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Theorem latmlem12 14189
Description: Add join to both sides of a lattice ordering. (ss2in 3396 analog.) (Contributed by NM, 10-Nov-2011.)
Hypotheses
Ref Expression
latmle.b  |-  B  =  ( Base `  K
)
latmle.l  |-  .<_  =  ( le `  K )
latmle.m  |-  ./\  =  ( meet `  K )
Assertion
Ref Expression
latmlem12  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  ./\  Z )  .<_  ( Y  ./\ 
W ) ) )

Proof of Theorem latmlem12
StepHypRef Expression
1 simp1 955 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  K  e.  Lat )
2 simp2l 981 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  X  e.  B )
3 simp2r 982 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Y  e.  B )
4 simp3l 983 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  Z  e.  B )
5 latmle.b . . . 4  |-  B  =  ( Base `  K
)
6 latmle.l . . . 4  |-  .<_  =  ( le `  K )
7 latmle.m . . . 4  |-  ./\  =  ( meet `  K )
85, 6, 7latmlem1 14187 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  ( X  .<_  Y  ->  ( X  ./\  Z )  .<_  ( Y  ./\  Z ) ) )
91, 2, 3, 4, 8syl13anc 1184 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  .<_  Y  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  Z ) ) )
10 simp3r 984 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  ->  W  e.  B )
115, 6, 7latmlem2 14188 . . 3  |-  ( ( K  e.  Lat  /\  ( Z  e.  B  /\  W  e.  B  /\  Y  e.  B
) )  ->  ( Z  .<_  W  ->  ( Y  ./\  Z )  .<_  ( Y  ./\  W ) ) )
121, 4, 10, 3, 11syl13anc 1184 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Z  .<_  W  -> 
( Y  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
135, 7latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  X  e.  B  /\  Z  e.  B )  ->  ( X  ./\  Z
)  e.  B )
141, 2, 4, 13syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( X  ./\  Z
)  e.  B )
155, 7latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  Z  e.  B )  ->  ( Y  ./\  Z
)  e.  B )
161, 3, 4, 15syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  ./\  Z
)  e.  B )
175, 7latmcl 14157 . . . 4  |-  ( ( K  e.  Lat  /\  Y  e.  B  /\  W  e.  B )  ->  ( Y  ./\  W
)  e.  B )
181, 3, 10, 17syl3anc 1182 . . 3  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( Y  ./\  W
)  e.  B )
195, 6lattr 14162 . . 3  |-  ( ( K  e.  Lat  /\  ( ( X  ./\  Z )  e.  B  /\  ( Y  ./\  Z )  e.  B  /\  ( Y  ./\  W )  e.  B ) )  -> 
( ( ( X 
./\  Z )  .<_  ( Y  ./\  Z )  /\  ( Y  ./\  Z )  .<_  ( Y  ./\ 
W ) )  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
201, 14, 16, 18, 19syl13anc 1184 . 2  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( ( X 
./\  Z )  .<_  ( Y  ./\  Z )  /\  ( Y  ./\  Z )  .<_  ( Y  ./\ 
W ) )  -> 
( X  ./\  Z
)  .<_  ( Y  ./\  W ) ) )
219, 12, 20syl2and 469 1  |-  ( ( K  e.  Lat  /\  ( X  e.  B  /\  Y  e.  B
)  /\  ( Z  e.  B  /\  W  e.  B ) )  -> 
( ( X  .<_  Y  /\  Z  .<_  W )  ->  ( X  ./\  Z )  .<_  ( Y  ./\ 
W ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   Basecbs 13148   lecple 13215   meetcmee 14079   Latclat 14151
This theorem is referenced by:  dalem10  29862  dalem55  29916  dalawlem3  30062  dalawlem7  30066  dalawlem11  30070  dalawlem12  30071  cdlemk51  31142
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
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  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-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  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-undef 6298  df-riota 6304  df-poset 14080  df-glb 14109  df-meet 14111  df-lat 14152
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