Theorem crap0 9137

Description: The real representation of complex numbers is apart from zero iff one of its terms is apart from zero. (Contributed by Jim Kingdon, 5-Mar-2020.)

Assertion

Ref	Expression
crap0	|- ( ( A e. RR /\ B e. RR ) -> ( ( A # 0 \/ B # 0 ) <-> ( A + ( _i x. B ) ) # 0 ) )

Proof of Theorem crap0

Step	Hyp	Ref	Expression
1		0re 8178	. . 3 |- 0 e. RR
2		apreim 8782	. . 3 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 e. RR /\ 0 e. RR ) ) -> ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A # 0 \/ B # 0 ) ) )
3	1, 1, 2	mpanr12 439	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A # 0 \/ B # 0 ) ) )
4		ax-icn 8126	. . . . . 6 |- _i e. CC
5	4	mul01i 8569	. . . . 5 |- ( _i x. 0 ) = 0
6	5	oveq2i 6028	. . . 4 |- ( 0 + ( _i x. 0 ) ) = ( 0 + 0 )
7		00id 8319	. . . 4 |- ( 0 + 0 ) = 0
8	6, 7	eqtri 2252	. . 3 |- ( 0 + ( _i x. 0 ) ) = 0
9	8	breq2i 4096	. 2 |- ( ( A + ( _i x. B ) ) # ( 0 + ( _i x. 0 ) ) <-> ( A + ( _i x. B ) ) # 0 )
10	3, 9	bitr3di 195	1 |- ( ( A e. RR /\ B e. RR ) -> ( ( A # 0 \/ B # 0 ) <-> ( A + ( _i x. B ) ) # 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   e. wcel 2202   class class class wbr 4088  (class class class)co 6017   RRcr 8030   0cc0 8031   _ici 8033   + caddc 8034   x. cmul 8036   # cap 8760

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-iota 5286  df-fun 5328  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-pnf 8215  df-mnf 8216  df-ltxr 8218  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761

This theorem is referenced by:  abs00ap  11622