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Theorem dia2dimlem7 30633
Description: Lemma for dia2dim 30640. Eliminate  ( F `  P )  =/=  P condition. (Contributed by NM, 8-Sep-2014.)
Hypotheses
Ref Expression
dia2dimlem7.l  |-  .<_  =  ( le `  K )
dia2dimlem7.j  |-  .\/  =  ( join `  K )
dia2dimlem7.m  |-  ./\  =  ( meet `  K )
dia2dimlem7.a  |-  A  =  ( Atoms `  K )
dia2dimlem7.h  |-  H  =  ( LHyp `  K
)
dia2dimlem7.t  |-  T  =  ( ( LTrn `  K
) `  W )
dia2dimlem7.r  |-  R  =  ( ( trL `  K
) `  W )
dia2dimlem7.y  |-  Y  =  ( ( DVecA `  K
) `  W )
dia2dimlem7.s  |-  S  =  ( LSubSp `  Y )
dia2dimlem7.pl  |-  .(+)  =  (
LSSum `  Y )
dia2dimlem7.n  |-  N  =  ( LSpan `  Y )
dia2dimlem7.i  |-  I  =  ( ( DIsoA `  K
) `  W )
dia2dimlem7.q  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
dia2dimlem7.k  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
dia2dimlem7.u  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
dia2dimlem7.v  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
dia2dimlem7.p  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
dia2dimlem7.f  |-  ( ph  ->  F  e.  T )
dia2dimlem7.rf  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
dia2dimlem7.uv  |-  ( ph  ->  U  =/=  V )
dia2dimlem7.ru  |-  ( ph  ->  ( R `  F
)  =/=  U )
dia2dimlem7.rv  |-  ( ph  ->  ( R `  F
)  =/=  V )
Assertion
Ref Expression
dia2dimlem7  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )

Proof of Theorem dia2dimlem7
StepHypRef Expression
1 dia2dimlem7.k . . . . 5  |-  ( ph  ->  ( K  e.  HL  /\  W  e.  H ) )
2 dia2dimlem7.f . . . . 5  |-  ( ph  ->  F  e.  T )
3 dia2dimlem7.p . . . . 5  |-  ( ph  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
4 eqid 2283 . . . . . 6  |-  ( Base `  K )  =  (
Base `  K )
5 dia2dimlem7.l . . . . . 6  |-  .<_  =  ( le `  K )
6 dia2dimlem7.a . . . . . 6  |-  A  =  ( Atoms `  K )
7 dia2dimlem7.h . . . . . 6  |-  H  =  ( LHyp `  K
)
8 dia2dimlem7.t . . . . . 6  |-  T  =  ( ( LTrn `  K
) `  W )
94, 5, 6, 7, 8ltrnideq 29737 . . . . 5  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  F  e.  T  /\  ( P  e.  A  /\  -.  P  .<_  W ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <-> 
( F `  P
)  =  P ) )
101, 2, 3, 9syl3anc 1182 . . . 4  |-  ( ph  ->  ( F  =  (  _I  |`  ( Base `  K ) )  <->  ( F `  P )  =  P ) )
11 dia2dimlem7.y . . . . . . . 8  |-  Y  =  ( ( DVecA `  K
) `  W )
12 eqid 2283 . . . . . . . 8  |-  ( 0g
`  Y )  =  ( 0g `  Y
)
134, 7, 8, 11, 12dva0g 30590 . . . . . . 7  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  ( 0g `  Y
)  =  (  _I  |`  ( Base `  K
) ) )
141, 13syl 15 . . . . . 6  |-  ( ph  ->  ( 0g `  Y
)  =  (  _I  |`  ( Base `  K
) ) )
157, 11dvalvec 30589 . . . . . . . 8  |-  ( ( K  e.  HL  /\  W  e.  H )  ->  Y  e.  LVec )
16 lveclmod 15859 . . . . . . . 8  |-  ( Y  e.  LVec  ->  Y  e. 
