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Theorem mullocprlem 6726
Description: Calculations for mullocpr 6727. (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 109 . . . . . . . 8  |-  ( ph  ->  E  e.  Q. )
4 mullocprlem.duq . . . . . . . . 9  |-  ( ph  ->  ( D  e.  Q.  /\  U  e.  Q. )
)
54simpld 109 . . . . . . . 8  |-  ( ph  ->  D  e.  Q. )
64simprd 111 . . . . . . . 8  |-  ( ph  ->  U  e.  Q. )
7 mulcomnqg 6539 . . . . . . . . 9  |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x  .Q  y
)  =  ( y  .Q  x ) )
87adantl 266 . . . . . . . 8  |-  ( (
ph  /\  ( x  e.  Q.  /\  y  e. 
Q. ) )  -> 
( x  .Q  y
)  =  ( y  .Q  x ) )
9 mulassnqg 6540 . . . . . . . . 9  |-  ( ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  (
( x  .Q  y
)  .Q  z )  =  ( x  .Q  ( y  .Q  z
) ) )
109adantl 266 . . . . . . . 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 5712 . . . . . . 7  |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) )  =  ( U  .Q  ( D  .Q  E
) ) )
121, 11breqtrd 3816 . . . . . 6  |-  ( ph  ->  ( U  .Q  Q
)  <Q  ( U  .Q  ( D  .Q  E
) ) )
13 mullocprlem.qr . . . . . . . 8  |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. )
)
1413simpld 109 . . . . . . 7  |-  ( ph  ->  Q  e.  Q. )
15 mulclnq 6532 . . . . . . . 8  |-  ( ( D  e.  Q.  /\  E  e.  Q. )  ->  ( D  .Q  E
)  e.  Q. )
165, 3, 15syl2anc 397 . . . . . . 7  |-  ( ph  ->  ( D  .Q  E
)  e.  Q. )
17 ltmnqg 6557 . . . . . . 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 1146 . . . . . 6  |-  ( ph  ->  ( Q  <Q  ( D  .Q  E )  <->  ( U  .Q  Q )  <Q  ( U  .Q  ( D  .Q  E ) ) ) )
1912, 18mpbird 160 . . . . 5  |-  ( ph  ->  Q  <Q  ( D  .Q  E ) )
2019adantr 265 . . . 4  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  Q  <Q  ( D  .Q  E ) )
21 mullocprlem.ab . . . . . . . 8  |-  ( ph  ->  ( A  e.  P.  /\  B  e.  P. )
)
2221simpld 109 . . . . . . 7  |-  ( ph  ->  A  e.  P. )
23 mullocprlem.du . . . . . . . 8  |-  ( ph  ->  ( D  e.  ( 1st `  A )  /\  U  e.  ( 2nd `  A ) ) )
2423simpld 109 . . . . . . 7  |-  ( ph  ->  D  e.  ( 1st `  A ) )
2522, 24jca 294 . . . . . 6  |-  ( ph  ->  ( A  e.  P.  /\  D  e.  ( 1st `  A ) ) )
2625adantr 265 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( A  e.  P.  /\  D  e.  ( 1st `  A
) ) )
2721simprd 111 . . . . . 6  |-  ( ph  ->  B  e.  P. )
2827anim1i 327 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( B  e.  P.  /\  E  e.  ( 1st `  B
) ) )
2914adantr 265 . . . . 5  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  Q  e.  Q. )
30 mulnqprl 6724 . . . . 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 1145 . . . 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 662 . 2  |-  ( (
ph  /\  E  e.  ( 1st `  B ) )  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
342simprd 111 . . . . . . 7  |-  ( ph  ->  T  e.  Q. )
35 mulcomnqg 6539 . . . . . . 7  |-  ( ( T  e.  Q.  /\  U  e.  Q. )  ->  ( T  .Q  U
)  =  ( U  .Q  T ) )
3634, 6, 35syl2anc 397 . . . . . 6  |-  ( ph  ->  ( T  .Q  U
)  =  ( U  .Q  T ) )
37 mullocprlem.tdudr . . . . . . 7  |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) ) 
<Q  ( D  .Q  R
) )
38 mulclnq 6532 . . . . . . . . . 10  |-  ( ( T  e.  Q.  /\  U  e.  Q. )  ->  ( T  .Q  U
)  e.  Q. )
3934, 6, 38syl2anc 397 . . . . . . . . 9  |-  ( ph  ->  ( T  .Q  U
)  e.  Q. )
4013simprd 111 . . . . . . . . 9  |-  ( ph  ->  R  e.  Q. )
