Theorem recextlem1 8412

Description: Lemma for recexap 8414. (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

Step	Hyp	Ref	Expression
1		simpl 108	. . 3 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		ax-icn 7715	. . . . 5 |- _i e. CC
3		mulcl 7747	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
4	2, 3	mpan 420	. . . 4 |- ( B e. CC -> ( _i x. B ) e. CC )
5	4	adantl 275	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
6		subcl 7961	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A - ( _i x. B ) ) e. CC )
7	4, 6	sylan2 284	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - ( _i x. B ) ) e. CC )
8	1, 5, 7	adddird 7791	. 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 ) ) ) ) )
9	1, 1, 5	subdid 8176	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( A - ( _i x. B ) ) ) = ( ( A x. A ) - ( A x. ( _i x. B ) ) ) )
10	5, 1, 5	subdid 8176	. . . 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 7749	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
12	4, 11	sylan2 284	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
13		ixi 8345	. . . . . . . . . 10 |- ( _i x. _i ) = -u 1
14	13	oveq1i 5784	. . . . . . . . 9 |- ( ( _i x. _i ) x. ( B x. B ) ) = ( -u 1 x. ( B x. B ) )
15		mulcl 7747	. . . . . . . . . 10 |- ( ( B e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
16	15	mulm1d 8172	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( -u 1 x. ( B x. B ) ) = -u ( B x. B ) )
17	14, 16	syl5req 2185	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. _i ) x. ( B x. B ) ) )
18		mul4 7894	. . . . . . . . 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 ) ) )
19	2, 2, 18	mpanl12 432	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> ( ( _i x. _i ) x. ( B x. B ) ) = ( ( _i x. B ) x. ( _i x. B ) ) )
20	17, 19	eqtrd 2172	. . . . . . 7 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
21	20	anidms 394	. . . . . 6 |- ( B e. CC -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
22	21	adantl 275	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
23	12, 22	oveq12d 5792	. . . 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 ) ) ) )
24	10, 23	eqtr4d 2175	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( _i x. B ) x. ( A - ( _i x. B ) ) ) = ( ( A x. ( _i x. B ) ) - -u ( B x. B ) ) )
25	9, 24	oveq12d 5792	. 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 7747	. . . . . 6 |- ( ( A e. CC /\ A e. CC ) -> ( A x. A ) e. CC )
27	26	anidms 394	. . . . 5 |- ( A e. CC -> ( A x. A ) e. CC )
28	27	adantr 274	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
29		mulcl 7747	. . . . 5 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
30	4, 29	sylan2 284	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
31	15	negcld 8060	. . . . . 6 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
32	31	anidms 394	. . . . 5 |- ( B e. CC -> -u ( B x. B ) e. CC )
33	32	adantl 275	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
34	28, 30, 33	npncand 8097	. . 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 ) ) )
35	15	anidms 394	. . . 4 |- ( B e. CC -> ( B x. B ) e. CC )
36		subneg 8011	. . . 4 |- ( ( ( A x. A ) e. CC /\ ( B x. B ) e. CC ) -> ( ( A x. A ) - -u ( B x. B ) ) = ( ( A x. A ) + ( B x. B ) ) )
37	27, 35, 36	syl2an 287	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) - -u ( B x. B ) ) = ( ( A x. A ) + ( B x. B ) ) )
38	34, 37	eqtrd 2172	. 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 ) ) )
39	8, 25, 38	3eqtrd 2176	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( _i x. B ) ) x. ( A - ( _i x. B ) ) ) = ( ( A x. A ) + ( B x. B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1331   e. wcel 1480  (class class class)co 5774   CCcc 7618   1c1 7621   _ici 7622   + caddc 7623   x. cmul 7625   - cmin 7933   -ucneg 7934

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-setind 4452  ax-resscn 7712  ax-1cn 7713  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-cnre 7731

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-sub 7935  df-neg 7936

This theorem is referenced by:  recexap  8414