LMod )
171, 15, 163syl 18 . . . . . . 7  |-  ( ph  ->  Y  e.  LMod )
18 dia2dimlem7.u . . . . . . . . . . 11  |-  ( ph  ->  ( U  e.  A  /\  U  .<_  W ) )
1918simpld 445 . . . . . . . . . 10  |-  ( ph  ->  U  e.  A )
204, 6atbase 28852 . . . . . . . . . 10  |-  ( U  e.  A  ->  U  e.  ( Base `  K
) )
2119, 20syl 15 . . . . . . . . 9  |-  ( ph  ->  U  e.  ( Base `  K ) )
2218simprd 449 . . . . . . . . 9  |-  ( ph  ->  U  .<_  W )
23 dia2dimlem7.i . . . . . . . . . 10  |-  I  =  ( ( DIsoA `  K
) `  W )
24 dia2dimlem7.s . . . . . . . . . 10  |-  S  =  ( LSubSp `  Y )
254, 5, 7, 11, 23, 24dialss 30609 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( U  e.  ( Base `  K
)  /\  U  .<_  W ) )  ->  (
I `  U )  e.  S )
261, 21, 22, 25syl12anc 1180 . . . . . . . 8  |-  ( ph  ->  ( I `  U
)  e.  S )
27 dia2dimlem7.v . . . . . . . . . . 11  |-  ( ph  ->  ( V  e.  A  /\  V  .<_  W ) )
2827simpld 445 . . . . . . . . . 10  |-  ( ph  ->  V  e.  A )
294, 6atbase 28852 . . . . . . . . . 10  |-  ( V  e.  A  ->  V  e.  ( Base `  K
) )
3028, 29syl 15 . . . . . . . . 9  |-  ( ph  ->  V  e.  ( Base `  K ) )
3127simprd 449 . . . . . . . . 9  |-  ( ph  ->  V  .<_  W )
324, 5, 7, 11, 23, 24dialss 30609 . . . . . . . . 9  |-  ( ( ( K  e.  HL  /\  W  e.  H )  /\  ( V  e.  ( Base `  K
)  /\  V  .<_  W ) )  ->  (
I `  V )  e.  S )
331, 30, 31, 32syl12anc 1180 . . . . . . . 8  |-  ( ph  ->  ( I `  V
)  e.  S )
34 dia2dimlem7.pl . . . . . . . . 9  |-  .(+)  =  (
LSSum `  Y )
3524, 34lsmcl 15836 . . . . . . . 8  |-  ( ( Y  e.  LMod  /\  (
I `  U )  e.  S  /\  (
I `  V )  e.  S )  ->  (
( I `  U
)  .(+)  ( I `  V ) )  e.  S )
3617, 26, 33, 35syl3anc 1182 . . . . . . 7  |-  ( ph  ->  ( ( I `  U )  .(+)  ( I `
 V ) )  e.  S )
3712, 24lss0cl 15704 . . . . . . 7  |-  ( ( Y  e.  LMod  /\  (
( I `  U
)  .(+)  ( I `  V ) )  e.  S )  ->  ( 0g `  Y )  e.  ( ( I `  U )  .(+)  ( I `
 V ) ) )
3817, 36, 37syl2anc 642 . . . . . 6  |-  ( ph  ->  ( 0g `  Y
)  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
3914, 38eqeltrrd 2358 . . . . 5  |-  ( ph  ->  (  _I  |`  ( Base `  K ) )  e.  ( ( I `
 U )  .(+)  ( I `  V ) ) )
40 eleq1a 2352 . . . . 5  |-  ( (  _I  |`  ( Base `  K ) )  e.  ( ( I `  U )  .(+)  ( I `
 V ) )  ->  ( F  =  (  _I  |`  ( Base `  K ) )  ->  F  e.  ( ( I `  U
)  .(+)  ( I `  V ) ) ) )
4139, 40syl 15 . . . 4  |-  ( ph  ->  ( F  =  (  _I  |`  ( Base `  K ) )  ->  F  e.  ( (
I `  U )  .(+)  ( I `  V
) ) ) )
4210, 41sylbird 226 . . 3  |-  ( ph  ->  ( ( F `  P )  =  P  ->  F  e.  ( ( I `  U
)  .(+)  ( I `  V ) ) ) )
4342imp 418 . 2  |-  ( (
ph  /\  ( F `  P )  =  P )  ->  F  e.  ( ( I `  U )  .(+)  ( I `
 V ) ) )
44 dia2dimlem7.j . . 3  |-  .\/  =  ( join `  K )
45 dia2dimlem7.m . . 3  |-  ./\  =  ( meet `  K )
46 dia2dimlem7.r . . 3  |-  R  =  ( ( trL `  K
) `  W )
47 dia2dimlem7.n . . 3  |-  N  =  ( LSpan `  Y )
