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Theorem lcfrlem35 38158
Description: Lemma for lcfr 38166. (Contributed by NM, 2-Mar-2015.)
Hypotheses
Ref Expression
lcfrlem17.h 𝐻 = (LHyp‘𝐾)
lcfrlem17.o = ((ocH‘𝐾)‘𝑊)
lcfrlem17.u 𝑈 = ((DVecH‘𝐾)‘𝑊)
lcfrlem17.v 𝑉 = (Base‘𝑈)
lcfrlem17.p + = (+g𝑈)
lcfrlem17.z 0 = (0g𝑈)
lcfrlem17.n 𝑁 = (LSpan‘𝑈)
lcfrlem17.a 𝐴 = (LSAtoms‘𝑈)
lcfrlem17.k (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
lcfrlem17.x (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.y (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
lcfrlem17.ne (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
lcfrlem22.b 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))
lcfrlem24.t · = ( ·𝑠𝑈)
lcfrlem24.s 𝑆 = (Scalar‘𝑈)
lcfrlem24.q 𝑄 = (0g𝑆)
lcfrlem24.r 𝑅 = (Base‘𝑆)
lcfrlem24.j 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
lcfrlem24.ib (𝜑𝐼𝐵)
lcfrlem24.l 𝐿 = (LKer‘𝑈)
lcfrlem25.d 𝐷 = (LDual‘𝑈)
lcfrlem28.jn (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)
lcfrlem29.i 𝐹 = (invr𝑆)
lcfrlem30.m = (-g𝐷)
lcfrlem30.c 𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))
Assertion
Ref Expression
lcfrlem35 (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶))
Distinct variable groups:   𝑣,𝑘,𝑤,𝑥,   + ,𝑘,𝑣,𝑤,𝑥   𝑅,𝑘,𝑣,𝑥   𝑆,𝑘   · ,𝑘,𝑣,𝑤,𝑥   𝑣,𝑉,𝑥   𝑘,𝑋,𝑣,𝑤,𝑥   𝑘,𝑌,𝑣,𝑤,𝑥   𝑥, 0
Allowed substitution hints:   𝜑(𝑥,𝑤,𝑣,𝑘)   𝐴(𝑥,𝑤,𝑣,𝑘)   𝐵(𝑥,𝑤,𝑣,𝑘)   𝐶(𝑥,𝑤,𝑣,𝑘)   𝐷(𝑥,𝑤,𝑣,𝑘)   𝑄(𝑥,𝑤,𝑣,𝑘)   𝑅(𝑤)   𝑆(𝑥,𝑤,𝑣)   𝑈(𝑥,𝑤,𝑣,𝑘)   𝐹(𝑥,𝑤,𝑣,𝑘)   𝐻(𝑥,𝑤,𝑣,𝑘)   𝐼(𝑥,𝑤,𝑣,𝑘)   𝐽(𝑥,𝑤,𝑣,𝑘)   𝐾(𝑥,𝑤,𝑣,𝑘)   𝐿(𝑥,𝑤,𝑣,𝑘)   (𝑥,𝑤,𝑣,𝑘)   𝑁(𝑥,𝑤,𝑣,𝑘)   𝑉(𝑤,𝑘)   𝑊(𝑥,𝑤,𝑣,𝑘)   0 (𝑤,𝑣,𝑘)

Proof of Theorem lcfrlem35
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 lcfrlem17.h . . . 4 𝐻 = (LHyp‘𝐾)
2 lcfrlem17.o . . . 4 = ((ocH‘𝐾)‘𝑊)
3 lcfrlem17.u . . . 4 𝑈 = ((DVecH‘𝐾)‘𝑊)
4 lcfrlem17.v . . . 4 𝑉 = (Base‘𝑈)
5 lcfrlem17.p . . . 4 + = (+g𝑈)
6 lcfrlem17.z . . . 4 0 = (0g𝑈)
7 lcfrlem17.n . . . 4 𝑁 = (LSpan‘𝑈)
8 lcfrlem17.a . . . 4 𝐴 = (LSAtoms‘𝑈)
9 lcfrlem17.k . . . 4 (𝜑 → (𝐾 ∈ HL ∧ 𝑊𝐻))
10 lcfrlem17.x . . . 4 (𝜑𝑋 ∈ (𝑉 ∖ { 0 }))
11 lcfrlem17.y . . . 4 (𝜑𝑌 ∈ (𝑉 ∖ { 0 }))
12 lcfrlem17.ne . . . 4 (𝜑 → (𝑁‘{𝑋}) ≠ (𝑁‘{𝑌}))
13 lcfrlem22.b . . . 4 𝐵 = ((𝑁‘{𝑋, 𝑌}) ∩ ( ‘{(𝑋 + 𝑌)}))
14 eqid 2772 . . . 4 (LSSum‘𝑈) = (LSSum‘𝑈)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 38146 . . 3 (𝜑 → (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) = ( ‘{(𝑋 + 𝑌)}))
16 lcfrlem24.t . . . . . 6 · = ( ·𝑠𝑈)
17 lcfrlem24.s . . . . . 6 𝑆 = (Scalar‘𝑈)
18 lcfrlem24.q . . . . . 6 𝑄 = (0g𝑆)
19 lcfrlem24.r . . . . . 6 𝑅 = (Base‘𝑆)
20 lcfrlem24.j . . . . . 6 𝐽 = (𝑥 ∈ (𝑉 ∖ { 0 }) ↦ (𝑣𝑉 ↦ (𝑘𝑅𝑤 ∈ ( ‘{𝑥})𝑣 = (𝑤 + (𝑘 · 𝑥)))))
21 lcfrlem24.ib . . . . . 6 (𝜑𝐼𝐵)
