lcfrlem14 - Mathbox for Norm Megill

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Theorem lcfrlem14 39497

Description: Lemma for lcfr 39526. (Contributed by NM, 10-Mar-2015.)

**Hypotheses**
Ref	Expression
lcf1o.h	⊢ 𝐻 = (LHyp‘𝐾)
lcf1o.o	⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
lcf1o.u	⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcf1o.v	⊢ 𝑉 = (Base‘𝑈)
lcf1o.a	⊢ + = (+_g‘𝑈)
lcf1o.t	⊢ · = ( ·_𝑠 ‘𝑈)
lcf1o.s	⊢ 𝑆 = (Scalar‘𝑈)
lcf1o.r	⊢ 𝑅 = (Base‘𝑆)
lcf1o.z	⊢ 0 = (0_g‘𝑈)
lcf1o.f	⊢ 𝐹 = (LFnl‘𝑈)
lcf1o.l	⊢ 𝐿 = (LKer‘𝑈)
lcf1o.d	⊢ 𝐷 = (LDual‘𝑈)
lcf1o.q	⊢ 𝑄 = (0_g‘𝐷)
lcf1o.c	⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
lcf1o.j	⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcflo.k	⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
lcfrlem10.x	⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem14.n	⊢ 𝑁 = (LSpan‘𝑈)

**Assertion**
Ref	Expression
lcfrlem14	⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋}))

Distinct variable groups: 𝑥,𝑤, ⊥ 𝑥, 0 𝑥,𝑣,𝑉 𝑥, · 𝑣,𝑘,𝑤,𝑥,𝑋 𝑥, + 𝑥,𝑅 + ,𝑘,𝑣,𝑤 ⊥ ,𝑘,𝑣 𝑅,𝑘,𝑣 𝑆,𝑘 · ,𝑘,𝑣,𝑤,𝑓 + ,𝑓 𝑓,𝐹 𝑓,𝐿 ⊥ ,𝑓 𝑅,𝑓 · ,𝑓 𝑓,𝑉,𝑥 𝑓,𝐽 𝑓,𝑋 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑣,𝑓,𝑘) 𝐶(𝑥,𝑤,𝑣,𝑓,𝑘) 𝐷(𝑥,𝑤,𝑣,𝑓,𝑘) 𝑄(𝑥,𝑤,𝑣,𝑓,𝑘) 𝑅(𝑤) 𝑆(𝑥,𝑤,𝑣,𝑓) 𝑈(𝑥,𝑤,𝑣,𝑓,𝑘) 𝐹(𝑥,𝑤,𝑣,𝑘) 𝐻(𝑥,𝑤,𝑣,𝑓,𝑘) 𝐽(𝑥,𝑤,𝑣,𝑘) 𝐾(𝑥,𝑤,𝑣,𝑓,𝑘) 𝐿(𝑥,𝑤,𝑣,𝑘) 𝑁(𝑥,𝑤,𝑣,𝑓,𝑘) 𝑉(𝑤,𝑘) 𝑊(𝑥,𝑤,𝑣,𝑓,𝑘) 0 (𝑤,𝑣,𝑓,𝑘)

