dochfln0 - Mathbox for Norm Megill

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Theorem dochfln0 37287

Description: The value of a functional is nonzero at a nonzero vector in the orthocomplement of its kernel. (Contributed by NM, 2-Jan-2015.)

**Hypotheses**
Ref	Expression
dochfln0.h	⊢ 𝐻 = (LHyp‘𝐾)
dochfln0.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
dochfln0.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
dochfln0.v	⊢ 𝑉 = (Base‘𝑈)
dochfln0.r	⊢ 𝑅 = (Scalar‘𝑈)
dochfln0.n	⊢ 𝑁 = (0_g‘𝑅)
dochfln0.z	⊢ 0 = (0_g‘𝑈)
dochfln0.f	⊢ 𝐹 = (LFnl‘𝑈)
dochfln0.l	⊢ 𝐿 = (LKer‘𝑈)
dochfln0.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
dochfln0.g	⊢ (𝜑 → 𝐺 ∈ 𝐹)
dochfln0.x	⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))

**Assertion**
Ref	Expression
dochfln0	⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁)

**Proof of Theorem dochfln0**
Step	Hyp	Ref	Expression
1		dochfln0.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		dochfln0.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		dochfln0.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		dochfln0.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
5		dochfln0.z	. . 3 ⊢ 0 = (0_g‘𝑈)
6		dochfln0.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
7		dochfln0.f	. . . . . . 7 ⊢ 𝐹 = (LFnl‘𝑈)
8		dochfln0.l	. . . . . . 7 ⊢ 𝐿 = (LKer‘𝑈)
9	1, 3, 6	dvhlmod 36920	. . . . . . 7 ⊢ (𝜑 → 𝑈 ∈ LMod)
10		dochfln0.g	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝐹)
11	4, 7, 8, 9, 10	lkrssv 34905	. . . . . 6 ⊢ (𝜑 → (𝐿‘𝐺) ⊆ 𝑉)
12	1, 3, 4, 2	dochssv 37165	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝐿‘𝐺) ⊆ 𝑉) → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
13	6, 11, 12	syl2anc 573	. . . . 5 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘𝐺)) ⊆ 𝑉)
14	13	ssdifd 3897	. . . 4 ⊢ (𝜑 → (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }) ⊆ (𝑉 ∖ { 0 }))
15		dochfln0.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ (( ⊥ ‘(𝐿‘𝐺)) ∖ { 0 }))
16	14, 15	sseldd 3753	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
17	1, 2, 3, 4, 5, 6, 16	dochnel 37203	. 2 ⊢ (𝜑 → ¬ 𝑋 ∈ ( ⊥ ‘{𝑋}))
18	15	eldifad 3735	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ( ⊥ ‘(𝐿‘𝐺)))
19	13, 18	sseldd 3753	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
20	19	biantrurd 522	. . . 4 ⊢ (𝜑 → ((𝐺‘𝑋) = 𝑁 ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁)))
21		dochfln0.r	. . . . . 6 ⊢ 𝑅 = (Scalar‘𝑈)
22		dochfln0.n	. . . . . 6 ⊢ 𝑁 = (0_g‘𝑅)
23	4, 21, 22, 7, 8	ellkr 34898	. . . . 5 ⊢ ((𝑈 ∈ LMod ∧ 𝐺 ∈ 𝐹) → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁)))
24	9, 10, 23	syl2anc 573	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ (𝑋 ∈ 𝑉 ∧ (𝐺‘𝑋) = 𝑁)))
25	1, 2, 3, 4, 5, 7, 8, 6, 10, 15	dochsnkr 37282	. . . . 5 ⊢ (𝜑 → (𝐿‘𝐺) = ( ⊥ ‘{𝑋}))
26	25	eleq2d 2836	. . . 4 ⊢ (𝜑 → (𝑋 ∈ (𝐿‘𝐺) ↔ 𝑋 ∈ ( ⊥ ‘{𝑋})))
27	20, 24, 26	3bitr2rd 297	. . 3 ⊢ (𝜑 → (𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) = 𝑁))
28	27	necon3bbid 2980	. 2 ⊢ (𝜑 → (¬ 𝑋 ∈ ( ⊥ ‘{𝑋}) ↔ (𝐺‘𝑋) ≠ 𝑁))
29	17, 28	mpbid 222	1 ⊢ (𝜑 → (𝐺‘𝑋) ≠ 𝑁)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 196 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∖ cdif 3720 ⊆ wss 3723 {csn 4316 ‘cfv 6031 Basecbs 16064 Scalarcsca 16152 0_gc0g 16308 LModclmod 19073 LFnlclfn 34866 LKerclk 34894 HLchlt 35159 LHypclh 35792 DVecHcdvh 36888 ocHcoch 37157

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-cnex 10194 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 ax-riotaBAD 34761

This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-fal 1637 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-int 4612 df-iun 4656 df-iin 4657 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-tpos 7504 df-undef 7551 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-1o 7713 df-oadd 7717 df-er 7896 df-map 8011 df-en 8110 df-dom 8111 df-sdom 8112 df-fin 8113 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-nn 11223 df-2 11281 df-3 11282 df-4 11283 df-5 11284 df-6 11285 df-n0 11495 df-z 11580 df-uz 11889 df-fz 12534 df-struct 16066 df-ndx 16067 df-slot 16068 df-base 16070 df-sets 16071 df-ress 16072 df-plusg 16162 df-mulr 16163 df-sca 16165 df-vsca 16166 df-0g 16310 df-preset 17136 df-poset 17154 df-plt 17166 df-lub 17182 df-glb 17183 df-join 17184 df-meet 17185 df-p0 17247 df-p1 17248 df-lat 17254 df-clat 17316 df-mgm 17450 df-sgrp 17492 df-mnd 17503 df-submnd 17544 df-grp 17633 df-minusg 17634 df-sbg 17635 df-subg 17799 df-cntz 17957 df-lsm 18258 df-cmn 18402 df-abl 18403 df-mgp 18698 df-ur 18710 df-ring 18757 df-oppr 18831 df-dvdsr 18849 df-unit 18850 df-invr 18880 df-dvr 18891 df-drng 18959 df-lmod 19075 df-lss 19143 df-lsp 19185 df-lvec 19316 df-lsatoms 34785 df-lshyp 34786 df-lfl 34867 df-lkr 34895 df-oposet 34985 df-ol 34987 df-oml 34988 df-covers 35075 df-ats 35076 df-atl 35107 df-cvlat 35131 df-hlat 35160 df-llines 35306 df-lplanes 35307 df-lvols 35308 df-lines 35309 df-psubsp 35311 df-pmap 35312 df-padd 35604 df-lhyp 35796 df-laut 35797 df-ldil 35912 df-ltrn 35913 df-trl 35968 df-tgrp 36552 df-tendo 36564 df-edring 36566 df-dveca 36812 df-disoa 36839 df-dvech 36889 df-dib 36949 df-dic 36983 df-dih 37039 df-doch 37158 df-djh 37205

This theorem is referenced by: dochkr1 37288 dochkr1OLDN 37289 lcfl6lem 37308

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