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Theorem mullocprlem 7342
Description: Calculations for mullocpr 7343. (Contributed by Jim Kingdon, 10-Dec-2019.)
Hypotheses
Ref Expression
mullocprlem.ab  |-  ( ph  ->  ( A  e.  P.  /\  B  e.  P. )
)
mullocprlem.uqedu  |-  ( ph  ->  ( U  .Q  Q
)  <Q  ( E  .Q  ( D  .Q  U
) ) )
mullocprlem.edutdu  |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) ) 
<Q  ( T  .Q  ( D  .Q  U ) ) )
mullocprlem.tdudr  |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) ) 
<Q  ( D  .Q  R
) )
mullocprlem.qr  |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. )
)
mullocprlem.duq  |-  ( ph  ->  ( D  e.  Q.  /\  U  e.  Q. )
)
mullocprlem.du  |-  ( ph  ->  ( D  e.  ( 1st `  A )  /\  U  e.  ( 2nd `  A ) ) )
mullocprlem.et  |-  ( ph  ->  ( E  e.  Q.  /\  T  e.  Q. )
)
Assertion
Ref Expression
mullocprlem  |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )

Proof of Theorem mullocprlem
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mullocprlem.uqedu . . . . . . 7  |-  ( ph  ->  ( U  .Q  Q
)  <Q  ( E  .Q  ( D  .Q  U
) ) )
2 mullocprlem.et . . . . . . . . 9  |-  ( ph  ->  ( E  e.  Q.  /\  T  e.  Q. )
)
32simpld 111 . . . . . . . 8  |-  ( ph  ->  E  e.  Q. )
4 mullocprlem.duq . . . . . . . . 9  |-  ( ph  ->  ( D  e.  Q.  /\  U  e.  Q. )
)
54simpld 111 . . . . . . . 8  |-  ( ph  ->  D  e.  Q. )
64simprd 113 . . . . . . . 8  |-  ( ph  ->  U  e.  Q. )
7 mulcomnqg 7155 . . . . . . . . 9  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  =  ( y  .Q  x ) )
87adantl 273 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
( x  .Q  y
)  =  ( y  .Q  x ) )
9 mulassnqg 7156 . . . . . . . . 9  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  .Q  y
)  .Q  z )  =  ( x  .Q  ( y  .Q  z
) ) )
109adantl 273 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q.  /\  z  e.  Q. ) )  ->  (
( x  .Q  y
)  .Q  z )  =  ( x  .Q  ( y  .Q  z
) ) )
113, 5, 6, 8, 10caov13d 5920 . . . . . . 7  |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) )  =  ( U  .Q  ( D  .Q  E
) ) )
121, 11breqtrd 3922 . . . . . 6  |-  ( ph  ->  ( U  .Q  Q
)  <Q  ( U  .Q  ( D  .Q  E
) ) )
13 mullocprlem.qr . . . . . . . 8  |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. )
)
1413simpld 111 . . . . . . 7  |-  ( ph  ->  Q  e.  Q. )
15 mulclnq 7148 . . . . . . . 8  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  .Q  E
)  e.  Q. )
165, 3, 15syl2anc 406 . . . . . . 7  |-  ( ph  ->  ( D  .Q  E
)  e.  Q. )
17 ltmnqg 7173 . . . . . . 7  |-  ( ( Q  e.  Q.  /\  ( D  .Q  E
)  e.  Q.  /\  U  e.  Q. )  ->  ( Q  <Q  ( D  .Q  E )  <->  ( U  .Q  Q )  <Q  ( U  .Q  ( D  .Q  E ) ) ) )
1814, 16, 6, 17syl3anc 1199 . . . . . 6  |-  ( ph  ->  ( Q  <Q  ( D  .Q  E )  <->  ( U  .Q  Q )  <Q  ( U  .Q  ( D  .Q  E ) ) ) )
1912, 18mpbird 166 . . . . 5  |-  ( ph  ->  Q  <Q  ( D  .Q  E ) )
2019adantr 272 . . . 4  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  Q  <Q  ( D  .Q  E ) )
21 mullocprlem.ab . . . . . . . 8  |-  ( ph  ->  ( A  e.  P.  /\  B  e.  P. )
)
2221simpld 111 . . . . . . 7  |-  ( ph  ->  A  e.  P. )
23 mullocprlem.du . . . . . . . 8  |-  ( ph  ->  ( D  e.  ( 1st `  A )  /\  U  e.  ( 2nd `  A ) ) )
2423simpld 111 . . . . . . 7  |-  ( ph  ->  D  e.  ( 1st `  A ) )
2522, 24jca 302 . . . . . 6  |-  ( ph  ->  ( A  e.  P.  /\  D  e.  ( 1st `  A ) ) )
2625adantr 272 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( A  e.  P.  /\  D  e.  ( 1st `  A
) ) )
2721simprd 113 . . . . . 6  |-  ( ph  ->  B  e.  P. )
2827anim1i 336 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( B  e.  P.  /\  E  e.  ( 1st `  B
) ) )
