MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  recextlem2 Structured version   Unicode version

Theorem recextlem2 10176
Description: Lemma for recex 10177. (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 6290 . . . . . . . . 9  |-  ( B  =  0  ->  (
_i  x.  B )  =  ( _i  x.  0 ) )
2 ax-icn 9547 . . . . . . . . . 10  |-  _i  e.  CC
32mul01i 9765 . . . . . . . . 9  |-  ( _i  x.  0 )  =  0
41, 3syl6eq 2524 . . . . . . . 8  |-  ( B  =  0  ->  (
_i  x.  B )  =  0 )
5 oveq12 6291 . . . . . . . 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 9750 . . . . . . 7  |-  ( 0  +  0 )  =  0
86, 7syl6eq 2524 . . . . . 6  |-  ( ( A  =  0  /\  B  =  0 )  ->  ( A  +  ( _i  x.  B
) )  =  0 )
98necon3ai 2695 . . . . 5  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  -.  ( A  =  0  /\  B  =  0
) )
10 neorian 2794 . . . . 5  |-  ( ( A  =/=  0  \/  B  =/=  0 )  <->  -.  ( A  =  0  /\  B  =  0 ) )
119, 10sylibr 212 . . . 4  |-  ( ( A  +  ( _i  x.  B ) )  =/=  0  ->  ( A  =/=  0  \/  B  =/=  0 ) )
12 remulcl 9573 . . . . . . . . 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 9573 . . . . . . . . 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 10069 . . . . . . . 8  |-  ( ( A  e.  RR  /\  A  =/=  0 )  -> 
0  <  ( A  x.  A ) )
19 msqge0 10070 . . . . . . . 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 10036 . . . . . 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 10070 . . . . . . . 8  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 msqgt0 10069 . . . . . . . 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 10035 . . . . . 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 1191 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) )  =/=  0 )  ->  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3433gt0ne0d 10113 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   class class class wbr 4447  (class class class)co 6282   RRcr 9487   0cc0 9488   _ici 9490    + caddc 9491    x. cmul 9493    < clt 9624    <_ cle 9625
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804
This theorem is referenced by:  recex  10177
  Copyright terms: Public domain W3C validator