41 ltmnqg 6557 . . . . . . . . 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 1146 . . . . . . . 8  |-  ( ph  ->  ( ( T  .Q  U )  <Q  R  <->  ( D  .Q  ( T  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4334, 5, 6, 8, 10caov12d 5710 . . . . . . . . 9  |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) )  =  ( D  .Q  ( T  .Q  U
) ) )
4443breq1d 3802 . . . . . . . 8  |-  ( ph  ->  ( ( T  .Q  ( D  .Q  U
) )  <Q  ( D  .Q  R )  <->  ( D  .Q  ( T  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4542, 44bitr4d 184 . . . . . . 7  |-  ( ph  ->  ( ( T  .Q  U )  <Q  R  <->  ( T  .Q  ( D  .Q  U
) )  <Q  ( D  .Q  R ) ) )
4637, 45mpbird 160 . . . . . 6  |-  ( ph  ->  ( T  .Q  U
)  <Q  R )
4736, 46eqbrtrrd 3814 . . . . 5  |-  ( ph  ->  ( U  .Q  T
)  <Q  R )
4847adantr 265 . . . 4  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( U  .Q  T )  <Q  R )
4923simprd 111 . . . . . . 7  |-  ( ph  ->  U  e.  ( 2nd `  A ) )
5022, 49jca 294 . . . . . 6  |-  ( ph  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A ) ) )
5150adantr 265 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( A  e.  P.  /\  U  e.  ( 2nd `  A
) ) )
5227anim1i 327 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  ( B  e.  P.  /\  T  e.  ( 2nd `  B
) ) )
5340adantr 265 . . . . 5  |-  ( (
ph  /\  T  e.  ( 2nd `  B ) )  ->  R  e.  Q. )
54 mulnqpru 6725 . . . . 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 1145 . . . 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 663 . 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 6532 . . . . . . 7  |-  ( ( D  e.  Q.  /\  U  e.  Q. )  ->  ( D  .Q  U
)  e.  Q. )
604, 59syl 14 . . . . . 6  |-  ( ph  ->  ( D  .Q  U
)  e.  Q. )
61 ltmnqg 6557 . . . . . 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 1146 . . . . 5  |-  ( ph  ->  ( E  <Q  T  <->  ( ( D  .Q  U )  .Q  E )  <Q  (
( D  .Q  U
)  .Q  T ) ) )
63 mulcomnqg 6539 . . . . . . 7  |-  ( ( ( D  .Q  U
)  e.  Q.  /\  E  e.  Q. )  ->  ( ( D  .Q  U )  .Q  E
)  =  ( E  .Q  ( D  .Q  U ) ) )
6460, 3, 63syl2anc 397 . . . . . 6  |-  ( ph  ->  ( ( D  .Q  U )  .Q  E
)  =  ( E  .Q  ( D  .Q  U ) ) )
65 mulcomnqg 6539 . . . . . . 7  |-  ( ( ( D  .Q  U
)  e.  Q.  /\  T  e.  Q. )  ->  ( ( D  .Q  U )  .Q  T
)  =  ( T  .Q  ( D  .Q  U ) ) )
6660, 34, 65syl2anc 397 . . . . . 6  |-  ( ph  ->  ( ( D  .Q  U )  .Q  T
)  =  ( T  .Q  ( D  .Q  U ) ) )
6764, 66breq12d 3805 . . . . 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 181 . . . 4  |-  ( ph  ->  ( E  <Q  T  <->  ( E  .Q  ( D  .Q  U
) )  <Q  ( T  .Q  ( D  .Q  U ) ) ) )
6958, 68mpbird 160 . . 3  |-  ( ph  ->  E  <Q  T )
70 prop 6631 . . . 4  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
71 prloc 6647 . . . 4  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  E  <Q  T )  ->  ( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7270, 71sylan 271 . . 3  |-  ( ( B  e.  P.  /\  E  <Q  T )  -> 
( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7327, 69, 72syl2anc 397 . 2  |-  ( ph  ->  ( E  e.  ( 1st `  B )  \/  T  e.  ( 2nd `  B ) ) )
7433, 57, 73mpjaodan 722 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    = wceq 1259    e. wcel 1409   <.cop 3406   class class class wbr 3792   ` cfv 4930  (class class class)co 5540   1stc1st 5793   2ndc2nd 5794   Q.cnq 6436    .Q cmq 6439    <Q cltq 6441   P.cnp 6447    .P. cmp 6450
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-mi 6462  df-lti 6463  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-inp 6622  df-imp 6625
This theorem is referenced by:  mullocpr  6727
  Copyright terms: Public domain W3C validator