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Theorem mulnqpru 7345
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 7177 . . . . . . 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.  ( 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 7251 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
4 elprnqu 7258 . . . . . . . . 9  |-  ( (
<. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
53, 4sylan 281 . . . . . . . 8  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  ->  G  e.  Q. )
65ad2antrr 479 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
7 prop 7251 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
8 elprnqu 7258 . . . . . . . . 9  |-  ( (
<. ( 1st `  B
) ,  ( 2nd `  B ) >.  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
97, 8sylan 281 . . . . . . . 8  |-  ( ( B  e.  P.  /\  H  e.  ( 2nd `  B ) )  ->  H  e.  Q. )
109ad2antlr 480 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
11 mulclnq 7152 . . . . . . 7  |-  ( ( G  e.  Q.  /\  H  e.  Q. )  ->  ( G  .Q  H
)  e.  Q. )
126, 10, 11syl2anc 408 . . . . . 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 109 . . . . . 6  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  X  e.  Q. )
14 recclnq 7168 . . . . . . 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 7159 . . . . . . 7  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  =  ( z  .Q  y ) )
1716adantl 275 . . . . . 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 5908 . . . . 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 7160 . . . . . . . 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 1201 . . . . . . 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 7169 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5758 . . . . . . . 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 7165 . . . . . . . 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 2154 . . . . . 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 3909 . . . . 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 187 . . . 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 7261 . . . . . 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 281 . . . . 5  |-  ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  -> 
( G  <Q  ( X  .Q  ( *Q `  H ) )  -> 
( X  .Q  ( *Q `  H ) )  e.  ( 2nd `  A
) ) )
3130ad2antrr 479 . . . 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 149 . . 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 7245 . . . . . . . . 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 7152 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genppreclu 7291 . . . . . . . 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 364 . . . . . . 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 255 . . . . 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 468 . . . 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 274 . . 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 7160 . . . . 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 1201 . . . 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 7159 . . . . . . 7  |-  ( ( ( *Q `  H
)  e.  Q.  /\  H  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  ( H  .Q  ( *Q `  H ) ) )
4515, 10, 44syl2anc 408 . . . . . 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 2150 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5758 . . . 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 7165 . . . . 5  |-  ( X  e.  Q.  ->  ( X  .Q  1Q )  =  X )
5049adantl 275 . . . 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 2154 . . 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 2186 . 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 148 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 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   <.cop 3500   class class class wbr 3899   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   Q.cnq 7056   1Qc1q 7057    .Q cmq 7059   *Qcrq 7060    <Q cltq 7061   P.cnp 7067    .P. cmp 7070
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-eprel 4181  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-1o 6281  df-oadd 6285  df-omul 6286  df-er 6397  df-ec 6399  df-qs 6403  df-ni 7080  df-mi 7082  df-lti 7083  df-mpq 7121  df-enq 7123  df-nqqs 7124  df-mqqs 7126  df-1nqqs 7127  df-rq 7128  df-ltnqqs 7129  df-inp 7242  df-imp 7245
This theorem is referenced by:  mullocprlem  7346  mulclpr  7348
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