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Theorem mulnqprl 7369
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 7202 . . . . . . 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 7276 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7282 . . . . . . . . 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 479 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
8 prop 7276 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
9 elprnql 7282 . . . . . . . . 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 480 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
12 mulclnq 7177 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
137, 11, 12syl2anc 408 . . . . . 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 7193 . . . . . . 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 7184 . . . . . . 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 5933 . . . . 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 7185 . . . . . . . 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 1216 . . . . . . 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 7194 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5783 . . . . . . . 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 7190 . . . . . . . 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 2174 . . . . . 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 3936 . . . . 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 7285 . . . . . 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 479 . . . 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 7270 . . . . . . . . 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 7177 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genpprecll 7315 . . . . . . . 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 364 . . . . . . 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 468 . . . 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 7185 . . . . 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 1216 . . . 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 7184 . . . . . . 7  |-  ( ( ( *Q `  H
)  e.  Q.  /\  H  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4515, 11, 44syl2anc 408 . . . . . 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 2170 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5783 . . . 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 7190 . . . . 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 2174 . . 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 2206 . 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 962    = wceq 1331    e. wcel 1480   <.cop 3525   class class class wbr 3924   ` cfv 5118  (class class class)co 5767   1stc1st 6029   2ndc2nd 6030   Q.cnq 7081   1Qc1q 7082    .Q cmq 7084   *Qcrq 7085    <Q cltq 7086   P.cnp 7092    .P. cmp 7095
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-mi 7107  df-lti 7108  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-inp 7267  df-imp 7270
This theorem is referenced by:  mullocprlem  7371  mulclpr  7373
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