2914adantr 272 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  Q  e.  Q. )
30 mulnqprl 7340 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  D  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  E  e.  ( 1st `  B
) ) )  /\  Q  e.  Q. )  ->  ( Q  <Q  ( D  .Q  E )  ->  Q  e.  ( 1st `  ( A  .P.  B
) ) ) )
3126, 28, 29, 30syl21anc 1198 . . . 4  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( Q  <Q  ( D  .Q  E
)  ->  Q  e.  ( 1st `  ( A  .P.  B ) ) ) )
3220, 31mpd 13 . . 3  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  Q  e.  ( 1st `  ( A  .P.  B ) ) )
3332orcd 705 . 2  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
342simprd 113 . . . . . . 7  |-  ( ph  ->  T  e.  Q. )
35 mulcomnqg 7155 . . . . . . 7  |-  ( ( T  e.  Q.  /\  U  e.  Q. )  ->  ( T  .Q  U
)  =  ( U  .Q  T ) )
3634, 6, 35syl2anc 406 . . . . . 6  |-  ( ph  ->  ( T  .Q  U
)  =  ( U  .Q  T ) )
37 mullocprlem.tdudr . . . . . . 7  |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) ) 
<Q  ( D  .Q  R
) )
38 mulclnq 7148 . . . . . . . . . 10  |-  ( ( T  e.  Q.  /\  U  e.  Q. )  ->  ( T  .Q  U
)  e.  Q. )
3934, 6, 38syl2anc 406 . . . . . . . . 9  |-  ( ph  ->  ( T  .Q  U
)  e.  Q. )
4013simprd 113 . . . . . . . . 9  |-  ( ph  ->  R  e.  Q. )
41 ltmnqg 7173 . . . . . . . . 9  |-  ( ( ( T  .Q  U
)  e.  Q.  /\  R  e.  Q.  /\  D  e.  Q. )  ->  (
( T  .Q  U
)  <Q  R  <->  ( D  .Q  ( T  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4239, 40, 5, 41syl3anc 1199 . . . . . . . 8  |-  ( ph  ->  ( ( T  .Q  U )  <Q  R  <->  ( D  .Q  ( T  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4334, 5, 6, 8, 10caov12d 5918 . . . . . . . . 9  |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) )  =  ( D  .Q  ( T  .Q  U
) ) )
4443breq1d 3907 . . . . . . . 8  |-  ( ph  ->  ( ( T  .Q  ( D  .Q  U
) )  <Q  ( D  .Q  R )  <->  ( D  .Q  ( T  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4542, 44bitr4d 190 . . . . . . 7  |-  ( ph  ->  ( ( T  .Q  U )  <Q  R  <->  ( T  .Q  ( D  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4637, 45mpbird 166 . . . . . 6  |-  ( ph  ->  ( T  .Q  U
)  <Q  R )
4736, 46eqbrtrrd 3920 . . . . 5  |-  ( ph  ->  ( U  .Q  T
)  <Q  R )
4847adantr 272 . . . 4  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( U  .Q  T )  <Q  R )
4923simprd 113 . . . . . . 7  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
5022, 49jca 302 . . . . . 6  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
5150adantr 272 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A
) ) )
5227anim1i 336 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B
) ) )
5340adantr 272 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  R  e.  Q. )
54 mulnqpru 7341 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  U  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  T  e.  ( 2nd `  B
) ) )  /\  R  e.  Q. )  ->  ( ( U  .Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  .P.  B
) ) ) )
5551, 52, 53, 54syl21anc 1198 . . . 4  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( ( U  .Q  T )  <Q  R  ->  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
5648, 55mpd 13 . . 3  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  R  e.  ( 2nd `  ( A  .P.  B ) ) )
5756olcd 706 . 2  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
58 mullocprlem.edutdu . . . 4  |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) ) 
<Q  ( T  .Q  ( D  .Q  U ) ) )
59 mulclnq 7148 . . . . . . 7  |-  ( ( D  e.  Q.  /\  U  e.  Q. )  ->  ( D  .Q  U
)  e.  Q. )
604, 59syl 14 . . . . . 6  |-  ( ph  ->  ( D  .Q  U
)  e.  Q. )
61 ltmnqg 7173 . . . . . 6  |-  ( ( E  e.  Q.  /\  T  e.  Q.  /\  ( D  .Q  U )  e. 
