ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  recexaplem2 Unicode version

Theorem recexaplem2 8425
Description: Lemma for recexap 8426. (Contributed by Jim Kingdon, 20-Feb-2020.)
Assertion
Ref Expression
recexaplem2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) ) #  0 )

Proof of Theorem recexaplem2
StepHypRef Expression
1 ax-icn 7727 . . . . . . . . . . 11  |-  _i  e.  CC
21mul01i 8165 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
32oveq2i 5785 . . . . . . . . 9  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
4 00id 7915 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
53, 4eqtr2i 2161 . . . . . . . 8  |-  0  =  ( 0  +  ( _i  x.  0 ) )
65breq2i 3937 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) ) #  0  <->  ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) ) )
7 0re 7778 . . . . . . . 8  |-  0  e.  RR
8 apreim 8377 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  e.  RR  /\  0  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
97, 7, 8mpanr12 435 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
106, 9syl5bb 191 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  0  <->  ( A #  0  \/  B #  0 ) ) )
1110pm5.32i 449 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) ) )
12 remulcl 7760 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 394 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 7760 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 394 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 336 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 apsqgt0 8375 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A #  0 )  ->  0  <  ( A  x.  A
) )
19 msqge0 8390 . . . . . . . . 9  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 336 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B ) ) )
2120an32s 557 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
22 addgtge0 8224 . . . . . . 7  |-  ( ( ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR )  /\  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
2317, 21, 22syl2anc 408 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 8390 . . . . . . . . 9  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 apsqgt0 8375 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B #  0 )  ->  0  <  ( B  x.  B
) )
2725, 26anim12i 336 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B #  0 ) )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
2827anassrs 397 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
29 addgegt0 8223 . . . . . . 7  |-  ( ( ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR )  /\  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3024, 28, 29syl2anc 408 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 786 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3211, 31sylbi 120 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1176 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3433olcd 723 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( (
( A  x.  A
)  +  ( B  x.  B ) )  <  0  \/  0  <  ( ( A  x.  A )  +  ( B  x.  B
) ) ) )
35 simp1 981 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  A  e.  RR )
3635, 35remulcld 7808 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( A  x.  A )  e.  RR )
37 simp2 982 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  B  e.  RR )
3837, 37remulcld 7808 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( B  x.  B )  e.  RR )
3936, 38readdcld 7807 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) )  e.  RR )
40 reaplt 8362 . . 3  |-  ( ( ( ( A  x.  A )  +  ( B  x.  B ) )  e.  RR  /\  0  e.  RR )  ->  ( ( ( A  x.  A )  +  ( B  x.  B
) ) #  0  <->  (
( ( A  x.  A )  +  ( B  x.  B ) )  <  0  \/  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) ) ) )
4139, 7, 40sylancl 409 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( (
( A  x.  A
)  +  ( B  x.  B ) ) #  0  <->  ( ( ( A  x.  A )  +  ( B  x.  B ) )  <  0  \/  0  < 
( ( A  x.  A )  +  ( B  x.  B ) ) ) ) )
4234, 41mpbird 166 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) ) #  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    /\ w3a 962    e. wcel 1480   class class class wbr 3929  (class class class)co 5774   RRcr 7631   0cc0 7632   _ici 7634    + caddc 7635    x. cmul 7637    < clt 7812    <_ cle 7813   # cap 8355
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356
This theorem is referenced by:  recexap  8426
  Copyright terms: Public domain W3C validator