760 df-lmod 20909 df-lss 20979 df-lsp 21019 df-lvec 21150 df-lsatoms 39564 df-lshyp 39565 df-lfl 39646 df-lkr 39674 df-ldual 39712 df-oposet 39764 df-ol 39766 df-oml 39767 df-covers 39854 df-ats 39855 df-atl 39886 df-cvlat 39910 df-hlat 39939 df-llines 40086 df-lplanes 40087 df-lvols 40088 df-lines 40089 df-psubsp 40091 df-pmap 40092 df-padd 40384 df-lhyp 40576 df-laut 40577 df-ldil 40692 df-ltrn 40693 df-trl 40747 df-tgrp 41331 df-tendo 41343 df-edring 41345 df-dveca 41591 df-disoa 41617 df-dvech 41667 df-dib 41727 df-dic 41761 df-dih 41817 df-doch 41936 df-djh 41983

This theorem is referenced by: mapdrvallem2 42233

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