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Theorem lcfrlem35 37380
Description: Lemma for lcfr 37388. (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 2770 . . . 4 (LSSum‘𝑈) = (LSSum‘𝑈)
151, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14lcfrlem23 37368 . . 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 37369 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) = ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))))
24 eqid 2770 . . . . . 6 (.r𝑆) = (.r𝑆)
25 lcfrlem29.i . . . . . 6 𝐹 = (invr𝑆)
26 eqid 2770 . . . . . 6 (LFnl‘𝑈) = (LFnl‘𝑈)
27 lcfrlem25.d . . . . . 6 𝐷 = (LDual‘𝑈)
28 eqid 2770 . . . . . 6 ( ·𝑠𝐷) = ( ·𝑠𝐷)
29 lcfrlem30.m . . . . . 6 = (-g𝐷)
301, 3, 9dvhlvec 36912 . . . . . 6 (𝜑𝑈 ∈ LVec)
31 eqid 2770 . . . . . . 7 (0g𝐷) = (0g𝐷)
32 eqid 2770 . . . . . . 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 37355 . . . . . 6 (𝜑 → (𝐽𝑋) ∈ (LFnl‘𝑈))
341, 2, 3, 4, 5, 16, 17, 19, 6, 26, 22, 27, 31, 32, 20, 9, 11lcfrlem10 37355 . . . . . 6 (𝜑 → (𝐽𝑌) ∈ (LFnl‘𝑈))
35 eqid 2770 . . . . . . . 8 (LSubSp‘𝑈) = (LSubSp‘𝑈)
361, 3, 9dvhlmod 36913 . . . . . . . 8 (𝜑𝑈 ∈ LMod)
371, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13lcfrlem22 37367 . . . . . . . 8 (𝜑𝐵𝐴)
3835, 8, 36, 37lsatlssel 34799 . . . . . . 7 (𝜑𝐵 ∈ (LSubSp‘𝑈))
394, 35lssel 19147 . . . . . . 7 ((𝐵 ∈ (LSubSp‘𝑈) ∧ 𝐼𝐵) → 𝐼𝑉)
4038, 21, 39syl2anc 565 . . . . . 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 37346 . . . . 5 (𝜑 → ((𝐿‘(𝐽𝑋)) ∩ (𝐿‘(𝐽𝑌))) ⊆ (𝐿𝐶))
4423, 43eqsstrd 3786 . . . 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 37373 . . . . . 6 (𝜑𝐼0 )
466, 7, 8, 30, 37, 21, 45lsatel 34807 . . . . 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 37375 . . . . . . 7 (𝜑𝐶 ∈ (LFnl‘𝑈))
4826, 22, 35lkrlss 34897 . . . . . . 7 ((𝑈 ∈ LMod ∧ 𝐶 ∈ (LFnl‘𝑈)) → (𝐿𝐶) ∈ (LSubSp‘𝑈))
4936, 47, 48syl2anc 565 . . . . . 6 (𝜑 → (𝐿𝐶) ∈ (LSubSp‘𝑈))
504, 17, 24, 18, 25, 26, 27, 28, 29, 30, 33, 34, 40, 41, 42, 22lcfrlem3 37347 . . . . . 6 (𝜑𝐼 ∈ (𝐿𝐶))
5135, 7, 36, 49, 50lspsnel5a 19208 . . . . 5 (𝜑 → (𝑁‘{𝐼}) ⊆ (𝐿𝐶))
5246, 51eqsstrd 3786 . . . 4 (𝜑𝐵 ⊆ (𝐿𝐶))
5335lsssssubg 19170 . . . . . . 7 (𝑈 ∈ LMod → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5436, 53syl 17 . . . . . 6 (𝜑 → (LSubSp‘𝑈) ⊆ (SubGrp‘𝑈))
5510eldifad 3733 . . . . . . . 8 (𝜑𝑋𝑉)
5611eldifad 3733 . . . . . . . 8 (𝜑𝑌𝑉)
57 prssi 4485 . . . . . . . 8 ((𝑋𝑉𝑌𝑉) → {𝑋, 𝑌} ⊆ 𝑉)
5855, 56, 57syl2anc 565 . . . . . . 7 (𝜑 → {𝑋, 𝑌} ⊆ 𝑉)
591, 3, 4, 35, 2dochlss 37157 . . . . . . 7 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ {𝑋, 𝑌} ⊆ 𝑉) → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
609, 58, 59syl2anc 565 . . . . . 6 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (LSubSp‘𝑈))
6154, 60sseldd 3751 . . . . 5 (𝜑 → ( ‘{𝑋, 𝑌}) ∈ (SubGrp‘𝑈))
6254, 38sseldd 3751 . . . . 5 (𝜑𝐵 ∈ (SubGrp‘𝑈))
6354, 49sseldd 3751 . . . . 5 (𝜑 → (𝐿𝐶) ∈ (SubGrp‘𝑈))
6414lsmlub 18284 . . . . 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 1475 . . . 4 (𝜑 → ((( ‘{𝑋, 𝑌}) ⊆ (𝐿𝐶) ∧ 𝐵 ⊆ (𝐿𝐶)) ↔ (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶)))
6644, 52, 65mpbi2and 683 . . 3 (𝜑 → (( ‘{𝑋, 𝑌})(LSSum‘𝑈)𝐵) ⊆ (𝐿𝐶))
6715, 66eqsstr3d 3787 . 2 (𝜑 → ( ‘{(𝑋 + 𝑌)}) ⊆ (𝐿𝐶))
68 eqid 2770 . . 3 (LSHyp‘𝑈) = (LSHyp‘𝑈)
691, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12lcfrlem17 37362 . . . 4 (𝜑 → (𝑋 + 𝑌) ∈ (𝑉 ∖ { 0 }))
701, 2, 3, 4, 6, 68, 9, 69dochsnshp 37256 . . 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 37379 . . . 4 (𝜑𝐶 ≠ (0g𝐷))