Q. )  ->  ( E  <Q  T  <->  ( ( D  .Q  U )  .Q  E )  <Q  (
( D  .Q  U
)  .Q  T ) ) )
623, 34, 60, 61syl3anc 1199 . . . . 5  |-  ( ph  ->  ( E  <Q  T  <->  ( ( D  .Q  U )  .Q  E )  <Q  (
( D  .Q  U
)  .Q  T ) ) )
63 mulcomnqg 7155 . . . . . . 7  |-  ( ( ( D  .Q  U
)  e.  Q.  /\  E  e.  Q. )  ->  ( ( D  .Q  U )  .Q  E
)  =  ( E  .Q  ( D  .Q  U ) ) )
6460, 3, 63syl2anc 406 . . . . . 6  |-  ( ph  ->  ( ( D  .Q  U )  .Q  E
)  =  ( E  .Q  ( D  .Q  U ) ) )
65 mulcomnqg 7155 . . . . . . 7  |-  ( ( ( D  .Q  U
)  e.  Q.  /\  T  e.  Q. )  ->  ( ( D  .Q  U )  .Q  T
)  =  ( T  .Q  ( D  .Q  U ) ) )
6660, 34, 65syl2anc 406 . . . . . 6  |-  ( ph  ->  ( ( D  .Q  U )  .Q  T
)  =  ( T  .Q  ( D  .Q  U ) ) )
6764, 66breq12d 3910 . . . . 5  |-  ( ph  ->  ( ( ( D  .Q  U )  .Q  E )  <Q  (
( D  .Q  U
)  .Q  T )  <-> 
( E  .Q  ( D  .Q  U ) ) 
<Q  ( T  .Q  ( D  .Q  U ) ) ) )
6862, 67bitrd 187 . . . 4  |-  ( ph  ->  ( E  <Q  T  <->  ( E  .Q  ( D  .Q  U
) )  <Q  ( T  .Q  ( D  .Q  U ) ) ) )
6958, 68mpbird 166 . . 3  |-  ( ph  ->  E  <Q  T )
70 prop 7247 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
71 prloc 7263 . . . 4  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  <Q  T )  ->  ( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7270, 71sylan 279 . . 3  |-  ( ( B  e.  P.  /\  E  <Q  T )  -> 
( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7327, 69, 72syl2anc 406 . 2  |-  ( ph  ->  ( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7433, 57, 73mpjaodan 770 1  |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 680    /\ w3a 945    = wceq 1314    e. wcel 1463   <.cop 3498   class class class wbr 3897   ` cfv 5091  (class class class)co 5740   1stc1st 6002   2ndc2nd 6003   Q.cnq 7052    .Q cmq 7055    <Q cltq 7057   P.cnp 7063    .P. cmp 7066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-eprel 4179  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-irdg 6233  df-1o 6279  df-oadd 6283  df-omul 6284  df-er 6395  df-ec 6397  df-qs 6401  df-ni 7076  df-mi 7078  df-lti 7079  df-mpq 7117  df-enq 7119  df-nqqs 7120  df-mqqs 7122  df-1nqqs 7123  df-rq 7124  df-ltnqqs 7125  df-inp 7238  df-imp 7241
This theorem is referenced by:  mullocpr  7343
  Copyright terms: Public domain W3C validator