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Theorem mulnqpru 7581
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 7413 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
21adantl 277 . . . . . 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 7487 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elprnqu 7494 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
53, 4sylan 283 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
65ad2antrr 488 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
7 prop 7487 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
8 elprnqu 7494 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
97, 8sylan 283 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
109ad2antlr 489 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
11 mulclnq 7388 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
126, 10, 11syl2anc 411 . . . . . 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 110 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
14 recclnq 7404 . . . . . . 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 7395 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
1716adantl 277 . . . . . 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 6057 . . . . 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 7396 . . . . . . . 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 1248 . . . . . . 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 7405 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5904 . . . . . . . 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 7401 . . . . . . . 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 2224 . . . . . 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 4025 . . . . 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 188 . . . 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 7497 . . . . . 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 283 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  -> 
( G  <Q  ( X  .Q  ( *Q `  H ) )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
3130ad2antrr 488 . . . 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 150 . . 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 7481 . . . . . . . . 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 7388 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genppreclu 7527 . . . . . . . 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 367 . . . . . . 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 83 . . . . . 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 257 . . . . 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 477 . . . 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 276 . . 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 45 . 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 7396 . . . . 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 1248 . . . 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 7395 . . . . . . 7  |-  ( ( ( *Q `  H
)  e.  Q.  /\  H  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4515, 10, 44syl2anc 411 . . . . . 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 2220 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5904 . . . 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 7401 . . . . 5  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
5049adantl 277 . . . 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 2224 . . 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 2256 . 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 149 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 104    <-> wb 105    /\ w3a 979    = wceq 1363    e. wcel 2158   <.cop 3607   class class class wbr 4015   ` cfv 5228  (class class class)co 5888   1stc1st 6152   2ndc2nd 6153   Q.cnq 7292   1Qc1q 7293    .Q cmq 7295   *Qcrq 7296    <Q cltq 7297   P.cnp 7303    .P. cmp 7306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-eprel 4301  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-ov 5891  df-oprab 5892  df-mpo 5893  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-1o 6430  df-oadd 6434  df-omul 6435  df-er 6548  df-ec 6550  df-qs 6554  df-ni 7316  df-mi 7318  df-lti 7319  df-mpq 7357  df-enq 7359  df-nqqs 7360  df-mqqs 7362  df-1nqqs 7363  df-rq 7364  df-ltnqqs 7365  df-inp 7478  df-imp 7481
This theorem is referenced by:  mullocprlem  7582  mulclpr  7584
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