Theorem recextlem1 8120

Description: Lemma for recexap 8122. (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 107	. . 3 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		ax-icn 7440	. . . . 5 |- _i e. CC
3		mulcl 7469	. . . . 5 |- ( ( _i e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
4	2, 3	mpan 415	. . . 4 |- ( B e. CC -> ( _i x. B ) e. CC )
5	4	adantl 271	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( _i x. B ) e. CC )
6		subcl 7681	. . . 4 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A - ( _i x. B ) ) e. CC )
7	4, 6	sylan2 280	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A - ( _i x. B ) ) e. CC )
8	1, 5, 7	adddird 7513	. 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 7892	. . 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 7892	. . . 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 7471	. . . . . 6 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
12	4, 11	sylan2 280	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) = ( ( _i x. B ) x. A ) )
13		ixi 8060	. . . . . . . . . 10 |- ( _i x. _i ) = -u 1
14	13	oveq1i 5662	. . . . . . . . 9 |- ( ( _i x. _i ) x. ( B x. B ) ) = ( -u 1 x. ( B x. B ) )
15		mulcl 7469	. . . . . . . . . 10 |- ( ( B e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
16	15	mulm1d 7888	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( -u 1 x. ( B x. B ) ) = -u ( B x. B ) )
17	14, 16	syl5req 2133	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. _i ) x. ( B x. B ) ) )
18		mul4 7614	. . . . . . . . 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 427	. . . . . . . 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 2120	. . . . . . 7 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
21	20	anidms 389	. . . . . 6 |- ( B e. CC -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
22	21	adantl 271	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. B ) x. ( _i x. B ) ) )
23	12, 22	oveq12d 5670	. . . 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 2123	. . 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 5670	. 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 7469	. . . . . 6 |- ( ( A e. CC /\ A e. CC ) -> ( A x. A ) e. CC )
27	26	anidms 389	. . . . 5 |- ( A e. CC -> ( A x. A ) e. CC )
28	27	adantr 270	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. A ) e. CC )
29		mulcl 7469	. . . . 5 |- ( ( A e. CC /\ ( _i x. B ) e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
30	4, 29	sylan2 280	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A x. ( _i x. B ) ) e. CC )
31	15	negcld 7780	. . . . . 6 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
32	31	anidms 389	. . . . 5 |- ( B e. CC -> -u ( B x. B ) e. CC )
33	32	adantl 271	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> -u ( B x. B ) e. CC )
34	28, 30, 33	npncand 7817	. . 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 389	. . . 4 |- ( B e. CC -> ( B x. B ) e. CC )
36		subneg 7731	. . . 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 283	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) - -u ( B x. B ) ) = ( ( A x. A ) + ( B x. B ) ) )
38	34, 37	eqtrd 2120	. 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 2124	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 102   = wceq 1289   e. wcel 1438  (class class class)co 5652   CCcc 7348   1c1 7351   _ici 7352   + caddc 7353   x. cmul 7355   - cmin 7653   -ucneg 7654

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-setind 4353  ax-resscn 7437  ax-1cn 7438  ax-icn 7440  ax-addcl 7441  ax-addrcl 7442  ax-mulcl 7443  ax-addcom 7445  ax-mulcom 7446  ax-addass 7447  ax-mulass 7448  ax-distr 7449  ax-i2m1 7450  ax-1rid 7452  ax-0id 7453  ax-rnegex 7454  ax-cnre 7456

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2841  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-iota 4980  df-fun 5017  df-fv 5023  df-riota 5608  df-ov 5655  df-oprab 5656  df-mpt2 5657  df-sub 7655  df-neg 7656

This theorem is referenced by:  recexap  8122