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Theorem recextlem1 9545
Description: Lemma for recex 9547. (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 443 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 ax-icn 8943 . . . . 5  |-  _i  e.  CC
3 mulcl 8968 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
42, 3mpan 651 . . . 4  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
54adantl 452 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
6 subcl 9198 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  -  ( _i  x.  B
) )  e.  CC )
74, 6sylan2 460 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
_i  x.  B )
)  e.  CC )
81, 5, 7adddird 9007 . 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 9382 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) ) )
105, 1, 5subdid 9382 . . . 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 8970 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  =  ( ( _i  x.  B
)  x.  A ) )
124, 11sylan2 460 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  x.  A ) )
13 ixi 9544 . . . . . . . . . 10  |-  ( _i  x.  _i )  = 
-u 1
1413oveq1i 5991 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( B  x.  B ) )  =  ( -u 1  x.  ( B  x.  B
) )
15 mulcl 8968 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1615mulm1d 9378 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( B  x.  B
) )  =  -u ( B  x.  B
) )
1714, 16syl5req 2411 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  _i )  x.  ( B  x.  B ) ) )
18 mul4 9128 . . . . . . . . 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 663 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
2017, 19eqtrd 2398 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2120anidms 626 . . . . . 6  |-  ( B  e.  CC  ->  -u ( B  x.  B )  =  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) )
2221adantl 452 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2312, 22oveq12d 5999 . . . 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 2401 . . 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 5999 . 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 8968 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  x.  A
)  e.  CC )
2726anidms 626 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  A )  e.  CC )
2827adantr 451 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
29 mulcl 8968 . . . . 5  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  e.  CC )
304, 29sylan2 460 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  e.  CC )
3115negcld 9291 . . . . . 6  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3231anidms 626 . . . . 5  |-  ( B  e.  CC  ->  -u ( B  x.  B )  e.  CC )
3332adantl 452 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3428, 30, 33npncand 9328 . . 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 626 . . . 4  |-  ( B  e.  CC  ->  ( B  x.  B )  e.  CC )
36 subneg 9243 . . . 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 463 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B )
)  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3834, 37eqtrd 2398 . 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 2402 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 358    = wceq 1647    e. wcel 1715  (class class class)co 5981   CCcc 8882   1c1 8885   _ici 8886    + caddc 8887    x. cmul 8889    - cmin 9184   -ucneg 9185
This theorem is referenced by:  recex  9547
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615  ax-resscn 8941  ax-1cn 8942  ax-icn 8943  ax-addcl 8944  ax-addrcl 8945  ax-mulcl 8946  ax-mulrcl 8947  ax-mulcom 8948  ax-addass 8949  ax-mulass 8950  ax-distr 8951  ax-i2m1 8952  ax-1ne0 8953  ax-1rid 8954  ax-rnegex 8955  ax-rrecex 8956  ax-cnre 8957  ax-pre-lttri 8958  ax-pre-lttrn 8959  ax-pre-ltadd 8960
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 936  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-po 4417  df-so 4418  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-riota 6446  df-er 6802  df-en 7007  df-dom 7008  df-sdom 7009  df-pnf 9016  df-mnf 9017  df-ltxr 9019  df-sub 9186  df-neg 9187
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