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Theorem mulnqprl 7491
Description: Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-Dec-2019.)
Assertion
Ref Expression
mulnqprl  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  ->  X  e.  ( 1st `  ( A  .P.  B
) ) ) )

Proof of Theorem mulnqprl
Dummy variables  v  w  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltmnqg 7324 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )  ->  (
y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
21adantl 275 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A
) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q.  /\  w  e.  Q. )
)  ->  ( y  <Q  z  <->  ( w  .Q  y )  <Q  (
w  .Q  z ) ) )
3 simpr 109 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
4 prop 7398 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7404 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
64, 5sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  ->  G  e.  Q. )
76ad2antrr 480 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
8 prop 7398 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
9 elprnql 7404 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
108, 9sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 1st `  B ) )  ->  H  e.  Q. )
1110ad2antlr 481 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
12 mulclnq 7299 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
137, 11, 12syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
14 recclnq 7315 . . . . . . 7  |-  ( H  e.  Q.  ->  ( *Q `  H )  e. 
Q. )
1511, 14syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( *Q `  H
)  e.  Q. )
16 mulcomnqg 7306 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
1716adantl 275 . . . . . 6  |-  ( ( ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A
) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  /\  ( y  e.  Q.  /\  z  e.  Q. )
)  ->  ( y  .Q  z )  =  ( z  .Q  y ) )
182, 3, 13, 15, 17caovord2d 5993 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  <->  ( X  .Q  ( *Q `  H
) )  <Q  (
( G  .Q  H
)  .Q  ( *Q
`  H ) ) ) )
19 mulassnqg 7307 . . . . . . . 8  |-  ( ( G  e.  Q.  /\  H  e.  Q.  /\  ( *Q `  H )  e. 
Q. )  ->  (
( G  .Q  H
)  .Q  ( *Q
`  H ) )  =  ( G  .Q  ( H  .Q  ( *Q `  H ) ) ) )
207, 11, 15, 19syl3anc 1220 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  .Q  ( *Q `  H ) )  =  ( G  .Q  ( H  .Q  ( *Q `  H ) ) ) )
21 recidnq 7316 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5843 . . . . . . . 8  |-  ( H  e.  Q.  ->  ( G  .Q  ( H  .Q  ( *Q `  H ) ) )  =  ( G  .Q  1Q ) )
2311, 22syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  ( H  .Q  ( *Q `  H ) ) )  =  ( G  .Q  1Q ) )
24 mulidnq 7312 . . . . . . . 8  |-  ( G  e.  Q.  ->  ( G  .Q  1Q )  =  G )
257, 24syl 14 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( G  .Q  1Q )  =  G )
2620, 23, 253eqtrd 2194 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( G  .Q  H )  .Q  ( *Q `  H ) )  =  G )
2726breq2d 3979 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  <Q  ( ( G  .Q  H )  .Q  ( *Q `  H
) )  <->  ( X  .Q  ( *Q `  H
) )  <Q  G ) )
2818, 27bitrd 187 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  <->  ( X  .Q  ( *Q `  H
) )  <Q  G ) )
29 prcdnql 7407 . . . . . 6  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 1st `  A ) )  -> 
( ( X  .Q  ( *Q `  H ) )  <Q  G  ->  ( X  .Q  ( *Q
`  H ) )  e.  ( 1st `  A
) ) )
304, 29sylan 281 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  -> 
( ( X  .Q  ( *Q `  H ) )  <Q  G  ->  ( X  .Q  ( *Q
`  H ) )  e.  ( 1st `  A
) ) )
3130ad2antrr 480 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  <Q  G  ->  ( X  .Q  ( *Q
`  H ) )  e.  ( 1st `  A
) ) )
3228, 31sylbid 149 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
) ) )
