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Theorem recexaplem2 8557
Description: Lemma for recexap 8558. (Contributed by Jim Kingdon, 20-Feb-2020.)
Assertion
Ref Expression
recexaplem2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) ) #  0 )

Proof of Theorem recexaplem2
StepHypRef Expression
1 ax-icn 7856 . . . . . . . . . . 11  |-  _i  e.  CC
21mul01i 8297 . . . . . . . . . 10  |-  ( _i  x.  0 )  =  0
32oveq2i 5861 . . . . . . . . 9  |-  ( 0  +  ( _i  x.  0 ) )  =  ( 0  +  0 )
4 00id 8047 . . . . . . . . 9  |-  ( 0  +  0 )  =  0
53, 4eqtr2i 2192 . . . . . . . 8  |-  0  =  ( 0  +  ( _i  x.  0 ) )
65breq2i 3995 . . . . . . 7  |-  ( ( A  +  ( _i  x.  B ) ) #  0  <->  ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) ) )
7 0re 7907 . . . . . . . 8  |-  0  e.  RR
8 apreim 8509 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  e.  RR  /\  0  e.  RR ) )  -> 
( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
97, 7, 8mpanr12 437 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  ( 0  +  ( _i  x.  0 ) )  <->  ( A #  0  \/  B #  0
) ) )
106, 9syl5bb 191 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  +  ( _i  x.  B
) ) #  0  <->  ( A #  0  \/  B #  0 ) ) )
1110pm5.32i 451 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  <-> 
( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) ) )
12 remulcl 7889 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  A  e.  RR )  ->  ( A  x.  A
)  e.  RR )
1312anidms 395 . . . . . . . . 9  |-  ( A  e.  RR  ->  ( A  x.  A )  e.  RR )
14 remulcl 7889 . . . . . . . . . 10  |-  ( ( B  e.  RR  /\  B  e.  RR )  ->  ( B  x.  B
)  e.  RR )
1514anidms 395 . . . . . . . . 9  |-  ( B  e.  RR  ->  ( B  x.  B )  e.  RR )
1613, 15anim12i 336 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B
)  e.  RR ) )
1716adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
18 apsqgt0 8507 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  A #  0 )  ->  0  <  ( A  x.  A
) )
19 msqge0 8522 . . . . . . . . 9  |-  ( B  e.  RR  ->  0  <_  ( B  x.  B
) )
2018, 19anim12i 336 . . . . . . . 8  |-  ( ( ( A  e.  RR  /\  A #  0 )  /\  B  e.  RR )  ->  ( 0  <  ( A  x.  A )  /\  0  <_  ( B  x.  B ) ) )
2120an32s 563 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  ( 0  < 
( A  x.  A
)  /\  0  <_  ( B  x.  B ) ) )
22 addgtge0 8356 . . . . . . 7  |-  ( ( ( ( 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 409 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
2416adantr 274 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( ( A  x.  A )  e.  RR  /\  ( B  x.  B )  e.  RR ) )
25 msqge0 8522 . . . . . . . . 9  |-  ( A  e.  RR  ->  0  <_  ( A  x.  A
) )
26 apsqgt0 8507 . . . . . . . . 9  |-  ( ( B  e.  RR  /\  B #  0 )  ->  0  <  ( B  x.  B
) )
2725, 26anim12i 336 . . . . . . . 8  |-  ( ( A  e.  RR  /\  ( B  e.  RR  /\  B #  0 ) )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
2827anassrs 398 . . . . . . 7  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  ( 0  <_ 
( A  x.  A
)  /\  0  <  ( B  x.  B ) ) )
29 addgegt0 8355 . . . . . . 7  |-  ( ( ( ( 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 409 . . . . . 6  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  B #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
3123, 30jaodan 792 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A #  0  \/  B #  0 ) )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3211, 31sylbi 120 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( A  +  ( _i  x.  B
) ) #  0 )  ->  0  <  (
( A  x.  A
)  +  ( B  x.  B ) ) )
33323impa 1189 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  0  <  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3433olcd 729 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( (
( A  x.  A
)  +  ( B  x.  B ) )  <  0  \/  0  <  ( ( A  x.  A )  +  ( B  x.  B
) ) ) )
35 simp1 992 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  A  e.  RR )
3635, 35remulcld 7937 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( A  x.  A )  e.  RR )
37 simp2 993 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  B  e.  RR )
3837, 37remulcld 7937 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( B  x.  B )  e.  RR )
3936, 38readdcld 7936 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( ( A  x.  A )  +  ( B  x.  B ) )  e.  RR )
40 reaplt 8494 . . 3  |-  ( ( ( ( A  x.  A )  +  ( B  x.  B ) )  e.  RR  /\  0  e.  RR )  ->  ( ( ( A  x.  A )  +  ( B  x.  B
) ) #  0  <->  (
( ( A  x.  A )  +  ( B  x.  B ) )  <  0  \/  0  <  ( ( A  x.  A )  +  ( B  x.  B ) ) ) ) )
4139, 7, 40sylancl 411 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  ( A  +  ( _i  x.  B ) ) #  0 )  ->  ( (
( A  x.  A
)  +  ( B  x.  B ) ) #  0  <->  ( ( ( A  x.  A )  +  ( B  x.  B ) )  <  0  \/  0  < 
( ( A  x.  A )  +  ( B  x.  B ) ) ) ) )
4234, 41mpbird 166 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:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    /\ w3a 973    e. wcel 2141   class class class wbr 3987  (class class class)co 5850   RRcr 7760   0cc0 7761   _ici 7763    + caddc 7764    x. cmul 7766    < clt 7941    <_ cle 7942   # cap 8487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488
This theorem is referenced by:  recexap  8558
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