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Theorem recextlem1 8120
Description: Lemma for recexap 8122. (Contributed by Eric Schmidt, 23-May-2007.)
Assertion
Ref Expression
recextlem1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  +  ( B  x.  B
) ) )

Proof of Theorem recextlem1
StepHypRef Expression
1 simpl 107 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 ax-icn 7440 . . . . 5  |-  _i  e.  CC
3 mulcl 7469 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
42, 3mpan 415 . . . 4  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
54adantl 271 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
6 subcl 7681 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  -  ( _i  x.  B
) )  e.  CC )
74, 6sylan2 280 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
_i  x.  B )
)  e.  CC )
81, 5, 7adddird 7513 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  ( A  -  ( _i  x.  B
) ) )  +  ( ( _i  x.  B )  x.  ( A  -  ( _i  x.  B ) ) ) ) )
91, 1, 5subdid 7892 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) ) )
105, 1, 5subdid 7892 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  B )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( ( _i  x.  B )  x.  A )  -  ( ( _i  x.  B )  x.  (
_i  x.  B )
) ) )
11 mulcom 7471 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  =  ( ( _i  x.  B
)  x.  A ) )
124, 11sylan2 280 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  x.  A ) )
13 ixi 8060 . . . . . . . . . 10  |-  ( _i  x.  _i )  = 
-u 1
1413oveq1i 5662 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( B  x.  B ) )  =  ( -u 1  x.  ( B  x.  B
) )
15 mulcl 7469 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1615mulm1d 7888 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( B  x.  B
) )  =  -u ( B  x.  B
) )
1714, 16syl5req 2133 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  _i )  x.  ( B  x.  B ) ) )
18 mul4 7614 . . . . . . . . 9  |-  ( ( ( _i  e.  CC  /\  _i  e.  CC )  /\  ( B  e.  CC  /\  B  e.  CC ) )  -> 
( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
192, 2, 18mpanl12 427 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
2017, 19eqtrd 2120 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2120anidms 389 . . . . . 6  |-  ( B  e.  CC  ->  -u ( B  x.  B )  =  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) )
2221adantl 271 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2312, 22oveq12d 5670 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( _i  x.  B
) )  -  -u ( B  x.  B )
)  =  ( ( ( _i  x.  B
)  x.  A )  -  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) ) )
2410, 23eqtr4d 2123 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  B )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  ( _i  x.  B ) )  -  -u ( B  x.  B
) ) )
259, 24oveq12d 5670 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  ( A  -  (
_i  x.  B )
) )  +  ( ( _i  x.  B
)  x.  ( A  -  ( _i  x.  B ) ) ) )  =  ( ( ( A  x.  A
)  -  ( A  x.  ( _i  x.  B ) ) )  +  ( ( A  x.  ( _i  x.  B ) )  -  -u ( B  x.  B
) ) ) )
26 mulcl 7469 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  x.  A
)  e.  CC )
2726anidms 389 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  A )  e.  CC )
2827adantr 270 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
29 mulcl 7469 . . . . 5  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  e.  CC )
304, 29sylan2 280 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  e.  CC )
3115negcld 7780 . . . . . 6  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3231anidms 389 . . . . 5  |-  ( B  e.  CC  ->  -u ( B  x.  B )  e.  CC )
3332adantl 271 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3428, 30, 33npncand 7817 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) )  +  ( ( A  x.  (
_i  x.  B )
)  -  -u ( B  x.  B )
) )  =  ( ( A  x.  A
)  -  -u ( B  x.  B )
) )
3515anidms 389 . . . 4  |-  ( B  e.  CC  ->  ( B  x.  B )  e.  CC )
36 subneg 7731 . . . 4  |-  ( ( ( A  x.  A
)  e.  CC  /\  ( B  x.  B
)  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B
) )  =  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
3727, 35, 36syl2an 283 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B )
)  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3834, 37eqtrd 2120 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) )  +  ( ( A  x.  (
_i  x.  B )
)  -  -u ( B  x.  B )
) )  =  ( ( A  x.  A
)  +  ( B  x.  B ) ) )
398, 25, 383eqtrd 2124 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  ( _i  x.  B
) )  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  +  ( B  x.  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438  (class class class)co 5652   CCcc 7348   1c1 7351   _ici 7352    + caddc 7353    x. cmul 7355    - cmin 7653   -ucneg 7654
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-neg 7656
This theorem is referenced by:  recexap  8122
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