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Theorem mulreap 11553
Description: A product with a real multiplier apart from zero is real iff the multiplicand is real. (Contributed by Jim Kingdon, 14-Jun-2020.)
Assertion
Ref Expression
mulreap |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( A e. RR <-> ( B x. A ) e. RR ) )
Proof of Theorem mulreap
StepHypRef Expression
1 rereb 11552 . . 3 |- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
213ad2ant1 1045 . 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( A e. RR <-> ( Re ` A ) = A ) )
3 recl 11542 . . . . 5 |- ( A e. CC -> ( Re ` A ) e. RR )
43recnd 8304 . . . 4 |- ( A e. CC -> ( Re ` A ) e. CC )
543ad2ant1 1045 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` A ) e. CC )
6 simp1 1024 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> A e. CC )
7 recn 8262 . . . . 5 |- ( B e. RR -> B e. CC )
87anim1i 340 . . . 4 |- ( ( B e. RR /\ B # 0 ) -> ( B e. CC /\ B # 0 ) )
983adant1 1042 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( B e. CC /\ B # 0 ) )
10 mulcanap 8941 . . 3 |- ( ( ( Re ` A ) e. CC /\ A e. CC /\ ( B e. CC /\ B # 0 ) ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( Re ` A ) = A ) )
115, 6, 9, 10syl3anc 1274 . 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( Re ` A ) = A ) )
127adantr 276 . . . . . . . . . 10 |- ( ( B e. RR /\ A e. CC ) -> B e. CC )
134adantl 277 . . . . . . . . . 10 |- ( ( B e. RR /\ A e. CC ) -> ( Re ` A ) e. CC )
14 ax-icn 8224 . . . . . . . . . . . 12 |- _i e. CC
15 imcl 11543 . . . . . . . . . . . . 13 |- ( A e. CC -> ( Im ` A ) e. RR )
1615recnd 8304 . . . . . . . . . . . 12 |- ( A e. CC -> ( Im ` A ) e. CC )
17 mulcl 8256 . . . . . . . . . . . 12 |- ( ( _i e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( Im ` A ) ) e. CC )
1814, 16, 17sylancr 414 . . . . . . . . . . 11 |- ( A e. CC -> ( _i x. ( Im ` A ) ) e. CC )
1918adantl 277 . . . . . . . . . 10 |- ( ( B e. RR /\ A e. CC ) -> ( _i x. ( Im ` A ) ) e. CC )
2012, 13, 19adddid 8300 . . . . . . . . 9 |- ( ( B e. RR /\ A e. CC ) -> ( B x. ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) = ( ( B x. ( Re ` A ) ) + ( B x. ( _i x. ( Im ` A ) ) ) ) )
21 replim 11548 . . . . . . . . . . 11 |- ( A e. CC -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
2221adantl 277 . . . . . . . . . 10 |- ( ( B e. RR /\ A e. CC ) -> A = ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) )
2322oveq2d 6068 . . . . . . . . 9 |- ( ( B e. RR /\ A e. CC ) -> ( B x. A ) = ( B x. ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) )
24 mul12 8404 . . . . . . . . . . . 12 |- ( ( _i e. CC /\ B e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( B x. ( Im ` A ) ) ) = ( B x. ( _i x. ( Im ` A ) ) ) )
2514, 24mp3an1 1361 . . . . . . . . . . 11 |- ( ( B e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( B x. ( Im ` A ) ) ) = ( B x. ( _i x. ( Im ` A ) ) ) )
267, 16, 25syl2an 289 . . . . . . . . . 10 |- ( ( B e. RR /\ A e. CC ) -> ( _i x. ( B x. ( Im ` A ) ) ) = ( B x. ( _i x. ( Im ` A ) ) ) )
2726oveq2d 6068 . . . . . . . . 9 |- ( ( B e. RR /\ A e. CC ) -> ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) = ( ( B x. ( Re ` A ) ) + ( B x. ( _i x. ( Im ` A ) ) ) ) )
2820, 23, 273eqtr4d 2277 . . . . . . . 8 |- ( ( B e. RR /\ A e. CC ) -> ( B x. A ) = ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) )
2928fveq2d 5676 . . . . . . 7 |- ( ( B e. RR /\ A e. CC ) -> ( Re ` ( B x. A ) ) = ( Re ` ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) ) )
30 remulcl 8257 . . . . . . . . 9 |- ( ( B e. RR /\ ( Re ` A ) e. RR ) -> ( B x. ( Re ` A ) ) e. RR )
313, 30sylan2 286 . . . . . . . 8 |- ( ( B e. RR /\ A e. CC ) -> ( B x. ( Re ` A ) ) e. RR )
32 remulcl 8257 . . . . . . . . 9 |- ( ( B e. RR /\ ( Im ` A ) e. RR ) -> ( B x. ( Im ` A ) ) e. RR )
3315, 32sylan2 286 . . . . . . . 8 |- ( ( B e. RR /\ A e. CC ) -> ( B x. ( Im ` A ) ) e. RR )
34 crre 11546 . . . . . . . 8 |- ( ( ( B x. ( Re ` A ) ) e. RR /\ ( B x. ( Im ` A ) ) e. RR ) -> ( Re ` ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) ) = ( B x. ( Re ` A ) ) )
3531, 33, 34syl2anc 411 . . . . . . 7 |- ( ( B e. RR /\ A e. CC ) -> ( Re ` ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) ) = ( B x. ( Re ` A ) ) )
3629, 35eqtr2d 2268 . . . . . 6 |- ( ( B e. RR /\ A e. CC ) -> ( B x. ( Re ` A ) ) = ( Re ` ( B x. A ) ) )
3736eqeq1d 2243 . . . . 5 |- ( ( B e. RR /\ A e. CC ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( Re ` ( B x. A ) ) = ( B x. A ) ) )
38 mulcl 8256 . . . . . . 7 |- ( ( B e. CC /\ A e. CC ) -> ( B x. A ) e. CC )
397, 38sylan 283 . . . . . 6 |- ( ( B e. RR /\ A e. CC ) -> ( B x. A ) e. CC )
40 rereb 11552 . . . . . 6 |- ( ( B x. A ) e. CC -> ( ( B x. A ) e. RR <-> ( Re ` ( B x. A ) ) = ( B x. A ) ) )
4139, 40syl 14 . . . . 5 |- ( ( B e. RR /\ A e. CC ) -> ( ( B x. A ) e. RR <-> ( Re ` ( B x. A ) ) = ( B x. A ) ) )
4237, 41bitr4d 191 . . . 4 |- ( ( B e. RR /\ A e. CC ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( B x. A ) e. RR ) )
4342ancoms 268 . . 3 |- ( ( A e. CC /\ B e. RR ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( B x. A ) e. RR ) )
44433adant3 1044 . 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( B x. A ) e. RR ) )
452, 11, 443bitr2d 216 1 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( A e. RR <-> ( B x. A ) e. RR ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   class class class wbr 4111   ` cfv 5354  (class class class)co 6052   CCcc 8127   RRcr 8128   0cc0 8129   _ici 8131   + caddc 8132   x. cmul 8134   # cap 8857   Recre 11529   Imcim 11530
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 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-mulrcl 8228  ax-addcom 8229  ax-mulcom 8230  ax-addass 8231  ax-mulass 8232  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-1rid 8236  ax-0id 8237  ax-rnegex 8238  ax-precex 8239  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245  ax-pre-mulgt0 8246  ax-pre-mulext 8247
This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-reap 8851  df-ap 8858  df-div 8949  df-2 9298  df-cj 11531  df-re 11532  df-im 11533
This theorem is referenced by: (None)
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