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Theorem lcfrlem35 37183
 Description: Lemma for lcfr 37191. (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 2651 . . . 4 (LSSum‘𝑈) = (LSSum‘𝑈)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 37171 . . 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 37172 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))))
24 eqid 2651 . . . . . 6 (.r𝑆) = (.r𝑆)
25 lcfrlem29.i . . . . . 6 𝐹 = (invr𝑆)
26 eqid 2651 . . . . . 6 (LFnl‘𝑈) = (LFnl‘𝑈)
27 lcfrlem25.d . . . . . 6 𝐷 = (LDual‘𝑈)
28 eqid 2651 . . . . . 6 ( ·𝑠𝐷) = ( ·𝑠𝐷)
29 lcfrlem30.m . . . . . 6 = (-g𝐷)
301, 3, 9dvhlvec 36715 . . . . . 6 (𝜑𝑈 ∈ LVec)
31 eqid 2651 . . . . . . 7 (0g𝐷) = (0g𝐷)
32 eqid 2651 . . . . . . 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 37158 . . . . . 6 (𝜑 → (𝐽𝑋) ∈ (LFnl‘𝑈))
341, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11lcfrlem10 37158 . . . . . 6 (𝜑 → (𝐽𝑌) ∈ (LFnl‘𝑈))
35 eqid 2651 . . . . . . . 8 (LSubSp‘𝑈) = (LSubSp‘𝑈)
361, 3, 9dvhlmod 36716 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
371, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 37170 . . . . . . . 8 (𝜑𝐵𝐴)
3835, 8, 36, 37lsatlssel 34602 . . . . . . 7 (𝜑𝐵 ∈ (LSubSp‘𝑈))
394, 35lssel 18986 . . . . . . 7 ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼𝐵) → 𝐼𝑉)
4038, 21, 39syl2anc 694 . . . . . 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 37149 . . . . 5 (𝜑 → ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))) ⊆ (𝐿𝐶))
4423, 43eqsstrd 3672 . . . 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 37176 . . . . . 6 (𝜑𝐼0 )
466, 7, 8, 30, 37, 21, 45lsatel 34610 . . . . 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 37178 . . . . . . 7 (𝜑𝐶 ∈ (LFnl‘𝑈))
4826, 22, 35lkrlss 34700 . . . . . . 7 ((𝑈 ∈ LMod ∧ 𝐶 ∈ (LFnl‘𝑈)) → (𝐿𝐶) ∈ (LSubSp‘𝑈))
4936, 47, 48syl2anc 694 . . . . . 6 (𝜑 → (𝐿𝐶) ∈ (LSubSp‘𝑈))
504, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem3 37150 . . . . . 6 (𝜑𝐼 ∈ (𝐿𝐶))
5135, 7, 36, 49, 50lspsnel5a 19044 . . . . 5 (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿𝐶))
5246, 51eqsstrd 3672 . . . 4 (𝜑𝐵 ⊆ (𝐿𝐶))
5335lsssssubg 19006 . . . . . . 7 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5436, 53syl 17 . . . . . 6 (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5510eldifad 3619 . . . . . . . 8 (𝜑𝑋𝑉)
5611eldifad 3619 . . . . . . . 8 (𝜑𝑌𝑉)
57 prssi 4385 . . . . . . . 8 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
5855, 56, 57syl2anc 694 . . . . . . 7 (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
591, 3, 4, 35, 2dochlss 36960 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
609, 58, 59syl2anc 694 . . . . . 6 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
6154, 60sseldd 3637 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
6254, 38sseldd 3637 . . . . 5 (𝜑𝐵 ∈ (SubGrp‘𝑈))
6354, 49sseldd 3637 . . . . 5 (𝜑 → (𝐿𝐶) ∈ (SubGrp‘𝑈))
6414lsmlub 18124 . . . . 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 1366 . . . 4 (𝜑 → ((( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶) ∧ 𝐵 ⊆ (𝐿𝐶)) ↔ (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶)))
6644, 52, 65mpbi2and 976 . . 3 (𝜑 → (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶))
