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Theorem mulreap 11607
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 11606 . . 3 |- ( A e. CC -> ( A e. RR <-> ( Re ` A ) = A ) )
213ad2ant1 1049 . 2 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( A e. RR <-> ( Re ` A ) = A ) )
3 recl 11596 . . . . 5 |- ( A e. CC -> ( Re ` A ) e. RR )
43recnd 8344 . . . 4 |- ( A e. CC -> ( Re ` A ) e. CC )
543ad2ant1 1049 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( Re ` A ) e. CC )
6 simp1 1028 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> A e. CC )
7 recn 8302 . . . . 5 |- ( B e. RR -> B e. CC )
87anim1i 340 . . . 4 |- ( ( B e. RR /\ B # 0 ) -> ( B e. CC /\ B # 0 ) )
983adant1 1046 . . 3 |- ( ( A e. CC /\ B e. RR /\ B # 0 ) -> ( B e. CC /\ B # 0 ) )
10 mulcanap 8983 . . 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 1278 . 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 8264 . . . . . . . . . . . 12 |- _i e. CC
15 imcl 11597 . . . . . . . . . . . . 13 |- ( A e. CC -> ( Im ` A ) e. RR )
1615recnd 8344 . . . . . . . . . . . 12 |- ( A e. CC -> ( Im ` A ) e. CC )
17 mulcl 8296 . . . . . . . . . . . 12 |- ( ( _i e. CC /\ ( Im ` A ) e. CC ) -> ( _i x. ( Im ` A ) ) e. CC )
1814, 16, 17sylancr 418 . . . . . . . . . . 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 8340 . . . . . . . . 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 11602 . . . . . . . . . . 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 6091 . . . . . . . . 9 |- ( ( B e. RR /\ A e. CC ) -> ( B x. A ) = ( B x. ( ( Re ` A ) + ( _i x. ( Im ` A ) ) ) ) )
24 mul12 8445 . . . . . . . . . . . 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 1365 . . . . . . . . . . 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 6091 . . . . . . . . 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 2281 . . . . . . . 8 |- ( ( B e. RR /\ A e. CC ) -> ( B x. A ) = ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) )
2928fveq2d 5694 . . . . . . 7 |- ( ( B e. RR /\ A e. CC ) -> ( Re ` ( B x. A ) ) = ( Re ` ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) ) )
30 remulcl 8297 . . . . . . . . 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 8297 . . . . . . . . 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 11600 . . . . . . . 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 415 . . . . . . 7 |- ( ( B e. RR /\ A e. CC ) -> ( Re ` ( ( B x. ( Re ` A ) ) + ( _i x. ( B x. ( Im ` A ) ) ) ) ) = ( B x. ( Re ` A ) ) )
3629, 35eqtr2d 2272 . . . . . 6 |- ( ( B e. RR /\ A e. CC ) -> ( B x. ( Re ` A ) ) = ( Re ` ( B x. A ) ) )
3736eqeq1d 2247 . . . . 5 |- ( ( B e. RR /\ A e. CC ) -> ( ( B x. ( Re ` A ) ) = ( B x. A ) <-> ( Re ` ( B x. A ) ) = ( B x. A ) ) )
38 mulcl 8296 . . . . . . 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 11606 . . . . . 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 1048 . 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   CCcc 8167   RRcr 8168   0cc0 8169   _ici 8171   + caddc 8172   x. cmul 8174   # cap 8899   Recre 11583   Imcim 11584
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287
This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-po 4436  df-iso 4437  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-2 9342  df-cj 11585  df-re 11586  df-im 11587
This theorem is referenced by: (None)
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