Theorem recextlem1 8726

Description: Lemma for recexap 8728. (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 8022	. . . . 5 |- _i e. CC
3		mulcl 8054	. . . . 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 8273	. . . 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 8100	. 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 8488	. . 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 8488	. . . 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 8056	. . . . . 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 8658	. . . . . . . . . 10 |- ( _i x. _i ) = -u 1
14	13	oveq1i 5956	. . . . . . . . 9 |- ( ( _i x. _i ) x. ( B x. B ) ) = ( -u 1 x. ( B x. B ) )
15		mulcl 8054	. . . . . . . . . 10 |- ( ( B e. CC /\ B e. CC ) -> ( B x. B ) e. CC )
16	15	mulm1d 8484	. . . . . . . . 9 |- ( ( B e. CC /\ B e. CC ) -> ( -u 1 x. ( B x. B ) ) = -u ( B x. B ) )
17	14, 16	eqtr2id 2251	. . . . . . . 8 |- ( ( B e. CC /\ B e. CC ) -> -u ( B x. B ) = ( ( _i x. _i ) x. ( B x. B ) ) )
18		mul4 8206	. . . . . . . . 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 2238	. . . . . . 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 5964	. . . 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 2241	. . 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 5964	. 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 8054	. . . . . 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 8054	. . . . 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 8372	. . . . . 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 8409	. . 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 8323	. . . 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 2238	. 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 2242	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 2176  (class class class)co 5946   CCcc 7925   1c1 7928   _ici 7929   + caddc 7930   x. cmul 7932   - cmin 8245   -ucneg 8246

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254  ax-setind 4586  ax-resscn 8019  ax-1cn 8020  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-cnre 8038

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4046  df-opab 4107  df-id 4341  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-iota 5233  df-fun 5274  df-fv 5280  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-sub 8247  df-neg 8248

This theorem is referenced by:  recexap  8728