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Theorem lcfrlem35 38873
Description: Lemma for lcfr 38881. (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 2798 . . . 4 (LSSum‘𝑈) = (LSSum‘𝑈)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 38861 . . 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 38862 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))))
24 eqid 2798 . . . . . 6 (.r𝑆) = (.r𝑆)
25 lcfrlem29.i . . . . . 6 𝐹 = (invr𝑆)
26 eqid 2798 . . . . . 6 (LFnl‘𝑈) = (LFnl‘𝑈)
27 lcfrlem25.d . . . . . 6 𝐷 = (LDual‘𝑈)
28 eqid 2798 . . . . . 6 ( ·𝑠𝐷) = ( ·𝑠𝐷)
29 lcfrlem30.m . . . . . 6 = (-g𝐷)
301, 3, 9dvhlvec 38405 . . . . . 6 (𝜑𝑈 ∈ LVec)
31 eqid 2798 . . . . . . 7 (0g𝐷) = (0g𝐷)
32 eqid 2798 . . . . . . 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 38848 . . . . . 6 (𝜑 → (𝐽𝑋) ∈ (LFnl‘𝑈))
341, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11lcfrlem10 38848 . . . . . 6 (𝜑 → (𝐽𝑌) ∈ (LFnl‘𝑈))
35 eqid 2798 . . . . . . . 8 (LSubSp‘𝑈) = (LSubSp‘𝑈)
361, 3, 9dvhlmod 38406 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
371, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 38860 . . . . . . . 8 (𝜑𝐵𝐴)
3835, 8, 36, 37lsatlssel 36293 . . . . . . 7 (𝜑𝐵 ∈ (LSubSp‘𝑈))
394, 35lssel 19702 . . . . . . 7 ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼𝐵) → 𝐼𝑉)
4038, 21, 39syl2anc 587 . . . . . 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 38839 . . . . 5 (𝜑 → ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))) ⊆ (𝐿𝐶))
4423, 43eqsstrd 3953 . . . 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 38866 . . . . . 6 (𝜑𝐼0 )
466, 7, 8, 30, 37, 21, 45lsatel 36301 . . . . 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 38868 . . . . . . 7 (𝜑𝐶 ∈ (LFnl‘𝑈))
4826, 22, 35lkrlss 36391 . . . . . . 7 ((𝑈 ∈ LMod ∧ 𝐶 ∈ (LFnl‘𝑈)) → (𝐿𝐶) ∈ (LSubSp‘𝑈))
4936, 47, 48syl2anc 587 . . . . . 6 (𝜑 → (𝐿𝐶) ∈ (LSubSp‘𝑈))
504, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem3 38840 . . . . . 6 (𝜑𝐼 ∈ (𝐿𝐶))
5135, 7, 36, 49, 50lspsnel5a 19761 . . . . 5 (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿𝐶))
5246, 51eqsstrd 3953 . . . 4 (𝜑𝐵 ⊆ (𝐿𝐶))
5335lsssssubg 19723 . . . . . . 7 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5436, 53syl 17 . . . . . 6 (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5510eldifad 3893 . . . . . . . 8 (𝜑𝑋𝑉)
5611eldifad 3893 . . . . . . . 8 (𝜑𝑌𝑉)
57 prssi 4714 . . . . . . . 8 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
5855, 56, 57syl2anc 587 . . . . . . 7 (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
591, 3, 4, 35, 2dochlss 38650 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
609, 58, 59syl2anc 587 . . . . . 6 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
6154, 60sseldd 3916 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
6254, 38sseldd 3916 . . . . 5 (𝜑𝐵 ∈ (SubGrp‘𝑈))
6354, 49sseldd 3916 . . . . 5 (𝜑 → (𝐿𝐶) ∈ (SubGrp‘𝑈))
6414lsmlub 18782 . . . . 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 1368 . . . 4 (𝜑 → ((( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶) ∧ 𝐵 ⊆ (𝐿𝐶)) ↔ (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶)))
6644, 52, 65mpbi2and 711 . . 3 (𝜑 → (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶))
6715, 66eqsstrrd 3954 . 2 (𝜑 → ( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶))
68 eqid 2798 . . 3 (LSHyp‘𝑈) = (LSHyp‘𝑈)
691, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 38855 . . . 4 (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
701, 2, 3, 4, 6, 68, 9, 69dochsnshp 38749 . . 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 38872 . . . 4 (𝜑𝐶 ≠ (0g𝐷))
7268, 26, 22, 27, 31, 30, 47lduallkr3 36458 . . . 4 (𝜑 → ((𝐿𝐶) ∈ (LSHyp‘𝑈) ↔ 𝐶 ≠ (0g𝐷)))
7371, 72mpbird 260 . . 3 (𝜑 → (𝐿𝐶) ∈ (LSHyp‘𝑈))
7468, 30, 70, 73lshpcmp 36284 . 2 (𝜑 → (( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶) ↔ ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶)))
7567, 74mpbid 235 1 (𝜑 → ( ‘{(𝑋 + 𝑌)}) = (𝐿𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wne 2987  wrex 3107  {crab 3110  cdif 3878  cin 3880  wss 3881  {csn 4525  {cpr 4527  cmpt 5110  cfv 6324  crio 7092  (class class class)co 7135  Basecbs 16475  +gcplusg 16557  .rcmulr 16558  Scalarcsca 16560   ·𝑠 cvsca 16561  0gc0g 16705  -gcsg 18097  SubGrpcsubg 18265  LSSumclsm 18751  invrcinvr 19417  LModclmod 19627  LSubSpclss 19696  LSpanclspn 19736  LSAtomsclsa 36270  LSHypclsh 36271  LFnlclfn 36353  LKerclk 36381  LDualcld 36419  HLchlt 36646  LHypclh 37280  DVecHcdvh 38374  ocHcoch 38643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603  ax-riotaBAD 36249
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-iin 4884  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-of 7389  df-om 7561  df-1st 7671  df-2nd 7672  df-tpos 7875  df-undef 7922  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-struct 16477  df-ndx 16478  df-slot 16479  df-base 16481  df-sets 16482  df-ress 16483  df-plusg 16570  df-mulr 16571  df-sca 16573  df-vsca 16574  df-0g 16707  df-mre 16849  df-mrc 16850  df-acs 16852  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-submnd 17949  df-grp 18098  df-minusg 18099  df-sbg 18100  df-subg 18268  df-cntz 18439  df-oppg 18466  df-lsm 18753  df-cmn 18900  df-abl 18901  df-mgp 19233  df-ur 19245  df-ring 19292  df-oppr 19369  df-dvdsr 19387  df-unit 19388  df-invr 19418  df-dvr 19429  df-drng 19497  df-lmod 19629  df-lss 19697  df-lsp 19737  df-lvec 19868  df-lsatoms 36272  df-lshyp 36273  df-lcv 36315  df-lfl 36354  df-lkr 36382  df-ldual 36420  df-oposet 36472  df-ol 36474  df-oml 36475  df-covers 36562  df-ats 36563  df-atl 36594  df-cvlat 36618  df-hlat 36647  df-llines 36794  df-lplanes 36795  df-lvols 36796  df-lines 36797  df-psubsp 36799  df-pmap 36800  df-padd 37092  df-lhyp 37284  df-laut 37285  df-ldil 37400  df-ltrn 37401  df-trl 37455  df-tgrp 38039  df-tendo 38051  df-edring 38053  df-dveca 38299  df-disoa 38325  df-dvech 38375  df-dib 38435  df-dic 38469  df-dih 38525  df-doch 38644  df-djh 38691
This theorem is referenced by:  lcfrlem36  38874
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