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Theorem recextlem1 8607
Description: Lemma for recexap 8609. (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 109 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  A  e.  CC )
2 ax-icn 7905 . . . . 5  |-  _i  e.  CC
3 mulcl 7937 . . . . 5  |-  ( ( _i  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
42, 3mpan 424 . . . 4  |-  ( B  e.  CC  ->  (
_i  x.  B )  e.  CC )
54adantl 277 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( _i  x.  B
)  e.  CC )
6 subcl 8155 . . . 4  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  -  ( _i  x.  B
) )  e.  CC )
74, 6sylan2 286 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  -  (
_i  x.  B )
)  e.  CC )
81, 5, 7adddird 7982 . 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 8370 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  ( A  -  ( _i  x.  B ) ) )  =  ( ( A  x.  A )  -  ( A  x.  (
_i  x.  B )
) ) )
105, 1, 5subdid 8370 . . . 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 7939 . . . . . 6  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  =  ( ( _i  x.  B
)  x.  A ) )
124, 11sylan2 286 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  =  ( ( _i  x.  B )  x.  A ) )
13 ixi 8539 . . . . . . . . . 10  |-  ( _i  x.  _i )  = 
-u 1
1413oveq1i 5884 . . . . . . . . 9  |-  ( ( _i  x.  _i )  x.  ( B  x.  B ) )  =  ( -u 1  x.  ( B  x.  B
) )
15 mulcl 7937 . . . . . . . . . 10  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( B  x.  B
)  e.  CC )
1615mulm1d 8366 . . . . . . . . 9  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( -u 1  x.  ( B  x.  B
) )  =  -u ( B  x.  B
) )
1714, 16eqtr2id 2223 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  _i )  x.  ( B  x.  B ) ) )
18 mul4 8088 . . . . . . . . 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 436 . . . . . . . 8  |-  ( ( B  e.  CC  /\  B  e.  CC )  ->  ( ( _i  x.  _i )  x.  ( B  x.  B )
)  =  ( ( _i  x.  B )  x.  ( _i  x.  B ) ) )
2017, 19eqtrd 2210 . . . . . . 7  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2120anidms 397 . . . . . 6  |-  ( B  e.  CC  ->  -u ( B  x.  B )  =  ( ( _i  x.  B )  x.  ( _i  x.  B
) ) )
2221adantl 277 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  =  ( ( _i  x.  B
)  x.  ( _i  x.  B ) ) )
2312, 22oveq12d 5892 . . . 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 2213 . . 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 5892 . 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 7937 . . . . . 6  |-  ( ( A  e.  CC  /\  A  e.  CC )  ->  ( A  x.  A
)  e.  CC )
2726anidms 397 . . . . 5  |-  ( A  e.  CC  ->  ( A  x.  A )  e.  CC )
2827adantr 276 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  A
)  e.  CC )
29 mulcl 7937 . . . . 5  |-  ( ( A  e.  CC  /\  ( _i  x.  B
)  e.  CC )  ->  ( A  x.  ( _i  x.  B
) )  e.  CC )
304, 29sylan2 286 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  (
_i  x.  B )
)  e.  CC )
3115negcld 8254 . . . . . 6  |-  ( ( B  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3231anidms 397 . . . . 5  |-  ( B  e.  CC  ->  -u ( B  x.  B )  e.  CC )
3332adantl 277 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  -> 
-u ( B  x.  B )  e.  CC )
3428, 30, 33npncand 8291 . . 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 397 . . . 4  |-  ( B  e.  CC  ->  ( B  x.  B )  e.  CC )
36 subneg 8205 . . . 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 289 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  x.  A )  -  -u ( B  x.  B )
)  =  ( ( A  x.  A )  +  ( B  x.  B ) ) )
3834, 37eqtrd 2210 . 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 2214 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 104    = wceq 1353    e. wcel 2148  (class class class)co 5874   CCcc 7808   1c1 7811   _ici 7812    + caddc 7813    x. cmul 7815    - cmin 8127   -ucneg 8128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-setind 4536  ax-resscn 7902  ax-1cn 7903  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-sub 8129  df-neg 8130
This theorem is referenced by:  recexap  8609
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