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Theorem mulnqpru 7567
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 7399 . . . . . . 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 7473 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elprnqu 7480 . . . . . . . . 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 7473 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
8 elprnqu 7480 . . . . . . . . 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 7374 . . . . . . 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 7390 . . . . . . 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 7381 . . . . . . 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 6043 . . . . 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 7382 . . . . . . . 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 1238 . . . . . . 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 7391 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5890 . . . . . . . 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 7387 . . . . . . . 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 2214 . . . . . 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 4013 . . . . 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 7483 . . . . . 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 7467 . . . . . . . . 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 7374 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genppreclu 7513 . . . . . . . 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 7382 . . . . 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 1238 . . . 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 7381 . . . . . . 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 2210 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5890 . . . 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 7387 . . . . 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 2214 . . 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 2246 . 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 978    = wceq 1353    e. wcel 2148   <.cop 3595   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   1stc1st 6138   2ndc2nd 6139   Q.cnq 7278   1Qc1q 7279    .Q cmq 7281   *Qcrq 7282    <Q cltq 7283   P.cnp 7289    .P. cmp 7292
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-eprel 4289  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-1o 6416  df-oadd 6420  df-omul 6421  df-er 6534  df-ec 6536  df-qs 6540  df-ni 7302  df-mi 7304  df-lti 7305  df-mpq 7343  df-enq 7345  df-nqqs 7346  df-mqqs 7348  df-1nqqs 7349  df-rq 7350  df-ltnqqs 7351  df-inp 7464  df-imp 7467
This theorem is referenced by:  mullocprlem  7568  mulclpr  7570
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