Theorem recexaplem2 8557

Description: Lemma for recexap 8558. (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 7856	. . . . . . . . . . 11 |- _i e. CC
2	1	mul01i 8297	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
3	2	oveq2i 5861	. . . . . . . . 9 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
4		00id 8047	. . . . . . . . 9 |- ( 0 + 0 ) = 0
5	3, 4	eqtr2i 2192	. . . . . . . 8 |- 0 = ( 0 + ( _i x. 0 ) )
6	5	breq2i 3995	. . . . . . 7 |- ( ( A + ( _i x. B ) ) # 0 <-> ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) )
7		0re 7907	. . . . . . . 8 |- 0 e. RR
8		apreim 8509	. . . . . . . 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 437	. . . . . . 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 451	. . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ ( A + ( _i x. B ) ) # 0 ) <-> ( ( A e. RR /\ B e. RR ) /\ ( A # 0 \/ B # 0 ) ) )
12		remulcl 7889	. . . . . . . . . 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 7889	. . . . . . . . . 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 8507	. . . . . . . . 9 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
19		msqge0 8522	. . . . . . . . 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 563	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
22		addgtge0 8356	. . . . . . 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 8522	. . . . . . . . 9 |- ( A e. RR -> 0 <_ ( A x. A ) )
26		apsqgt0 8507	. . . . . . . . 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 8355	. . . . . . 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 792	. . . . 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 1189	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
34	33	olcd 729	. 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 992	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> A e. RR )
36	35, 35	remulcld 7937	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( A x. A ) e. RR )
37		simp2 993	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> B e. RR )
38	37, 37	remulcld 7937	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( B x. B ) e. RR )
39	36, 38	readdcld 7936	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) e. RR )
40		reaplt 8494	. . 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 411	. 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 703   /\ w3a 973   e. wcel 2141   class class class wbr 3987  (class class class)co 5850   RRcr 7760   0cc0 7761   _ici 7763   + caddc 7764   x. cmul 7766   < clt 7941   <_ cle 7942   # cap 8487

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-mulrcl 7860  ax-addcom 7861  ax-mulcom 7862  ax-addass 7863  ax-mulass 7864  ax-distr 7865  ax-i2m1 7866  ax-0lt1 7867  ax-1rid 7868  ax-0id 7869  ax-rnegex 7870  ax-precex 7871  ax-cnre 7872  ax-pre-ltirr 7873  ax-pre-ltwlin 7874  ax-pre-lttrn 7875  ax-pre-apti 7876  ax-pre-ltadd 7877  ax-pre-mulgt0 7878  ax-pre-mulext 7879

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-po 4279  df-iso 4280  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-xr 7945  df-ltxr 7946  df-le 7947  df-sub 8079  df-neg 8080  df-reap 8481  df-ap 8488

This theorem is referenced by:  recexap  8558