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Theorem recextlem2 10079
Description: Lemma for recex 10080. (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 6209 . . . . . . . . 9  |-  ( B  =  0  ->  (
_i  x.  B )  =  ( _i  x.  0 ) )
2 ax-icn 9453 . . . . . . . . . 10  |-  _i  e.  CC
32mul01i 9671 . . . . . . . . 9  |-  ( _i  x.  0 )  =  0
41, 3syl6eq 2511 . . . . . . . 8  |-  ( B  =  0  ->  (
_i  x.  B )  =  0 )
5 oveq12 6210 . . . . . . . 8  |-  ( ( A  =  0  /\  ( _i  x.  B
)  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
64, 5sylan2 474 . . . . . . 7  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  ( 0  +  0 ) )
7 00id 9656 . . . . . . 7  |-  ( 0  +  0 )  =  0
86, 7syl6eq 2511 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  0 )
98necon3ai 2680 . . . . 5  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
10 neorian 2779 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
119, 10sylibr 212 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  ( A  =/=  0  \/  B  =/=  0 ) )
12 remulcl 9479 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 645 . . . . . . . 8  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 9479 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 645 . . . . . . . 8  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 566 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 msqgt0 9972 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
0  <  ( A  x.  A ) )
19 msqge0 9973 . . . . . . . 8  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 566 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  A  =/=  0 )  /\  B  e.  RR )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
2120an32s 802 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B
) ) )
22 addgtge0 9939 . . . . . 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 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 9973 . . . . . . . 8  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 msqgt0 9972 . . . . . . . 8  |-  ( ( B  e.  RR  /\  B  =/=  0 )  -> 
0  <  ( B  x.  B ) )
2725, 26anim12i 566 . . . . . . 7  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B  =/=  0 ) )  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
2827anassrs 648 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  ( 0  <_  ( A  x.  A )  /\  0  <  ( B  x.  B
) ) )
29 addgegt0 9938 . . . . . 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 661 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 783 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  =/=  0  \/  B  =/=  0 ) )  -> 
0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3211, 31sylan2 474 . . 3  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) )  =/=  0
)  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1183 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3433gt0ne0d 10016 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 368    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758    =/= wne 2648   class class class wbr 4401  (class class class)co 6201   RRcr 9393   0cc0 9394   _ici 9396    + caddc 9397    x. cmul 9399    < clt 9530    <_ cle 9531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470  ax-pre-mulgt0 9471
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-sub 9709  df-neg 9710
This theorem is referenced by:  recex  10080
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