lcfrlem14 - Mathbox for Norm Megill

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Theorem lcfrlem14 39307

Description: Lemma for lcfr 39336. (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 39304	. . . 4 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
19		lcfrlem14.n	. . . . 5 ⊢ 𝑁 = (LSpan‘𝑈)
20	17	eldifad 3878	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝑉)
21	20	snssd 4722	. . . . 5 ⊢ (𝜑 → {𝑋} ⊆ 𝑉)
22	1, 3, 2, 4, 19, 16, 21	dochocsp 39130	. . . 4 ⊢ (𝜑 → ( ⊥ ‘(𝑁‘{𝑋})) = ( ⊥ ‘{𝑋}))
23	18, 22	eqtr4d 2780	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘(𝑁‘{𝑋})))
24	23	fveq2d 6721	. 2 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))))
25		eqid 2737	. . . . 5 ⊢ ((DIsoH‘𝐾)‘𝑊) = ((DIsoH‘𝐾)‘𝑊)
26	1, 3, 4, 19, 25	dihlsprn 39082	. . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝑋 ∈ 𝑉) → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
27	16, 20, 26	syl2anc 587	. . 3 ⊢ (𝜑 → (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊))
28	1, 25, 2	dochoc 39118	. . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑁‘{𝑋}) ∈ ran ((DIsoH‘𝐾)‘𝑊)) → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
29	16, 27, 28	syl2anc 587	. 2 ⊢ (𝜑 → ( ⊥ ‘( ⊥ ‘(𝑁‘{𝑋}))) = (𝑁‘{𝑋}))
30	24, 29	eqtrd 2777	1 ⊢ (𝜑 → ( ⊥ ‘(𝐿‘(𝐽‘𝑋))) = (𝑁‘{𝑋}))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ∃wrex 3062 {crab 3065 ∖ cdif 3863 {csn 4541 ↦ cmpt 5135 ran crn 5552 ‘cfv 6380 ℩crio 7169 (class class class)co 7213 Basecbs 16760 +_gcplusg 16802 Scalarcsca 16805 ·_𝑠 cvsca 16806 0_gc0g 16944 LSpanclspn 20008 LFnlclfn 36808 LKerclk 36836 LDualcld 36874 HLchlt 37101 LHypclh 37735 DVecHcdvh 38829 DIsoHcdih 38979 ocHcoch 39098

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-1cn 10787 ax-icn 10788 ax-addcl 10789 ax-addrcl 10790 ax-mulcl 10791 ax-mulrcl 10792 ax-mulcom 10793 ax-addass 10794 ax-mulass 10795 ax-distr 10796 ax-i2m1 10797 ax-1ne0 10798 ax-1rid 10799 ax-rnegex 10800 ax-rrecex 10801 ax-cnre 10802 ax-pre-lttri 10803 ax-pre-lttrn 10804 ax-pre-ltadd 10805 ax-pre-mulgt0 10806 ax-riotaBAD 36704

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-iin 4907 df-br 5054 df-opab 5116 df-mpt 5136 df-tr 5162 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-riota 7170 df-ov 7216 df-oprab 7217 df-mpo 7218 df-om 7645 df-1st 7761 df-2nd 7762 df-tpos 7968 df-undef 8015 df-wrecs 8047 df-recs 8108 df-rdg 8146 df-1o 8202 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-fin 8630 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-sub 11064 df-neg 11065 df-nn 11831 df-2 11893 df-3 11894 df-4 11895 df-5 11896 df-6 11897 df-n0 12091 df-z 12177 df-uz 12439 df-fz 13096 df-struct 16700 df-sets 16717 df-slot 16735 df-ndx 16745 df-base 16761 df-ress 16785 df-plusg 16815 df-mulr 16816 df-sca 16818 df-vsca 16819 df-0g 16946 df-proset 17802 df-poset 17820 df-plt 17836 df-lub 17852 df-glb 17853 df-join 17854 df-meet 17855 df-p0 17931 df-p1 17932 df-lat 17938 df-clat 18005 df-mgm 18114 df-sgrp 18163 df-mnd 18174 df-submnd 18219 df-grp 18368 df-minusg 18369 df-sbg 18370 df-subg 18540 df-cntz 18711 df-lsm 19025 df-cmn 19172 df-abl 19173 df-mgp 19505 df-ur 19517 df-ring 19564 df-oppr 19641 df-dvdsr 19659 df-unit 19660 df-invr 19690 df-dvr 19701 df-drng 19769 df-lmod 19901 df-lss 19969 df-lsp 20009 df-lvec 20140 df-lsatoms 36727 df-lshyp 36728 df-lfl 36809 df-lkr 36837 df-oposet 36927 df-ol 36929 df-oml 36930 df-covers 37017 df-ats 37018 df-atl 37049 df-cvlat 37073 df-hlat 37102 df-llines 37249 df-lplanes 37250 df-lvols 37251 df-lines 37252 df-psubsp 37254 df-pmap 37255 df-padd 37547 df-lhyp 37739 df-laut 37740 df-ldil 37855 df-ltrn 37856 df-trl 37910 df-tgrp 38494 df-tendo 38506 df-edring 38508 df-dveca 38754 df-disoa 38780 df-dvech 38830 df-dib 38890 df-dic 38924 df-dih 38980 df-doch 39099 df-djh 39146

This theorem is referenced by: lcfrlem15 39308 lcfrlem31 39324

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