7268, 26, 22, 27, 31, 30, 47lduallkr3 34964 . . . 4 (𝜑 → ((𝐿𝐶) ∈ (LSHyp‘𝑈) ↔ 𝐶 ≠ (0g𝐷)))
7371, 72mpbird 247 . . 3 (𝜑 → (𝐿𝐶) ∈ (LSHyp‘𝑈))
7468, 30, 70, 73lshpcmp 34790 . 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 382   = wceq 1630  wcel 2144  wne 2942  wrex 3061  {crab 3064  cdif 3718  cin 3720  wss 3721  {csn 4314  {cpr 4316  cmpt 4861  cfv 6031  crio 6752  (class class class)co 6792  Basecbs 16063  +gcplusg 16148  .rcmulr 16149  Scalarcsca 16151   ·𝑠 cvsca 16152  0gc0g 16307  -gcsg 17631  SubGrpcsubg 17795  LSSumclsm 18255  invrcinvr 18878  LModclmod 19072  LSubSpclss 19141  LSpanclspn 19183  LSAtomsclsa 34776  LSHypclsh 34777  LFnlclfn 34859  LKerclk 34887  LDualcld 34925  HLchlt 35152  LHypclh 35785  DVecHcdvh 36881  ocHcoch 37150
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194  ax-1cn 10195  ax-icn 10196  ax-addcl 10197  ax-addrcl 10198  ax-mulcl 10199  ax-mulrcl 10200  ax-mulcom 10201  ax-addass 10202  ax-mulass 10203  ax-distr 10204  ax-i2m1 10205  ax-1ne0 10206  ax-1rid 10207  ax-rnegex 10208  ax-rrecex 10209  ax-cnre 10210  ax-pre-lttri 10211  ax-pre-lttrn 10212  ax-pre-ltadd 10213  ax-pre-mulgt0 10214  ax-riotaBAD 34754
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-fal 1636  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-nel 3046  df-ral 3065  df-rex 3066  df-reu 3067  df-rmo 3068  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-int 4610  df-iun 4654  df-iin 4655  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6753  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-of 7043  df-om 7212  df-1st 7314  df-2nd 7315  df-tpos 7503  df-undef 7550  df-wrecs 7558  df-recs 7620  df-rdg 7658  df-1o 7712  df-oadd 7716  df-er 7895  df-map 8010  df-en 8109  df-dom 8110  df-sdom 8111  df-fin 8112  df-pnf 10277  df-mnf 10278  df-xr 10279  df-ltxr 10280  df-le 10281  df-sub 10469  df-neg 10470  df-nn 11222  df-2 11280  df-3 11281  df-4 11282  df-5 11283  df-6 11284  df-n0 11494  df-z 11579  df-uz 11888  df-fz 12533  df-struct 16065  df-ndx 16066  df-slot 16067  df-base 16069  df-sets 16070  df-ress 16071  df-plusg 16161  df-mulr 16162  df-sca 16164  df-vsca 16165  df-0g 16309  df-mre 16453  df-mrc 16454  df-acs 16456  df-preset 17135  df-poset 17153  df-plt 17165  df-lub 17181  df-glb 17182  df-join 17183  df-meet 17184  df-p0 17246  df-p1 17247  df-lat 17253  df-clat 17315  df-mgm 17449  df-sgrp 17491  df-mnd 17502  df-submnd 17543  df-grp 17632  df-minusg 17633  df-sbg 17634  df-subg 17798  df-cntz 17956  df-oppg 17982  df-lsm 18257  df-cmn 18401  df-abl 18402  df-mgp 18697  df-ur 18709  df-ring 18756  df-oppr 18830  df-dvdsr 18848  df-unit 18849  df-invr 18879  df-dvr 18890  df-drng 18958  df-lmod 19074  df-lss 19142  df-lsp 19184  df-lvec 19315  df-lsatoms 34778  df-lshyp 34779  df-lcv 34821  df-lfl 34860  df-lkr 34888  df-ldual 34926  df-oposet 34978  df-ol 34980  df-oml 34981  df-covers 35068  df-ats 35069  df-atl 35100  df-cvlat 35124  df-hlat 35153  df-llines 35299  df-lplanes 35300  df-lvols 35301  df-lines 35302  df-psubsp 35304  df-pmap 35305  df-padd 35597  df-lhyp 35789  df-laut 35790  df-ldil 35905  df-ltrn 35906  df-trl 35961  df-tgrp 36545  df-tendo 36557  df-edring 36559  df-dveca 36805  df-disoa 36832  df-dvech 36882  df-dib 36942  df-dic 36976  df-dih 37032  df-doch 37151  df-djh 37198
This theorem is referenced by:  lcfrlem36  37381
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