Theorem recexaplem2 8274

Description: Lemma for recexap 8275. (Contributed by Jim Kingdon, 20-Feb-2020.)

Assertion

Ref	Expression
recexaplem2	|- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) # 0 )

Proof of Theorem recexaplem2

Step	Hyp	Ref	Expression
1		ax-icn 7590	. . . . . . . . . . 11 |- _i e. CC
2	1	mul01i 8020	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
3	2	oveq2i 5717	. . . . . . . . 9 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
4		00id 7774	. . . . . . . . 9 |- ( 0 + 0 ) = 0
5	3, 4	eqtr2i 2121	. . . . . . . 8 |- 0 = ( 0 + ( _i x. 0 ) )
6	5	breq2i 3883	. . . . . . 7 |- ( ( A + ( _i x. B ) ) # 0 <-> ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) )
7		0re 7638	. . . . . . . 8 |- 0 e. RR
8		apreim 8231	. . . . . . . 8 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 e. RR /\ 0 e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A # 0 \/ B # 0 ) ) )
9	7, 7, 8	mpanr12 433	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A # 0 \/ B # 0 ) ) )
10	6, 9	syl5bb 191	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) # 0 <-> ( A # 0 \/ B # 0 ) ) )
11	10	pm5.32i 445	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) <-> ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) )
12		remulcl 7620	. . . . . . . . . 10 |- ( ( A e. RR /\ A e. RR ) -> ( A x. A ) e. RR )
13	12	anidms 392	. . . . . . . . 9 |- ( A e. RR -> ( A x. A ) e. RR )
14		remulcl 7620	. . . . . . . . . 10 |- ( ( B e. RR /\ B e. RR ) -> ( B x. B ) e. RR )
15	14	anidms 392	. . . . . . . . 9 |- ( B e. RR -> ( B x. B ) e. RR )
16	13, 15	anim12i 334	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
17	16	adantr 272	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
18		apsqgt0 8229	. . . . . . . . 9 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
19		msqge0 8244	. . . . . . . . 9 |- ( B e. RR -> 0 <_ ( B x. B ) )
20	18, 19	anim12i 334	. . . . . . . 8 |- ( ( ( A e. RR /\ A # 0 ) /\ B e. RR ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
21	20	an32s 538	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
22		addgtge0 8079	. . . . . . 7 |- ( ( ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) /\ ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
23	17, 21, 22	syl2anc 406	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
24	16	adantr 272	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
25		msqge0 8244	. . . . . . . . 9 |- ( A e. RR -> 0 <_ ( A x. A ) )
26		apsqgt0 8229	. . . . . . . . 9 |- ( ( B e. RR /\ B # 0 ) -> 0 < ( B x. B ) )
27	25, 26	anim12i 334	. . . . . . . 8 |- ( ( A e. RR /\ ( B e. RR /\ B # 0 ) ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
28	27	anassrs 395	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
29		addgegt0 8078	. . . . . . 7 |- ( ( ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) /\ ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
30	24, 28, 29	syl2anc 406	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
31	23, 30	jaodan 752	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
32	11, 31	sylbi 120	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
33	32	3impa 1144	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
34	33	olcd 694	. 2 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( ( A x. A ) + ( B x. B ) ) < 0 \/ 0 < ( ( A x. A ) + ( B x. B ) ) ) )
35		simp1 949	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> A e. RR )
36	35, 35	remulcld 7668	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( A x. A ) e. RR )
37		simp2 950	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> B e. RR )
38	37, 37	remulcld 7668	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( B x. B ) e. RR )
39	36, 38	readdcld 7667	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) e. RR )
40		reaplt 8216	. . 3 |- ( ( ( ( A x. A ) + ( B x. B ) ) e. RR /\ 0 e. RR ) -> ( ( ( A x. A ) + ( B x. B ) ) # 0 <-> ( ( ( A x. A ) + ( B x. B ) ) < 0 \/ 0 < ( ( A x. A ) + ( B x. B ) ) ) ) )
41	39, 7, 40	sylancl 407	. 2 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( ( A x. A ) + ( B x. B ) ) # 0 <-> ( ( ( A x. A ) + ( B x. B ) ) < 0 \/ 0 < ( ( A x. A ) + ( B x. B ) ) ) ) )
42	34, 41	mpbird 166	1 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) # 0 )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 670   /\ w3a 930   e. wcel 1448   class class class wbr 3875  (class class class)co 5706   RRcr 7499   0cc0 7500   _ici 7502   + caddc 7503   x. cmul 7505   < clt 7672   <_ cle 7673   # cap 8209

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-ltwlin 7608  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612  ax-pre-mulext 7613

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-po 4156  df-iso 4157  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210

This theorem is referenced by:  recexap  8275