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Theorem recexaplem2 7707
Description: Lemma for recexap 7708. (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 7037 . . . . . . . . . . 11  |-  _i  e.  CC
21mul01i 7460 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
32oveq2i 5551 . . . . . . . . 9  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
4 00id 7215 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
53, 4eqtr2i 2077 . . . . . . . 8  |-  0  =  ( 0  +  ( _i  x.  0 ) )
65breq2i 3800 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) ) #  0  <->  ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) ) )
7 0re 7085 . . . . . . . 8  |-  0  e.  RR
8 apreim 7668 . . . . . . . 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 423 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
106, 9syl5bb 185 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  0  <->  ( A #  0  \/  B #  0 ) ) )
1110pm5.32i 435 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) ) )
12 remulcl 7067 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 383 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 7067 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 383 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 325 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 265 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 apsqgt0 7666 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A #  0 )  ->  0  <  ( A  x.  A
) )
19 msqge0 7681 . . . . . . . . 9  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 325 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B ) ) )
2120an32s 510 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
22 addgtge0 7519 . . . . . . 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 397 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 265 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 7681 . . . . . . . . 9  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 apsqgt0 7666 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B #  0 )  ->  0  <  ( B  x.  B
) )
2725, 26anim12i 325 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B #  0 ) )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
2827anassrs 386 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
29 addgegt0 7518 . . . . . . 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 397 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 721 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3211, 31sylbi 118 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1110 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3433olcd 663 . 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 915 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  A  e.  RR )
3635, 35remulcld 7115 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( A  x.  A )  e.  RR )
37 simp2 916 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  B  e.  RR )
3837, 37remulcld 7115 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( B  x.  B )  e.  RR )
3936, 38readdcld 7114 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) )  e.  RR )
40 reaplt 7653 . . 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 398 . 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 160 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 101    <-> wb 102    \/ wo 639    /\ w3a 896    e. wcel 1409   class class class wbr 3792  (class class class)co 5540   RRcr 6946   0cc0 6947   _ici 6949    + caddc 6950    x. cmul 6952    < clt 7119    <_ cle 7120   # cap 7646
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-nul 3911  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290  ax-iinf 4339  ax-cnex 7033  ax-resscn 7034  ax-1cn 7035  ax-1re 7036  ax-icn 7037  ax-addcl 7038  ax-addrcl 7039  ax-mulcl 7040  ax-mulrcl 7041  ax-addcom 7042  ax-mulcom 7043  ax-addass 7044  ax-mulass 7045  ax-distr 7046  ax-i2m1 7047  ax-1rid 7049  ax-0id 7050  ax-rnegex 7051  ax-precex 7052  ax-cnre 7053  ax-pre-ltirr 7054  ax-pre-ltwlin 7055  ax-pre-lttrn 7056  ax-pre-apti 7057  ax-pre-ltadd 7058  ax-pre-mulgt0 7059  ax-pre-mulext 7060
This theorem depends on definitions:  df-bi 114  df-dc 754  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-int 3644  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-eprel 4054  df-id 4058  df-po 4061  df-iso 4062  df-iord 4131  df-on 4133  df-suc 4136  df-iom 4342  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-riota 5496  df-ov 5543  df-oprab 5544  df-mpt2 5545  df-1st 5795  df-2nd 5796  df-recs 5951  df-irdg 5988  df-1o 6032  df-2o 6033  df-oadd 6036  df-omul 6037  df-er 6137  df-ec 6139  df-qs 6143  df-ni 6460  df-pli 6461  df-mi 6462  df-lti 6463  df-plpq 6500  df-mpq 6501  df-enq 6503  df-nqqs 6504  df-plqqs 6505  df-mqqs 6506  df-1nqqs 6507  df-rq 6508  df-ltnqqs 6509  df-enq0 6580  df-nq0 6581  df-0nq0 6582  df-plq0 6583  df-mq0 6584  df-inp 6622  df-i1p 6623  df-iplp 6624  df-iltp 6626  df-enr 6869  df-nr 6870  df-ltr 6873  df-0r 6874  df-1r 6875  df-0 6954  df-1 6955  df-r 6957  df-lt 6960  df-pnf 7121  df-mnf 7122  df-xr 7123  df-ltxr 7124  df-le 7125  df-sub 7247  df-neg 7248  df-reap 7640  df-ap 7647
This theorem is referenced by:  recexap  7708
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