48 dia2dimlem7.q . . 3  |-  Q  =  ( ( P  .\/  U )  ./\  ( ( F `  P )  .\/  V ) )
491adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( K  e.  HL  /\  W  e.  H ) )
5018adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( U  e.  A  /\  U  .<_  W ) )
5127adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( V  e.  A  /\  V  .<_  W ) )
523adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( P  e.  A  /\  -.  P  .<_  W ) )
532anim1i 551 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( F  e.  T  /\  ( F `  P )  =/=  P ) )
54 dia2dimlem7.rf . . . 4  |-  ( ph  ->  ( R `  F
)  .<_  ( U  .\/  V ) )
5554adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( R `  F )  .<_  ( U 
.\/  V ) )
56 dia2dimlem7.uv . . . 4  |-  ( ph  ->  U  =/=  V )
5756adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  U  =/=  V )
58 dia2dimlem7.ru . . . 4  |-  ( ph  ->  ( R `  F
)  =/=  U )
5958adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( R `  F )  =/=  U
)
60 dia2dimlem7.rv . . . 4  |-  ( ph  ->  ( R `  F
)  =/=  V )
6160adantr 451 . . 3  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  ( R `  F )  =/=  V
)
625, 44, 45, 6, 7, 8, 46, 11, 24, 34, 47, 23, 48, 49, 50, 51, 52, 53, 55, 57, 59, 61dia2dimlem6 30632 . 2  |-  ( (
ph  /\  ( F `  P )  =/=  P
)  ->  F  e.  ( ( I `  U )  .(+)  ( I `
 V ) ) )
6343, 62pm2.61dane 2524 1  |-  ( ph  ->  F  e.  ( ( I `  U ) 
.(+)  ( I `  V ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023    _I cid 4304    |` cres 4691   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   0gc0g 13400   joincjn 14078   meetcmee 14079   LSSumclsm 14945   LModclmod 15627   LSubSpclss 15689   LSpanclspn 15728   LVecclvec 15855   Atomscatm 28826   HLchlt 28913   LHypclh 29546   LTrncltrn 29663   trLctrl 29720   DVecAcdveca 30564   DIsoAcdia 30591
This theorem is referenced by:  dia2dimlem8  30634
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-fal 1311  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-undef 6298  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-0g 13404  df-poset 14080  df-plt 14092  df-lub 14108  df-glb 14109  df-join 14110  df-meet 14111  df-p0 14145  df-p1 14146  df-lat 14152  df-clat 14214  df-mnd 14367  df-submnd 14416  df-grp 14489  df-minusg 14490  df-sbg 14491  df-subg 14618  df-cntz 14793  df-lsm 14947  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-lmod 15629  df-lss 15690  df-lsp 15729  df-lvec 15856  df-oposet 28739  df-ol 28741  df-oml 28742  df-covers 28829  df-ats 28830  df-atl 28861  df-cvlat 28885  df-hlat 28914  df-llines 29060  df-lplanes 29061  df-lvols 29062  df-lines 29063  df-psubsp 29065  df-pmap 29066  df-padd 29358  df-lhyp 29550  df-laut 29551  df-ldil 29666  df-ltrn 29667  df-trl 29721  df-tgrp 30305  df-tendo 30317  df-edring 30319  df-dveca 30565  df-disoa 30592
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