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Theorem lneq2at 35064
Description: A line equals the join of any two of its distinct points (atoms). (Contributed by NM, 29-Apr-2012.)
Hypotheses
Ref Expression
lneq2at.b |- B = ( Base ` K )
lneq2at.l |- .<_ = ( le ` K )
lneq2at.j |- .\/ = ( join ` K )
lneq2at.a |- A = ( Atoms ` K )
lneq2at.n |- N = ( Lines ` K )
lneq2at.m |- M = ( pmap ` K )
Assertion
Ref Expression
lneq2at |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> X = ( P .\/ Q ) )
Proof of Theorem lneq2at
Dummy variables s r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp11 1091 . . . 4 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> K e. HL )
2 simp12 1092 . . . 4 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> X e. B )
31, 2jca 554 . . 3 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( K e. HL /\ X e. B ) )
4 simp13 1093 . . 3 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( M ` X ) e. N )
5 lneq2at.b . . . . 5 |- B = ( Base ` K )
6 lneq2at.j . . . . 5 |- .\/ = ( join ` K )
7 lneq2at.a . . . . 5 |- A = ( Atoms ` K )
8 lneq2at.n . . . . 5 |- N = ( Lines ` K )
9 lneq2at.m . . . . 5 |- M = ( pmap ` K )
105, 6, 7, 8, 9isline3 35062 . . . 4 |- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N <-> E. r e. A E. s e. A ( r =/= s /\ X = ( r .\/ s ) ) ) )
1110biimpd 219 . . 3 |- ( ( K e. HL /\ X e. B ) -> ( ( M ` X ) e. N -> E. r e. A E. s e. A ( r =/= s /\ X = ( r .\/ s ) ) ) )
123, 4, 11sylc 65 . 2 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> E. r e. A E. s e. A ( r =/= s /\ X = ( r .\/ s ) ) )
13 simp3r 1090 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> X = ( r .\/ s ) )
14 simp111 1190 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> K e. HL )
15 simp121 1193 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> P e. A )
16 simp122 1194 . . . . . . . . 9 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> Q e. A )
1715, 16jca 554 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( P e. A /\ Q e. A ) )
18 simp2 1062 . . . . . . . 8 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( r e. A /\ s e. A ) )
1914, 17, 183jca 1242 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) )
20 simp123 1195 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> P =/= Q )
2119, 20jca 554 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) )
22 hllat 34650 . . . . . . . . . . 11 |- ( K e. HL -> K e. Lat )
231, 22syl 17 . . . . . . . . . 10 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> K e. Lat )
24 simp21 1094 . . . . . . . . . . . 12 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> P e. A )
255, 7atbase 34576 . . . . . . . . . . . 12 |- ( P e. A -> P e. B )
2624, 25syl 17 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> P e. B )
27 simp22 1095 . . . . . . . . . . . 12 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> Q e. A )
285, 7atbase 34576 . . . . . . . . . . . 12 |- ( Q e. A -> Q e. B )
2927, 28syl 17 . . . . . . . . . . 11 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> Q e. B )
3026, 29, 23jca 1242 . . . . . . . . . 10 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( P e. B /\ Q e. B /\ X e. B ) )
3123, 30jca 554 . . . . . . . . 9 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) )
32 simp3 1063 . . . . . . . . 9 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( P .<_ X /\ Q .<_ X ) )
33 lneq2at.l . . . . . . . . . . 11 |- .<_ = ( le ` K )
345, 33, 6latjle12 17062 . . . . . . . . . 10 |- ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) -> ( ( P .<_ X /\ Q .<_ X ) <-> ( P .\/ Q ) .<_ X ) )
3534biimpd 219 . . . . . . . . 9 |- ( ( K e. Lat /\ ( P e. B /\ Q e. B /\ X e. B ) ) -> ( ( P .<_ X /\ Q .<_ X ) -> ( P .\/ Q ) .<_ X ) )
3631, 32, 35sylc 65 . . . . . . . 8 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( P .\/ Q ) .<_ X )
37363ad2ant1 1082 . . . . . . 7 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( P .\/ Q ) .<_ X )
3837, 13breqtrd 4679 . . . . . 6 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( P .\/ Q ) .<_ ( r .\/ s ) )
39 simpl1 1064 . . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> K e. HL )
40 simpl2l 1114 . . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> P e. A )
41 simpl2r 1115 . . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> Q e. A )
42 simpr 477 . . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> P =/= Q )
43 simpl3 1066 . . . . . . . 8 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> ( r e. A /\ s e. A ) )
4433, 6, 7ps-1 34763 . . . . . . . 8 |- ( ( K e. HL /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( r e. A /\ s e. A ) ) -> ( ( P .\/ Q ) .<_ ( r .\/ s ) <-> ( P .\/ Q ) = ( r .\/ s ) ) )
4539, 40, 41, 42, 43, 44syl131anc 1339 . . . . . . 7 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> ( ( P .\/ Q ) .<_ ( r .\/ s ) <-> ( P .\/ Q ) = ( r .\/ s ) ) )
4645biimpd 219 . . . . . 6 |- ( ( ( K e. HL /\ ( P e. A /\ Q e. A ) /\ ( r e. A /\ s e. A ) ) /\ P =/= Q ) -> ( ( P .\/ Q ) .<_ ( r .\/ s ) -> ( P .\/ Q ) = ( r .\/ s ) ) )
4721, 38, 46sylc 65 . . . . 5 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> ( P .\/ Q ) = ( r .\/ s ) )
4813, 47eqtr4d 2659 . . . 4 |- ( ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) /\ ( r e. A /\ s e. A ) /\ ( r =/= s /\ X = ( r .\/ s ) ) ) -> X = ( P .\/ Q ) )
49483exp 1264 . . 3 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( ( r e. A /\ s e. A ) -> ( ( r =/= s /\ X = ( r .\/ s ) ) -> X = ( P .\/ Q ) ) ) )
5049rexlimdvv 3037 . 2 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> ( E. r e. A E. s e. A ( r =/= s /\ X = ( r .\/ s ) ) -> X = ( P .\/ Q ) ) )
5112, 50mpd 15 1 |- ( ( ( K e. HL /\ X e. B /\ ( M ` X ) e. N ) /\ ( P e. A /\ Q e. A /\ P =/= Q ) /\ ( P .<_ X /\ Q .<_ X ) ) -> X = ( P .\/ Q ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   E.wrex 2913   class class class wbr 4653   ` cfv 5888  (class class class)co 6650   Basecbs 15857   lecple 15948   joincjn 16944   Latclat 17045   Atomscatm 34550   HLchlt 34637   Linesclines 34780   pmapcpmap 34783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-preset 16928  df-poset 16946  df-plt 16958  df-lub 16974  df-glb 16975  df-join 16976  df-meet 16977  df-p0 17039  df-lat 17046  df-clat 17108  df-oposet 34463  df-ol 34465  df-oml 34466  df-covers 34553  df-ats 34554  df-atl 34585  df-cvlat 34609  df-hlat 34638  df-lines 34787  df-pmap 34790
This theorem is referenced by:  lnjatN  35066  lncmp  35069  cdlema1N  35077
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