Theorem recexaplem2 8891

Description: Lemma for recexap 8892. (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 8187	. . . . . . . . . . 11 |- _i e. CC
2	1	mul01i 8629	. . . . . . . . . 10 |- ( _i x. 0 ) = 0
3	2	oveq2i 6039	. . . . . . . . 9 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
4		00id 8379	. . . . . . . . 9 |- ( 0 + 0 ) = 0
5	3, 4	eqtr2i 2253	. . . . . . . 8 |- 0 = ( 0 + ( _i x. 0 ) )
6	5	breq2i 4101	. . . . . . 7 |- ( ( A + ( _i x. B ) ) # 0 <-> ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) )
7		0re 8239	. . . . . . . 8 |- 0 e. RR
8		apreim 8842	. . . . . . . 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 8220	. . . . . . . . . 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 8220	. . . . . . . . . 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 8840	. . . . . . . . 9 |- ( ( A e. RR /\ A # 0 ) -> 0 < ( A x. A ) )
19		msqge0 8855	. . . . . . . . 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 570	. . . . . . 7 |- ( ( ( A e. RR /\ B e. RR ) /\ A # 0 ) -> ( 0 < ( A x. A ) /\ 0 <_ ( B x. B ) ) )
22		addgtge0 8689	. . . . . . 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 8855	. . . . . . . . 9 |- ( A e. RR -> 0 <_ ( A x. A ) )
26		apsqgt0 8840	. . . . . . . . 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 8688	. . . . . . 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 805	. . . . 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 1221	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> 0 < ( ( A x. A ) + ( B x. B ) ) )
34	33	olcd 742	. 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 1024	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> A e. RR )
36	35, 35	remulcld 8269	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( A x. A ) e. RR )
37		simp2 1025	. . . . 5 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> B e. RR )
38	37, 37	remulcld 8269	. . . 4 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( B x. B ) e. RR )
39	36, 38	readdcld 8268	. . 3 |- ( ( A e. RR /\ B e. RR /\ ( A + ( _i x. B ) ) # 0 ) -> ( ( A x. A ) + ( B x. B ) ) e. RR )
40		reaplt 8827	. . 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 716   /\ w3a 1005   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   RRcr 8091   0cc0 8092   _ici 8094   + caddc 8095   x. cmul 8097   < clt 8273   <_ cle 8274   # cap 8820

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-po 4399  df-iso 4400  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-iota 5293  df-fun 5335  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821

This theorem is referenced by:  recexap  8892