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Theorem recextlem1 9608
Description: Lemma for recex 9610. (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 444 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 ax-icn 9005 . . . . 5  |-  _i  e.  CC
3 mulcl 9030 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
42, 3mpan 652 . . . 4  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
54adantl 453 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
6 subcl 9261 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  -  ( _i  x.  B
) )  e.  CC )
74, 6sylan2 461 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
_i  x.  B )
)  e.  CC )
81, 5, 7adddird 9069 . 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 9445 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) ) )
105, 1, 5subdid 9445 . . . 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 9032 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  =  ( ( _i  x.  B
)  x.  A ) )
124, 11sylan2 461 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  x.  A ) )
13 ixi 9607 . . . . . . . . . 10  |-  ( _i  x.  _i )  = 
-u 1
1413oveq1i 6050 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( B  x.  B ) )  =  ( -u 1  x.  ( B  x.  B
) )
15 mulcl 9030 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1615mulm1d 9441 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( B  x.  B
) )  =  -u ( B  x.  B
) )
1714, 16syl5req 2449 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  _i )  x.  ( B  x.  B ) ) )
18 mul4 9191 . . . . . . . . 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 664 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
2017, 19eqtrd 2436 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2120anidms 627 . . . . . 6  |-  ( B  e.  CC  ->  -u ( B  x.  B )  =  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) )
2221adantl 453 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2312, 22oveq12d 6058 . . . 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 2439 . . 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 6058 . 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 9030 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  x.  A
)  e.  CC )
2726anidms 627 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  A )  e.  CC )
2827adantr 452 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
29 mulcl 9030 . . . . 5  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  e.  CC )
304, 29sylan2 461 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  e.  CC )
3115negcld 9354 . . . . . 6  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3231anidms 627 . . . . 5  |-  ( B  e.  CC  ->  -u ( B  x.  B )  e.  CC )
3332adantl 453 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3428, 30, 33npncand 9391 . . 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 627 . . . 4  |-  ( B  e.  CC  ->  ( B  x.  B )  e.  CC )
36 subneg 9306 . . . 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 464 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B )
)  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3834, 37eqtrd 2436 . 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 2440 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 359    = wceq 1649    e. wcel 1721  (class class class)co 6040   CCcc 8944   1c1 8947   _ici 8948    + caddc 8949    x. cmul 8951    - cmin 9247   -ucneg 9248
This theorem is referenced by:  recex  9610
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-ltxr 9081  df-sub 9249  df-neg 9250
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