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Theorem recextlem2 10242
Description: Lemma for recex 10243. (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 6313 . . . . . . . . 9  |-  ( B  =  0  ->  (
_i  x.  B )  =  ( _i  x.  0 ) )
2 ax-icn 9597 . . . . . . . . . 10  |-  _i  e.  CC
32mul01i 9822 . . . . . . . . 9  |-  ( _i  x.  0 )  =  0
41, 3syl6eq 2486 . . . . . . . 8  |-  ( B  =  0  ->  (
_i  x.  B )  =  0 )
5 oveq12 6314 . . . . . . . 8  |-  ( ( A  =  0  /\  ( _i  x.  B
)  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
64, 5sylan2 476 . . . . . . 7  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
7 00id 9807 . . . . . . 7  |-  ( 0  +  0 )  =  0
86, 7syl6eq 2486 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  0 )
98necon3ai 2659 . . . . 5  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
10 neorian 2758 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
119, 10sylibr 215 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  ( A  =/=  0  \/  B  =/=  0 ) )
12 remulcl 9623 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 649 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 9623 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 649 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 568 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 msqgt0 10133 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
0  <  ( A  x.  A ) )
19 msqge0 10134 . . . . . . . 8  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 568 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
2120an32s 811 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B
) ) )
22 addgtge0 10101 . . . . . 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 665 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 466 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 10134 . . . . . . . 8  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 msqgt0 10133 . . . . . . . 8  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
0  <  ( B  x.  B ) )
2725, 26anim12i 568 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
2827anassrs 652 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
29 addgegt0 10100 . . . . . 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 665 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 792 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =/=  0  \/  B  =/=  0 ) )  -> 
0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3211, 31sylan2 476 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) )  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1200 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3433gt0ne0d 10177 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  (
( A  x.  A
)  +  ( B  x.  B ) )  =/=  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870    =/= wne 2625   class class class wbr 4426  (class class class)co 6305   RRcr 9537   0cc0 9538   _ici 9540    + caddc 9541    x. cmul 9543    < clt 9674    <_ cle 9675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862
This theorem is referenced by:  recex  10243
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