**Proof of Theorem lcfrlem14**
Step	Hyp	Ref	Expression
1		lcf1o.h	. . . . 5 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcf1o.o	. . . . 5 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
3		lcf1o.u	. . . . 5 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
4		lcf1o.v	. . . . 5 ⊢ 𝑉 = (Base‘𝑈)
5		lcf1o.a	. . . . 5 ⊢ + = (+_g‘𝑈)
6		lcf1o.t	. . . . 5 ⊢ · = ( ·_𝑠 ‘𝑈)
7		lcf1o.s	. . . . 5 ⊢ 𝑆 = (Scalar‘𝑈)
8		lcf1o.r	. . . . 5 ⊢ 𝑅 = (Base‘𝑆)
9		lcf1o.z	. . . . 5 ⊢ 0 = (0_g‘𝑈)
10		lcf1o.f	. . . . 5 ⊢ 𝐹 = (LFnl‘𝑈)
11		lcf1o.l	. . . . 5 ⊢ 𝐿 = (LKer‘𝑈)
12		lcf1o.d	. . . . 5 ⊢ 𝐷 = (LDual‘𝑈)
13		lcf1o.q	. . . . 5 ⊢ 𝑄 = (0_g‘𝐷)
14		lcf1o.c	. . . . 5 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
15		lcf1o.j	. . . . 5 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
16		lcflo.k	. . . . 5 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
17		lcfrlem10.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17	lcfrlem11 39494	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
19		lcfrlem14.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
20	17	eldifad 3895	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	20	snssd 4739	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
22	1, 3, 2, 4, 19, 16, 21	dochocsp 39320	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
23	18, 22	eqtr4d 2781	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘(𝑁‘{𝑋})))
24	23	fveq2d 6760	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))))
25		eqid 2738	. . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
26	1, 3, 4, 19, 25	dihlsprn 39272	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	16, 20, 26	syl2anc 583	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
28	1, 25, 2	dochoc 39308	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
29	16, 27, 28	syl2anc 583	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
30	24, 29	eqtrd 2778	1 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2108 ∃wrex 3064 {crab 3067 ∖ cdif 3880 {csn 4558 ↦ cmpt 5153 ran crn 5581 ‘cfv 6418 ℩crio 7211 (class class class)co 7255 Basecbs 16840 +_gcplusg 16888 Scalarcsca 16891 ·_𝑠 cvsca 16892 0_gc0g 17067 LSpanclspn 20148 LFnlclfn 36998 LKerclk 37026 LDualcld 37064 HLchlt 37291 LHypclh 37925 DVecHcdvh 39019 DIsoHcdih 39169 ocHcoch 39288

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-rep 5205 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879 ax-riotaBAD 36894

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-int 4877 df-iun 4923 df-iin 4924 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-tpos 8013 df-undef 8060 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-1o 8267 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-fin 8695 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-2 11966 df-3 11967 df-4 11968 df-5 11969 df-6 11970 df-n0 12164 df-z 12250 df-uz 12512 df-fz 13169 df-struct 16776 df-sets 16793 df-slot 16811 df-ndx 16823 df-base 16841 df-ress 16868 df-plusg 16901 df-mulr 16902 df-sca 16904 df-vsca 16905 df-0g 17069 df-proset 17928 df-poset 17946 df-plt 17963 df-lub 17979 df-glb 17980 df-join 17981 df-meet 17982 df-p0 18058 df-p1 18059 df-lat 18065 df-clat 18132 df-mgm 18241 df-sgrp 18290 df-mnd 18301 df-submnd 18346 df-grp 18495 df-minusg 18496 df-sbg 18497 df-subg 18667 df-cntz 18838 df-lsm 19156 df-cmn 19303 df-abl 19304 df-mgp 19636 df-ur 19653 df-ring 19700 df-oppr 19777 df-dvdsr 19798 df-unit 19799 df-invr 19829 df-dvr 19840 df-drng 19908 df-lmod 20040 df-lss 20109 df-lsp 20149 df-lvec 20280 df-lsatoms 36917 df-lshyp 36918 df-lfl 36999 df-lkr 37027 df-oposet 37117 df-ol 37119 df-oml 37120 df-covers 37207 df-ats 37208 df-atl 37239 df-cvlat 37263 df-hlat 37292 df-llines 37439 df-lplanes 37440 df-lvols 37441 df-lines 37442 df-psubsp 37444 df-pmap 37445 df-padd 37737 df-lhyp 37929 df-laut 37930 df-ldil 38045 df-ltrn 38046 df-trl 38100 df-tgrp 38684 df-tendo 38696 df-edring 38698 df-dveca 38944 df-disoa 38970 df-dvech 39020 df-dib 39080 df-dic 39114 df-dih 39170 df-doch 39289 df-djh 39336

This theorem is referenced by: lcfrlem15 39498 lcfrlem31 39514

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