22 lcfrlem24.l . . . . . 6 𝐿 = (LKer‘𝑈)
231, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22lcfrlem24 38147 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))))
24 eqid 2772 . . . . . 6 (.r𝑆) = (.r𝑆)
25 lcfrlem29.i . . . . . 6 𝐹 = (invr𝑆)
26 eqid 2772 . . . . . 6 (LFnl‘𝑈) = (LFnl‘𝑈)
27 lcfrlem25.d . . . . . 6 𝐷 = (LDual‘𝑈)
28 eqid 2772 . . . . . 6 ( ·𝑠𝐷) = ( ·𝑠𝐷)
29 lcfrlem30.m . . . . . 6 = (-g𝐷)
301, 3, 9dvhlvec 37690 . . . . . 6 (𝜑𝑈 ∈ LVec)
31 eqid 2772 . . . . . . 7 (0g𝐷) = (0g𝐷)
32 eqid 2772 . . . . . . 7 {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)} = {𝑓 ∈ (LFnl‘𝑈) ∣ ( ‘( ‘(𝐿𝑓))) = (𝐿𝑓)}
331, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 10lcfrlem10 38133 . . . . . 6 (𝜑 → (𝐽𝑋) ∈ (LFnl‘𝑈))
341, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11lcfrlem10 38133 . . . . . 6 (𝜑 → (𝐽𝑌) ∈ (LFnl‘𝑈))
35 eqid 2772 . . . . . . . 8 (LSubSp‘𝑈) = (LSubSp‘𝑈)
361, 3, 9dvhlmod 37691 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
371, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 38145 . . . . . . . 8 (𝜑𝐵𝐴)
3835, 8, 36, 37lsatlssel 35578 . . . . . . 7 (𝜑𝐵 ∈ (LSubSp‘𝑈))
394, 35lssel 19425 . . . . . . 7 ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼𝐵) → 𝐼𝑉)
4038, 21, 39syl2anc 576 . . . . . 6 (𝜑𝐼𝑉)
41 lcfrlem28.jn . . . . . 6 (𝜑 → ((𝐽𝑌)‘𝐼) ≠ 𝑄)
42 lcfrlem30.c . . . . . 6 𝐶 = ((𝐽𝑋) (((𝐹‘((𝐽𝑌)‘𝐼))(.r𝑆)((𝐽𝑋)‘𝐼))( ·𝑠𝐷)(𝐽𝑌)))
434, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem2 38124 . . . . 5 (𝜑 → ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))) ⊆ (𝐿𝐶))
4423, 43eqsstrd 3889 . . . 4 (𝜑 → ( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶))
451, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41lcfrlem28 38151 . . . . . 6 (𝜑𝐼0 )
466, 7, 8, 30, 37, 21, 45lsatel 35586 . . . . 5 (𝜑𝐵 = (𝑁‘{𝐼}))
471, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42lcfrlem30 38153 . . . . . . 7 (𝜑𝐶 ∈ (LFnl‘𝑈))
4826, 22, 35lkrlss 35676 . . . . . . 7 ((𝑈 ∈ LMod ∧ 𝐶 ∈ (LFnl‘𝑈)) → (𝐿𝐶) ∈ (LSubSp‘𝑈))
4936, 47, 48syl2anc 576 . . . . . 6 (𝜑 → (𝐿𝐶) ∈ (LSubSp‘𝑈))
504, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem3 38125 . . . . . 6 (𝜑𝐼 ∈ (𝐿𝐶))
5135, 7, 36, 49, 50lspsnel5a 19484 . . . . 5 (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿𝐶))
5246, 51eqsstrd 3889 . . . 4 (𝜑𝐵 ⊆ (𝐿𝐶))
5335lsssssubg 19446 . . . . . . 7 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5436, 53syl 17 . . . . . 6 (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5510eldifad 3835 . . . . . . . 8 (𝜑𝑋𝑉)
5611eldifad 3835 . . . . . . . 8 (𝜑𝑌𝑉)
57 prssi 4622 . . . . . . . 8 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
