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Theorem recextlem1 10078
Description: Lemma for recex 10080. (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 457 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 ax-icn 9453 . . . . 5  |-  _i  e.  CC
3 mulcl 9478 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
42, 3mpan 670 . . . 4  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
54adantl 466 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
6 subcl 9721 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  -  ( _i  x.  B
) )  e.  CC )
74, 6sylan2 474 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
_i  x.  B )
)  e.  CC )
81, 5, 7adddird 9523 . 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 9912 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) ) )
105, 1, 5subdid 9912 . . . 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 9480 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  =  ( ( _i  x.  B
)  x.  A ) )
124, 11sylan2 474 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  x.  A ) )
13 ixi 10077 . . . . . . . . . 10  |-  ( _i  x.  _i )  = 
-u 1
1413oveq1i 6211 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( B  x.  B ) )  =  ( -u 1  x.  ( B  x.  B
) )
15 mulcl 9478 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1615mulm1d 9908 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( B  x.  B
) )  =  -u ( B  x.  B
) )
1714, 16syl5req 2508 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  _i )  x.  ( B  x.  B ) ) )
18 mul4 9650 . . . . . . . . 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 682 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
2017, 19eqtrd 2495 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2120anidms 645 . . . . . 6  |-  ( B  e.  CC  ->  -u ( B  x.  B )  =  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) )
2221adantl 466 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2312, 22oveq12d 6219 . . . 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 2498 . . 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 6219 . 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 9478 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  x.  A
)  e.  CC )
2726anidms 645 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  A )  e.  CC )
2827adantr 465 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
29 mulcl 9478 . . . . 5  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  e.  CC )
304, 29sylan2 474 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  e.  CC )
3115negcld 9818 . . . . . 6  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3231anidms 645 . . . . 5  |-  ( B  e.  CC  ->  -u ( B  x.  B )  e.  CC )
3332adantl 466 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3428, 30, 33npncand 9855 . . 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 645 . . . 4  |-  ( B  e.  CC  ->  ( B  x.  B )  e.  CC )
36 subneg 9770 . . . 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 477 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B )
)  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3834, 37eqtrd 2495 . 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 2499 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758  (class class class)co 6201   CCcc 9392   1c1 9395   _ici 9396    + caddc 9397    x. cmul 9399    - cmin 9707   -ucneg 9708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-mulcom 9458  ax-addass 9459  ax-mulass 9460  ax-distr 9461  ax-i2m1 9462  ax-1ne0 9463  ax-1rid 9464  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469  ax-pre-ltadd 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-ltxr 9535  df-sub 9709  df-neg 9710
This theorem is referenced by:  recex  10080
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