Theorem recexaplem2 8611

Description: Lemma for recexap 8612. (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 7908	. . . . . . . . . . 11 |- _i e. CC
2	1	mul01i 8350	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
3	2	oveq2i 5888	. . . . . . . . 9 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
4		00id 8100	. . . . . . . . 9 |- ( 0 + 0 ) = 0
5	3, 4	eqtr2i 2199	. . . . . . . 8 |- 0 = ( 0 + ( _i x. 0 ) )
6	5	breq2i 4013	. . . . . . 7 |- ( ( A + ( _i x. B ) ) # 0 <-> ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) )
7		0re 7959	. . . . . . . 8 |- 0 e. RR
8		apreim 8562	. . . . . . . 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 439	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A # 0 \/ B # 0 ) ) )
10	6, 9	bitrid 192	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) # 0 <-> ( A # 0 \/ B # 0 ) ) )
11	10	pm5.32i 454	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) <-> ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) )
12		remulcl 7941	. . . . . . . . . 10 |- ( ( A e. RR /\ A e. RR ) -> ( A x. A ) e. RR )
13	12	anidms 397	. . . . . . . . 9 |- ( A e. RR -> ( A x. A ) e. RR )
14		remulcl 7941	. . . . . . . . . 10 |- ( ( B e. RR /\ B e. RR ) -> ( B x. B ) e. RR )
15	14	anidms 397	. . . . . . . . 9 |- ( B e. RR -> ( B x. B ) e. RR )
16	13, 15	anim12i 338	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
17	16	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
18		apsqgt0 8560	. . . . . . . . 9 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
19		msqge0 8575	. . . . . . . . 9 |- ( B e. RR -> 0 <_ ( B x. B ) )
20	18, 19	anim12i 338	. . . . . . . 8 |- ( ( ( A e. RR /\ A # 0 ) /\ B e. RR ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
21	20	an32s 568	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
22		addgtge0 8409	. . . . . . 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 411	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
24	16	adantr 276	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
25		msqge0 8575	. . . . . . . . 9 |- ( A e. RR -> 0 <_ ( A x. A ) )
26		apsqgt0 8560	. . . . . . . . 9 |- ( ( B e. RR /\ B # 0 ) -> 0 < ( B x. B ) )
27	25, 26	anim12i 338	. . . . . . . 8 |- ( ( A e. RR /\ ( B e. RR /\ B # 0 ) ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
28	27	anassrs 400	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
29		addgegt0 8408	. . . . . . 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 411	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
31	23, 30	jaodan 797	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
32	11, 31	sylbi 121	. . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
33	32	3impa 1194	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
34	33	olcd 734	. 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 997	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> A e. RR )
36	35, 35	remulcld 7990	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( A x. A ) e. RR )
37		simp2 998	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> B e. RR )
38	37, 37	remulcld 7990	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( B x. B ) e. RR )
39	36, 38	readdcld 7989	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) e. RR )
40		reaplt 8547	. . 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 413	. 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 167	1 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) # 0 )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708   /\ w3a 978   e. wcel 2148   class class class wbr 4005  (class class class)co 5877   RRcr 7812   0cc0 7813   _ici 7815   + caddc 7816   x. cmul 7818   < clt 7994   <_ cle 7995   # cap 8540

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541

This theorem is referenced by:  recexap  8612