lcfrlem12N - Mathbox for Norm Megill

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Theorem lcfrlem12N 41599

Description: Lemma for lcfr 41630. (Contributed by NM, 23-Feb-2015.) (New usage is discouraged.)

**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 }))
lcfrlem12.b	⊢ 𝐵 = (0_g‘𝑆)
lcfrlem12.y	⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋}))

**Assertion**
Ref	Expression
lcfrlem12N	⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)

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

**Proof of Theorem lcfrlem12N**
Step	Hyp	Ref	Expression
1		lcf1o.h	. . 3 ⊢ 𝐻 = (LHyp‘𝐾)
2		lcf1o.u	. . 3 ⊢ 𝑈 = ((DVecH‘𝐾)‘𝑊)
3		lcflo.k	. . 3 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻))
4	1, 2, 3	dvhlmod 41155	. 2 ⊢ (𝜑 → 𝑈 ∈ LMod)
5		lcf1o.o	. . 3 ⊢ ⊥ = ((ocH‘𝐾)‘𝑊)
6		lcf1o.v	. . 3 ⊢ 𝑉 = (Base‘𝑈)
7		lcf1o.a	. . 3 ⊢ + = (+_g‘𝑈)
8		lcf1o.t	. . 3 ⊢ · = ( ·_𝑠 ‘𝑈)
9		lcf1o.s	. . 3 ⊢ 𝑆 = (Scalar‘𝑈)
10		lcf1o.r	. . 3 ⊢ 𝑅 = (Base‘𝑆)
11		lcf1o.z	. . 3 ⊢ 0 = (0_g‘𝑈)
12		lcf1o.f	. . 3 ⊢ 𝐹 = (LFnl‘𝑈)
13		lcf1o.l	. . 3 ⊢ 𝐿 = (LKer‘𝑈)
14		lcf1o.d	. . 3 ⊢ 𝐷 = (LDual‘𝑈)
15		lcf1o.q	. . 3 ⊢ 𝑄 = (0_g‘𝐷)
16		lcf1o.c	. . 3 ⊢ 𝐶 = {𝑓 ∈ 𝐹 ∣ ( ⊥ ‘( ⊥ ‘(𝐿‘𝑓))) = (𝐿‘𝑓)}
17		lcf1o.j	. . 3 ⊢ 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣 ∈ 𝑉 ↦ (℩𝑘 ∈ 𝑅 ∃𝑤 ∈ ( ⊥ ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
18		lcfrlem10.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ (𝑉 ∖ { 0 }))
19	1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18	lcfrlem10 41597	. 2 ⊢ (𝜑 → (𝐽‘𝑋) ∈ 𝐹)
20		lcfrlem12.y	. . 3 ⊢ (𝜑 → 𝑌 ∈ ( ⊥ ‘{𝑋}))
21	1, 5, 2, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 3, 18	lcfrlem11 41598	. . 3 ⊢ (𝜑 → (𝐿‘(𝐽‘𝑋)) = ( ⊥ ‘{𝑋}))
22	20, 21	eleqtrrd 2834	. 2 ⊢ (𝜑 → 𝑌 ∈ (𝐿‘(𝐽‘𝑋)))
23		lcfrlem12.b	. . 3 ⊢ 𝐵 = (0_g‘𝑆)
24	9, 23, 12, 13	lkrf0 39138	. 2 ⊢ ((𝑈 ∈ LMod ∧ (𝐽‘𝑋) ∈ 𝐹 ∧ 𝑌 ∈ (𝐿‘(𝐽‘𝑋))) → ((𝐽‘𝑋)‘𝑌) = 𝐵)
25	4, 19, 22, 24	syl3anc 1373	1 ⊢ (𝜑 → ((𝐽‘𝑋)‘𝑌) = 𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2111 ∃wrex 3056 {crab 3395 ∖ cdif 3899 {csn 4576 ↦ cmpt 5172 ‘cfv 6481 ℩crio 7302 (class class class)co 7346 Basecbs 17120 +_gcplusg 17161 Scalarcsca 17164 ·_𝑠 cvsca 17165 0_gc0g 17343 LModclmod 20794 LFnlclfn 39102 LKerclk 39130 LDualcld 39168 HLchlt 39395 LHypclh 40029 DVecHcdvh 41123 ocHcoch 41392

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11062 ax-resscn 11063 ax-1cn 11064 ax-icn 11065 ax-addcl 11066 ax-addrcl 11067 ax-mulcl 11068 ax-mulrcl 11069 ax-mulcom 11070 ax-addass 11071 ax-mulass 11072 ax-distr 11073 ax-i2m1 11074 ax-1ne0 11075 ax-1rid 11076 ax-rnegex 11077 ax-rrecex 11078 ax-cnre 11079 ax-pre-lttri 11080 ax-pre-lttrn 11081 ax-pre-ltadd 11082 ax-pre-mulgt0 11083 ax-riotaBAD 38998

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-tpos 8156 df-undef 8203 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-er 8622 df-map 8752 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-pnf 11148 df-mnf 11149 df-xr 11150 df-ltxr 11151 df-le 11152 df-sub 11346 df-neg 11347 df-nn 12126 df-2 12188 df-3 12189 df-4 12190 df-5 12191 df-6 12192 df-n0 12382 df-z 12469 df-uz 12733 df-fz 13408 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-sca 17177 df-vsca 17178 df-0g 17345 df-proset 18200 df-poset 18219 df-plt 18234 df-lub 18250 df-glb 18251 df-join 18252 df-meet 18253 df-p0 18329 df-p1 18330 df-lat 18338 df-clat 18405 df-mgm 18548 df-sgrp 18627 df-mnd 18643 df-submnd 18692 df-grp 18849 df-minusg 18850 df-sbg 18851 df-subg 19036 df-cntz 19230 df-lsm 19549 df-cmn 19695 df-abl 19696 df-mgp 20060 df-rng 20072 df-ur 20101 df-ring 20154 df-oppr 20256 df-dvdsr 20276 df-unit 20277 df-invr 20307 df-dvr 20320 df-drng 20647 df-lmod 20796 df-lss 20866 df-lsp 20906 df-lvec 21038 df-lsatoms 39021 df-lshyp 39022 df-lfl 39103 df-lkr 39131 df-oposet 39221 df-ol 39223 df-oml 39224 df-covers 39311 df-ats 39312 df-atl 39343 df-cvlat 39367 df-hlat 39396 df-llines 39543 df-lplanes 39544 df-lvols 39545 df-lines 39546 df-psubsp 39548 df-pmap 39549 df-padd 39841 df-lhyp 40033 df-laut 40034 df-ldil 40149 df-ltrn 40150 df-trl 40204 df-tgrp 40788 df-tendo 40800 df-edring 40802 df-dveca 41048 df-disoa 41074 df-dvech 41124 df-dib 41184 df-dic 41218 df-dih 41274 df-doch 41393 df-djh 41440

This theorem is referenced by: (None)

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