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Theorem mulnqpru 6724
Description: Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqpru  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  ->  X  e.  ( 2nd `  ( A  .P.  B
) ) ) )

Proof of Theorem mulnqpru
Dummy variables  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 6556 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
21adantl 266 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A
) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )
)  ->  ( y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
3 prop 6630 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elprnqu 6637 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
53, 4sylan 271 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
65ad2antrr 465 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
7 prop 6630 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
8 elprnqu 6637 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
97, 8sylan 271 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
109ad2antlr 466 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
11 mulclnq 6531 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
126, 10, 11syl2anc 397 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
13 simpr 107 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
14 recclnq 6547 . . . . . . 7  |-  ( H  e.  Q.  ->  ( *Q `  H )  e. 
Q. )
1510, 14syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( *Q `  H
)  e.  Q. )
16 mulcomnqg 6538 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
1716adantl 266 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A
) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q. )
)  ->  ( y  .Q  z )  =  ( z  .Q  y ) )
182, 12, 13, 15, 17caovord2d 5697 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  <->  ( ( G  .Q  H )  .Q  ( *Q `  H
) )  <Q  ( X  .Q  ( *Q `  H ) ) ) )
19 mulassnqg 6539 . . . . . . . 8  |-  ( ( G  e.  Q.  /\  H  e.  Q.  /\  ( *Q `  H )  e. 
Q. )  ->  (
( G  .Q  H
)  .Q  ( *Q
`  H ) )  =  ( G  .Q  ( H  .Q  ( *Q `  H ) ) ) )
206, 10, 15, 19syl3anc 1146 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  .Q  ( *Q `  H ) )  =  ( G  .Q  ( H  .Q  ( *Q `  H ) ) ) )
21 recidnq 6548 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5555 . . . . . . . 8  |-  ( H  e.  Q.  ->  ( G  .Q  ( H  .Q  ( *Q `  H ) ) )  =  ( G  .Q  1Q ) )
2310, 22syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  ( H  .Q  ( *Q `  H ) ) )  =  ( G  .Q  1Q ) )
24 mulidnq 6544 . . . . . . . 8  |-  ( G  e.  Q.  ->  ( G  .Q  1Q )  =  G )
256, 24syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  1Q )  =  G )
2620, 23, 253eqtrd 2092 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  .Q  ( *Q `  H ) )  =  G )
2726breq1d 3801 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( G  .Q  H )  .Q  ( *Q `  H
) )  <Q  ( X  .Q  ( *Q `  H ) )  <->  G  <Q  ( X  .Q  ( *Q
`  H ) ) ) )
2818, 27bitrd 181 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  <->  G  <Q  ( X  .Q  ( *Q
`  H ) ) ) )
29 prcunqu 6640 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  -> 
( G  <Q  ( X  .Q  ( *Q `  H ) )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
303, 29sylan 271 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  -> 
( G  <Q  ( X  .Q  ( *Q `  H ) )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
3130ad2antrr 465 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( G  <Q  ( X  .Q  ( *Q `  H ) )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
3228, 31sylbid 143 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
33 df-imp 6624 . . . . . . . . 9  |-  .P.  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v
)  /\  x  =  ( y  .Q  z
) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e.  Q.  (
y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v
)  /\  x  =  ( y  .Q  z
) ) } >. )
34 mulclnq 6531 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genppreclu 6670 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
)  /\  H  e.  ( 2nd `  B ) )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
3635exp4b 353 . . . . . . 7  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
)  ->  ( H  e.  ( 2nd `  B
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) ) ) )
3736com34 81 . . . . . 6  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( H  e.  ( 2nd `  B )  ->  (
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) ) ) )
3837imp32 248 . . . . 5  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
3938adantlr 454 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
4039adantr 265 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A )  ->  (
( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
4132, 40syld 44 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  -> 
( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) ) ) )
42 mulassnqg 6539 . . . . 5  |-  ( ( X  e.  Q.  /\  ( *Q `  H )  e.  Q.  /\  H  e.  Q. )  ->  (
( X  .Q  ( *Q `  H ) )  .Q  H )  =  ( X  .Q  (
( *Q `  H
)  .Q  H ) ) )
4313, 15, 10, 42syl3anc 1146 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  =  ( X  .Q  ( ( *Q `  H )  .Q  H
) ) )
44 mulcomnqg 6538 . . . . . . 7  |-  ( ( ( *Q `  H
)  e.  Q.  /\  H  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4515, 10, 44syl2anc 397 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4610, 21syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
4745, 46eqtrd 2088 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5555 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  H
)  .Q  H ) )  =  ( X  .Q  1Q ) )
49 mulidnq 6544 . . . . 5  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
5049adantl 266 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
5143, 48, 503eqtrd 2092 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  =  X )
5251eleq1d 2122 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 2nd `  ( A  .P.  B ) )  <-> 
X  e.  ( 2nd `  ( A  .P.  B
) ) ) )
5341, 52sylibd 142 1  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  <Q  X  ->  X  e.  ( 2nd `  ( A  .P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    e. wcel 1409   <.cop 3405   class class class wbr 3791   ` cfv 4929  (class class class)co 5539   1stc1st 5792   2ndc2nd 5793   Q.cnq 6435   1Qc1q 6436    .Q cmq 6438   *Qcrq 6439    <Q cltq 6440   P.cnp 6446    .P. cmp 6449
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 3899  ax-sep 3902  ax-nul 3910  ax-pow 3954  ax-pr 3971  ax-un 4197  ax-setind 4289  ax-iinf 4338
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 2787  df-csb 2880  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-int 3643  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-tr 3882  df-eprel 4053  df-id 4057  df-iord 4130  df-on 4132  df-suc 4135  df-iom 4341  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-ov 5542  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795  df-recs 5950  df-irdg 5987  df-1o 6031  df-oadd 6035  df-omul 6036  df-er 6136  df-ec 6138  df-qs 6142  df-ni 6459  df-mi 6461  df-lti 6462  df-mpq 6500  df-enq 6502  df-nqqs 6503  df-mqqs 6505  df-1nqqs 6506  df-rq 6507  df-ltnqqs 6508  df-inp 6621  df-imp 6624
This theorem is referenced by:  mullocprlem  6725  mulclpr  6727
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