33 df-imp 7392 . . . . . . . . 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 7299 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genpprecll 7437 . . . . . . . 8  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( ( ( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
)  /\  H  e.  ( 1st `  B ) )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) )
3635exp4b 365 . . . . . . 7  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
)  ->  ( H  e.  ( 1st `  B
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) ) ) )
3736com34 83 . . . . . 6  |-  ( A  e.  P.  ->  ( B  e.  P.  ->  ( H  e.  ( 1st `  B )  ->  (
( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) ) ) )
3837imp32 255 . . . . 5  |-  ( ( A  e.  P.  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) )
3938adantlr 469 . . . 4  |-  ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A
)  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) )
4039adantr 274 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  e.  ( 1st `  A )  ->  (
( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) )
4132, 40syld 45 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  -> 
( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) ) ) )
42 mulassnqg 7307 . . . . 5  |-  ( ( X  e.  Q.  /\  ( *Q `  H )  e.  Q.  /\  H  e.  Q. )  ->  (
( X  .Q  ( *Q `  H ) )  .Q  H )  =  ( X  .Q  (
( *Q `  H
)  .Q  H ) ) )
433, 15, 11, 42syl3anc 1220 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  =  ( X  .Q  ( ( *Q `  H )  .Q  H
) ) )
44 mulcomnqg 7306 . . . . . . 7  |-  ( ( ( *Q `  H
)  e.  Q.  /\  H  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4515, 11, 44syl2anc 409 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4611, 21syl 14 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
4745, 46eqtrd 2190 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5843 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  (
( *Q `  H
)  .Q  H ) )  =  ( X  .Q  1Q ) )
49 mulidnq 7312 . . . . 5  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
5049adantl 275 . . . 4  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  .Q  1Q )  =  X )
5143, 48, 503eqtrd 2194 . . 3  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( X  .Q  ( *Q `  H ) )  .Q  H )  =  X )
5251eleq1d 2226 . 2  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( ( X  .Q  ( *Q `  H ) )  .Q  H )  e.  ( 1st `  ( A  .P.  B ) )  <-> 
X  e.  ( 1st `  ( A  .P.  B
) ) ) )
5341, 52sylibd 148 1  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( X  <Q  ( G  .Q  H )  ->  X  e.  ( 1st `  ( A  .P.  B
) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1335    e. wcel 2128   <.cop 3564   class class class wbr 3967   ` cfv 5173  (class class class)co 5827   1stc1st 6089   2ndc2nd 6090   Q.cnq 7203   1Qc1q 7204    .Q cmq 7206   *Qcrq 7207    <Q cltq 7208   P.cnp 7214    .P. cmp 7217
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4082  ax-sep 4085  ax-nul 4093  ax-pow 4138  ax-pr 4172  ax-un 4396  ax-setind 4499  ax-iinf 4550
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3396  df-pw 3546  df-sn 3567  df-pr 3568  df-op 3570  df-uni 3775  df-int 3810  df-iun 3853  df-br 3968  df-opab 4029  df-mpt 4030  df-tr 4066  df-eprel 4252  df-id 4256  df-iord 4329  df-on 4331  df-suc 4334  df-iom 4553  df-xp 4595  df-rel 4596  df-cnv 4597  df-co 4598  df-dm 4599  df-rn 4600  df-res 4601  df-ima 4602  df-iota 5138  df-fun 5175  df-fn 5176  df-f 5177  df-f1 5178  df-fo 5179  df-f1o 5180  df-fv 5181  df-ov 5830  df-oprab 5831  df-mpo 5832  df-1st 6091  df-2nd 6092  df-recs 6255  df-irdg 6320  df-1o 6366  df-oadd 6370  df-omul 6371  df-er 6483  df-ec 6485  df-qs 6489  df-ni 7227  df-mi 7229  df-lti 7230  df-mpq 7268  df-enq 7270  df-nqqs 7271  df-mqqs 7273  df-1nqqs 7274  df-rq 7275  df-ltnqqs 7276  df-inp 7389  df-imp 7392
This theorem is referenced by:  mullocprlem  7493  mulclpr  7495
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