Theorem recextlem1 8695

Description: Lemma for recexap 8697. (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 109	. . 3 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		ax-icn 7991	. . . . 5 |- _i e. CC
3		mulcl 8023	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
4	2, 3	mpan 424	. . . 4 |- ( B e. CC -> ( _i x. B ) e. CC )
5	4	adantl 277	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
6		subcl 8242	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A - ( _i x. B ) ) e. CC )
7	4, 6	sylan2 286	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - ( _i x. B ) ) e. CC )
8	1, 5, 7	adddird 8069	. 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 8457	. . 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 8457	. . . 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 8025	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
12	4, 11	sylan2 286	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
13		ixi 8627	. . . . . . . . . 10 |- ( _i x. _i ) = -u 1
14	13	oveq1i 5935	. . . . . . . . 9 |- ( ( _i x. _i ) x. ( B x. B ) ) = ( -u 1 x. ( B x. B ) )
15		mulcl 8023	. . . . . . . . . 10 |- ( ( B e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
16	15	mulm1d 8453	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( -u 1 x. ( B x. B ) ) = -u ( B x. B ) )
17	14, 16	eqtr2id 2242	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. _i ) x. ( B x. B ) ) )
18		mul4 8175	. . . . . . . . 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 436	. . . . . . . 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 2229	. . . . . . 7 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
21	20	anidms 397	. . . . . 6 |- ( B e. CC -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
22	21	adantl 277	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
23	12, 22	oveq12d 5943	. . . 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 2232	. . 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 5943	. 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 8023	. . . . . 6 |- ( ( A e. CC /\ A e. CC ) -> ( A x. A ) e. CC )
27	26	anidms 397	. . . . 5 |- ( A e. CC -> ( A x. A ) e. CC )
28	27	adantr 276	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
29		mulcl 8023	. . . . 5 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
30	4, 29	sylan2 286	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
31	15	negcld 8341	. . . . . 6 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
32	31	anidms 397	. . . . 5 |- ( B e. CC -> -u ( B x. B ) e. CC )
33	32	adantl 277	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
34	28, 30, 33	npncand 8378	. . 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 397	. . . 4 |- ( B e. CC -> ( B x. B ) e. CC )
36		subneg 8292	. . . 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 289	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) - -u ( B x. B ) ) = ( ( A x. A ) + ( B x. B ) ) )
38	34, 37	eqtrd 2229	. 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 2233	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 104   = wceq 1364   e. wcel 2167  (class class class)co 5925   CCcc 7894   1c1 7897   _ici 7898   + caddc 7899   x. cmul 7901   - cmin 8214   -ucneg 8215

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-setind 4574  ax-resscn 7988  ax-1cn 7989  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-cnre 8007

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-sub 8216  df-neg 8217

This theorem is referenced by:  recexap  8697