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Theorem recextlem2 9399
Description: Lemma for recex 9400. (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 5866 . . . . . . . . 9  |-  ( B  =  0  ->  (
_i  x.  B )  =  ( _i  x.  0 ) )
2 ax-icn 8796 . . . . . . . . . 10  |-  _i  e.  CC
32mul01i 9002 . . . . . . . . 9  |-  ( _i  x.  0 )  =  0
41, 3syl6eq 2331 . . . . . . . 8  |-  ( B  =  0  ->  (
_i  x.  B )  =  0 )
5 oveq12 5867 . . . . . . . 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 8987 . . . . . . 7  |-  ( 0  +  0 )  =  0
86, 7syl6eq 2331 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  0 )
98necon3ai 2486 . . . . 5  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
10 neorian 2533 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
119, 10sylibr 203 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  ( A  =/=  0  \/  B  =/=  0 ) )
12 remulcl 8822 . . . . . . . . 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 8822 . . . . . . . . 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 9294 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
0  <  ( A  x.  A ) )
19 msqge0 9295 . . . . . . . 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 9262 . . . . . 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 9295 . . . . . . . 8  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 msqgt0 9294 . . . . . . . 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 9261 . . . . . 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 9337 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 1623    e. wcel 1684    =/= wne 2446   class class class wbr 4023  (class class class)co 5858   RRcr 8736   0cc0 8737   _ici 8739    + caddc 8740    x. cmul 8742    < clt 8867    <_ cle 8868
This theorem is referenced by:  recex  9400
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040
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