Theorem recexaplem2 8549

Description: Lemma for recexap 8550. (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 7848	. . . . . . . . . . 11 |- _i e. CC
2	1	mul01i 8289	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
3	2	oveq2i 5853	. . . . . . . . 9 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
4		00id 8039	. . . . . . . . 9 |- ( 0 + 0 ) = 0
5	3, 4	eqtr2i 2187	. . . . . . . 8 |- 0 = ( 0 + ( _i x. 0 ) )
6	5	breq2i 3990	. . . . . . 7 |- ( ( A + ( _i x. B ) ) # 0 <-> ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) )
7		0re 7899	. . . . . . . 8 |- 0 e. RR
8		apreim 8501	. . . . . . . 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 436	. . . . . . 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 450	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) <-> ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) )
12		remulcl 7881	. . . . . . . . . 10 |- ( ( A e. RR /\ A e. RR ) -> ( A x. A ) e. RR )
13	12	anidms 395	. . . . . . . . 9 |- ( A e. RR -> ( A x. A ) e. RR )
14		remulcl 7881	. . . . . . . . . 10 |- ( ( B e. RR /\ B e. RR ) -> ( B x. B ) e. RR )
15	14	anidms 395	. . . . . . . . 9 |- ( B e. RR -> ( B x. B ) e. RR )
16	13, 15	anim12i 336	. . . . . . . 8 |- ( ( A e. RR /\ B e. RR ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
17	16	adantr 274	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
18		apsqgt0 8499	. . . . . . . . 9 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
19		msqge0 8514	. . . . . . . . 9 |- ( B e. RR -> 0 <_ ( B x. B ) )
20	18, 19	anim12i 336	. . . . . . . 8 |- ( ( ( A e. RR /\ A # 0 ) /\ B e. RR ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
21	20	an32s 558	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
22		addgtge0 8348	. . . . . . 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 409	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
24	16	adantr 274	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( ( A x. A ) e. RR /\ ( B x. B ) e. RR ) )
25		msqge0 8514	. . . . . . . . 9 |- ( A e. RR -> 0 <_ ( A x. A ) )
26		apsqgt0 8499	. . . . . . . . 9 |- ( ( B e. RR /\ B # 0 ) -> 0 < ( B x. B ) )
27	25, 26	anim12i 336	. . . . . . . 8 |- ( ( A e. RR /\ ( B e. RR /\ B # 0 ) ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
28	27	anassrs 398	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> ( 0 <_ ( A x. A ) /\ 0 < ( B x. B ) ) )
29		addgegt0 8347	. . . . . . 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 409	. . . . . 6 |- ( ( ( A e. RR /\ B e. RR ) /\ B # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
31	23, 30	jaodan 787	. . . . 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 1184	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
34	33	olcd 724	. 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 987	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> A e. RR )
36	35, 35	remulcld 7929	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( A x. A ) e. RR )
37		simp2 988	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> B e. RR )
38	37, 37	remulcld 7929	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( B x. B ) e. RR )
39	36, 38	readdcld 7928	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) e. RR )
40		reaplt 8486	. . 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 410	. 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 698   /\ w3a 968   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   RRcr 7752   0cc0 7753   _ici 7755   + caddc 7756   x. cmul 7758   < clt 7933   <_ cle 7934   # cap 8479

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480

This theorem is referenced by:  recexap  8550