Theorem recextlem1 8759

Description: Lemma for recexap 8761. (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 8055	. . . . 5 |- _i e. CC
3		mulcl 8087	. . . . 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 8306	. . . 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 8133	. 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 8521	. . 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 8521	. . . 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 8089	. . . . . 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 8691	. . . . . . . . . 10 |- ( _i x. _i ) = -u 1
14	13	oveq1i 5977	. . . . . . . . 9 |- ( ( _i x. _i ) x. ( B x. B ) ) = ( -u 1 x. ( B x. B ) )
15		mulcl 8087	. . . . . . . . . 10 |- ( ( B e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
16	15	mulm1d 8517	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( -u 1 x. ( B x. B ) ) = -u ( B x. B ) )
17	14, 16	eqtr2id 2253	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. _i ) x. ( B x. B ) ) )
18		mul4 8239	. . . . . . . . 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 2240	. . . . . . 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 5985	. . . 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 2243	. . 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 5985	. 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 8087	. . . . . 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 8087	. . . . 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 8405	. . . . . 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 8442	. . 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 8356	. . . 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 2240	. 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 2244	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 1373   e. wcel 2178  (class class class)co 5967   CCcc 7958   1c1 7961   _ici 7962   + caddc 7963   x. cmul 7965   - cmin 8278   -ucneg 8279

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-setind 4603  ax-resscn 8052  ax-1cn 8053  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-cnre 8071

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-iota 5251  df-fun 5292  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-sub 8280  df-neg 8281

This theorem is referenced by:  recexap  8761