6715, 66eqsstr3d 3673 . 2 (𝜑 → ( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶))
68 eqid 2651 . . 3 (LSHyp‘𝑈) = (LSHyp‘𝑈)
691, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 37165 . . . 4 (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
701, 2, 3, 4, 6, 68, 9, 69dochsnshp 37059 . . 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 37182 . . . 4 (𝜑𝐶 ≠ (0g𝐷))
7268, 26, 22, 27, 31, 30, 47lduallkr3 34767 . . . 4 (𝜑 → ((𝐿𝐶) ∈ (LSHyp‘𝑈) ↔ 𝐶 ≠ (0g𝐷)))
7371, 72mpbird 247 . . 3 (𝜑 → (𝐿𝐶) ∈ (LSHyp‘𝑈))
7468, 30, 70, 73lshpcmp 34593 . 2 (𝜑 → (( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶) ↔ ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶)))
7567, 74mpbid 222 1 (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823  ∃wrex 2942  {crab 2945   ∖ cdif 3604   ∩ cin 3606   ⊆ wss 3607  {csn 4210  {cpr 4212   ↦ cmpt 4762  ‘cfv 5926  ℩crio 6650  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  .rcmulr 15989  Scalarcsca 15991   ·𝑠 cvsca 15992  0gc0g 16147  -gcsg 17471  SubGrpcsubg 17635  LSSumclsm 18095  invrcinvr 18717  LModclmod 18911  LSubSpclss 18980  LSpanclspn 19019  LSAtomsclsa 34579  LSHypclsh 34580  LFnlclfn 34662  LKerclk 34690  LDualcld 34728  HLchlt 34955  LHypclh 35588  DVecHcdvh 36684  ocHcoch 36953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051  ax-riotaBAD 34557 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-of 6939  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-undef 7444  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-oadd 7609  df-er 7787  df-map 7901  df-en 7998  df-dom 7999  df-sdom 8000  df-fin 8001  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-n0 11331  df-z 11416  df-uz 11726  df-fz 12365  df-struct 15906  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-ress 15912  df-plusg 16001  df-mulr 16002  df-sca 16004  df-vsca 16005  df-0g 16149  df-mre 16293  df-mrc 16294  df-acs 16296  df-preset 16975  df-poset 16993  df-plt 17005  df-lub 17021  df-glb 17022  df-join 17023  df-meet 17024  df-p0 17086  df-p1 17087  df-lat 17093  df-clat 17155  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-submnd 17383  df-grp 17472  df-minusg 17473  df-sbg 17474  df-subg 17638  df-cntz 17796  df-oppg 17822  df-lsm 18097  df-cmn 18241  df-abl 18242  df-mgp 18536  df-ur 18548  df-ring 18595  df-oppr 18669  df-dvdsr 18687  df-unit 18688  df-invr 18718  df-dvr 18729  df-drng 18797  df-lmod 18913  df-lss 18981  df-lsp 19020  df-lvec 19151  df-lsatoms 34581  df-lshyp 34582  df-lcv 34624  df-lfl 34663  df-lkr 34691  df-ldual 34729  df-oposet 34781  df-ol 34783  df-oml 34784  df-covers 34871  df-ats 34872  df-atl 34903  df-cvlat 34927  df-hlat 34956  df-llines 35102  df-lplanes 35103  df-lvols 35104  df-lines 35105  df-psubsp 35107  df-pmap 35108  df-padd 35400  df-lhyp 35592  df-laut 35593  df-ldil 35708  df-ltrn 35709  df-trl 35764  df-tgrp 36348  df-tendo 36360  df-edring 36362  df-dveca 36608  df-disoa 36635  df-dvech 36685  df-dib 36745  df-dic 36779  df-dih 36835  df-doch 36954  df-djh 37001 This theorem is referenced by:  lcfrlem36  37184
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