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Theorem mulnqprl 7530
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 7363 . . . . . . 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 7437 . . . . . . . . 9  |-  ( A  e.  P.  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  P. )
5 elprnql 7443 . . . . . . . . 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 485 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  G  e.  Q. )
8 prop 7437 . . . . . . . . 9  |-  ( B  e.  P.  ->  <. ( 1st `  B ) ,  ( 2nd `  B
) >.  e.  P. )
9 elprnql 7443 . . . . . . . . 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 486 . . . . . . 7  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  H  e.  Q. )
12 mulclnq 7338 . . . . . . 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 7354 . . . . . . 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 7345 . . . . . . 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 6022 . . . . 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 7346 . . . . . . . 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 1233 . . . . . . 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 7355 . . . . . . . . 9  |-  ( H  e.  Q.  ->  ( H  .Q  ( *Q `  H ) )  =  1Q )
2221oveq2d 5869 . . . . . . . 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 7351 . . . . . . . 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 2207 . . . . . 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 4001 . . . . 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 7446 . . . . . 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 485 . . . 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 7431 . . . . . . . . 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 7338 . . . . . . . . 9  |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y  .Q  z
)  e.  Q. )
3533, 34genpprecll 7476 . . . . . . . 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 474 . . . 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 7346 . . . . 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 1233 . . . 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 7345 . . . . . . 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 2203 . . . . 5  |-  ( ( ( ( A  e. 
P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B
) ) )  /\  X  e.  Q. )  ->  ( ( *Q `  H )  .Q  H
)  =  1Q )
4847oveq2d 5869 . . . 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 7351 . . . . 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 2207 . . 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 2239 . 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 973    = wceq 1348    e. wcel 2141   <.cop 3586   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   1Qc1q 7243    .Q cmq 7245   *Qcrq 7246    <Q cltq 7247   P.cnp 7253    .P. cmp 7256
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-mi 7268  df-lti 7269  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-inp 7428  df-imp 7431
This theorem is referenced by:  mullocprlem  7532  mulclpr  7534
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