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Theorem recextlem2 9415
Description: Lemma for recex 9416. (Contributed by Eric Schmidt, 23-May-2007.)
Assertion
Ref Expression
recextlem2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  (
( A  x.  A
)  +  ( B  x.  B ) )  =/=  0 )

Proof of Theorem recextlem2
StepHypRef Expression
1 oveq2 5882 . . . . . . . . 9  |-  ( B  =  0  ->  (
_i  x.  B )  =  ( _i  x.  0 ) )
2 ax-icn 8812 . . . . . . . . . 10  |-  _i  e.  CC
32mul01i 9018 . . . . . . . . 9  |-  ( _i  x.  0 )  =  0
41, 3syl6eq 2344 . . . . . . . 8  |-  ( B  =  0  ->  (
_i  x.  B )  =  0 )
5 oveq12 5883 . . . . . . . 8  |-  ( ( A  =  0  /\  ( _i  x.  B
)  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
64, 5sylan2 460 . . . . . . 7  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
7 00id 9003 . . . . . . 7  |-  ( 0  +  0 )  =  0
86, 7syl6eq 2344 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  0 )
98necon3ai 2499 . . . . 5  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
10 neorian 2546 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
119, 10sylibr 203 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  ( A  =/=  0  \/  B  =/=  0 ) )
12 remulcl 8838 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 626 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 8838 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 626 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 549 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 msqgt0 9310 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
0  <  ( A  x.  A ) )
19 msqge0 9311 . . . . . . . 8  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 549 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
2120an32s 779 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B
) ) )
22 addgtge0 9278 . . . . . 6  |-  ( ( ( ( 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 642 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 451 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 9311 . . . . . . . 8  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 msqgt0 9310 . . . . . . . 8  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
0  <  ( B  x.  B ) )
2725, 26anim12i 549 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
2827anassrs 629 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
29 addgegt0 9277 . . . . . 6  |-  ( ( ( ( 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 642 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 760 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =/=  0  \/  B  =/=  0 ) )  -> 
0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3211, 31sylan2 460 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) )  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1146 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3433gt0ne0d 9353 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:   -. wn 3    -> wi 4    \/ wo 357    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    =/= wne 2459   class class class wbr 4039  (class class class)co 5874   RRcr 8752   0cc0 8753   _ici 8755    + caddc 8756    x. cmul 8758    < clt 8883    <_ cle 8884
This theorem is referenced by:  recex  9416
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056
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