5855, 56, 57syl2anc 576 . . . . . . 7 (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
591, 3, 4, 35, 2dochlss 37935 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
609, 58, 59syl2anc 576 . . . . . 6 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
6154, 60sseldd 3853 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
6254, 38sseldd 3853 . . . . 5 (𝜑𝐵 ∈ (SubGrp‘𝑈))
6354, 49sseldd 3853 . . . . 5 (𝜑 → (𝐿𝐶) ∈ (SubGrp‘𝑈))
6414lsmlub 18543 . . . . 5 ((( ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈) ∧ 𝐵 ∈ (SubGrp‘𝑈) ∧ (𝐿𝐶) ∈ (SubGrp‘𝑈)) → ((( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶) ∧ 𝐵 ⊆ (𝐿𝐶)) ↔ (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶)))
6561, 62, 63, 64syl3anc 1351 . . . 4 (𝜑 → ((( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶) ∧ 𝐵 ⊆ (𝐿𝐶)) ↔ (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶)))
6644, 52, 65mpbi2and 699 . . 3 (𝜑 → (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶))
6715, 66eqsstr3d 3890 . 2 (𝜑 → ( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶))
68 eqid 2772 . . 3 (LSHyp‘𝑈) = (LSHyp‘𝑈)
691, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 38140 . . . 4 (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
701, 2, 3, 4, 6, 68, 9, 69dochsnshp 38034 . . 3 (𝜑 → ( ‘{(𝑋 + 𝑌)}) ∈ (LSHyp‘𝑈))
711, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 27, 41, 25, 29, 42lcfrlem34 38157 . . . 4 (𝜑𝐶 ≠ (0g𝐷))
7268, 26, 22, 27, 31, 30, 47lduallkr3 35743 . . . 4 (𝜑 → ((𝐿𝐶) ∈ (LSHyp‘𝑈) ↔ 𝐶 ≠ (0g𝐷)))
7371, 72mpbird 249 . . 3 (𝜑 → (𝐿𝐶) ∈ (LSHyp‘𝑈))
7468, 30, 70, 73lshpcmp 35569 . 2 (𝜑 → (( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶) ↔ ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶)))
7567, 74mpbid 224 1 (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2050  wne 2961  wrex 3083  {crab 3086  cdif 3820  cin 3822  wss 3823  {csn 4435  {cpr 4437  cmpt 5002  cfv 6182  crio 6930  (class class class)co 6970  Basecbs 16333  +gcplusg 16415  .rcmulr 16416  Scalarcsca 16418   ·𝑠 cvsca 16419  0gc0g 16563  -gcsg 17887  SubGrpcsubg 18051  LSSumclsm 18514  invrcinvr 19138  LModclmod 19350  LSubSpclss 19419  LSpanclspn 19459  LSAtomsclsa 35555  LSHypclsh 35556  LFnlclfn 35638  LKerclk 35666  LDualcld 35704  HLchlt 35931  LHypclh 36565  DVecHcdvh 37659  ocHcoch 37928
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-rep 5043  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273  ax-cnex 10385  ax-resscn 10386  ax-1cn 10387  ax-icn 10388  ax-addcl 10389  ax-addrcl 10390  ax-mulcl 10391  ax-mulrcl 10392  ax-mulcom 10393  ax-addass 10394  ax-mulass 10395  ax-distr 10396  ax-i2m1 10397  ax-1ne0 10398  ax-1rid 10399  ax-rnegex 10400  ax-rrecex 10401  ax-cnre 10402  ax-pre-lttri 10403  ax-pre-lttrn 10404  ax-pre-ltadd 10405  ax-pre-mulgt0 10406  ax-riotaBAD 35534
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-reu 3089  df-rmo 3090  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-pss 3839  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4707  df-int 4744  df-iun 4788  df-iin 4789  df-br 4924  df-opab 4986  df-mpt 5003  df-tr 5025  df-id 5306  df-eprel 5311  df-po 5320  df-so 5321  df-fr 5360  df-we 5362  df-xp 5407  df-rel 5408  df-cnv 5409  df-co 5410  df-dm 5411  df-rn 5412  df-res 5413  df-ima 5414  df-pred 5980  df-ord 6026  df-on 6027  df-lim 6028  df-suc 6029  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-f1 6187  df-fo 6188  df-f1o 6189  df-fv 6190  df-riota 6931  df-ov 6973  df-oprab 6974  df-mpo 6975  df-of 7221  df-om 7391  df-1st 7495  df-2nd 7496  df-tpos 7689  df-undef 7736  df-wrecs 7744  df-recs 7806  df-rdg 7844  df-1o 7899  df-oadd 7903  df-er 8083  df-map 8202  df-en 8301  df-dom 8302  df-sdom 8303  df-fin 8304  df-pnf 10470  df-mnf 10471  df-xr 10472  df-ltxr 10473  df-le 10474  df-sub 10666  df-neg 10667  df-nn 11434  df-2 11497  df-3 11498  df-4 11499  df-5 11500  df-6 11501  df-n0 11702  df-z 11788  df-uz 12053  df-fz 12703  df-struct 16335  df-ndx 16336  df-slot 16337  df-base 16339  df-sets 16340  df-ress 16341  df-plusg 16428  df-mulr 16429  df-sca 16431  df-vsca 16432  df-0g 16565  df-mre 16709  df-mrc 16710  df-acs 16712  df-proset 17390  df-poset 17408  df-plt 17420  df-lub 17436  df-glb 17437  df-join 17438  df-meet 17439  df-p0 17501  df-p1 17502  df-lat 17508  df-clat 17570  df-mgm 17704  df-sgrp 17746  df-mnd 17757  df-submnd 17798  df-grp 17888  df-minusg 17889  df-sbg 17890  df-subg 18054  df-cntz 18212  df-oppg 18239  df-lsm 18516  df-cmn 18662  df-abl 18663  df-mgp 18957  df-ur 18969  df-ring 19016  df-oppr 19090  df-dvdsr 19108  df-unit 19109  df-invr 19139  df-dvr 19150  df-drng 19221  df-lmod 19352  df-lss 19420  df-lsp 19460  df-lvec 19591  df-lsatoms 35557  df-lshyp 35558  df-lcv 35600  df-lfl 35639  df-lkr 35667  df-ldual 35705  df-oposet 35757  df-ol 35759  df-oml 35760  df-covers 35847  df-ats 35848  df-atl 35879  df-cvlat 35903  df-hlat 35932  df-llines 36079  df-lplanes 36080  df-lvols 36081  df-lines 36082  df-psubsp 36084  df-pmap 36085  df-padd 36377  df-lhyp 36569  df-laut 36570  df-ldil 36685  df-ltrn 36686  df-trl 36740  df-tgrp 37324  df-tendo 37336  df-edring 37338  df-dveca 37584  df-disoa 37610  df-dvech 37660  df-dib 37720  df-dic 37754  df-dih 37810  df-doch 37929  df-djh 37976
This theorem is referenced by:  lcfrlem36  38159
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