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Theorem nzadd 9074
Description: The sum of a real number not being an integer and an integer is not an integer. Note that "not being an integer" in this case means "the negation of is an integer" rather than "is apart from any integer" (given excluded middle, those two would be equivalent). (Contributed by AV, 19-Jul-2021.)
Assertion
Ref Expression
nzadd  |-  ( ( A  e.  ( RR 
\  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ( RR 
\  ZZ ) )

Proof of Theorem nzadd
StepHypRef Expression
1 eldif 3050 . . 3  |-  ( A  e.  ( RR  \  ZZ )  <->  ( A  e.  RR  /\  -.  A  e.  ZZ ) )
2 zre 9026 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  RR )
3 readdcl 7714 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
42, 3sylan2 284 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  RR )
54adantlr 468 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B )  e.  RR )
6 zsubcl 9063 . . . . . . . . . . 11  |-  ( ( ( A  +  B
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  +  B )  -  B
)  e.  ZZ )
76expcom 115 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  (
( A  +  B
)  e.  ZZ  ->  ( ( A  +  B
)  -  B )  e.  ZZ ) )
87adantl 275 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  ZZ  ->  ( ( A  +  B )  -  B
)  e.  ZZ ) )
9 recn 7721 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
10 zcn 9027 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  CC )
11 pncan 7936 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  B
)  =  A )
129, 10, 11syl2an 287 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  -  B
)  =  A )
1312eleq1d 2186 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( ( A  +  B )  -  B )  e.  ZZ  <->  A  e.  ZZ ) )
148, 13sylibd 148 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  ZZ  ->  A  e.  ZZ ) )
1514con3d 605 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( -.  A  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) )
1615ex 114 . . . . . 6  |-  ( A  e.  RR  ->  ( B  e.  ZZ  ->  ( -.  A  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) ) )
1716com23 78 . . . . 5  |-  ( A  e.  RR  ->  ( -.  A  e.  ZZ  ->  ( B  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) ) )
1817imp31 254 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  -.  ( A  +  B )  e.  ZZ )
195, 18jca 304 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  RR  /\  -.  ( A  +  B )  e.  ZZ ) )
201, 19sylanb 282 . 2  |-  ( ( A  e.  ( RR 
\  ZZ )  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  RR  /\ 
-.  ( A  +  B )  e.  ZZ ) )
21 eldif 3050 . 2  |-  ( ( A  +  B )  e.  ( RR  \  ZZ )  <->  ( ( A  +  B )  e.  RR  /\  -.  ( A  +  B )  e.  ZZ ) )
2220, 21sylibr 133 1  |-  ( ( A  e.  ( RR 
\  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  ( RR 
\  ZZ ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    \ cdif 3038  (class class class)co 5742   CCcc 7586   RRcr 7587    + caddc 7591    - cmin 7901   ZZcz 9022
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-inn 8689  df-n0 8946  df-z 9023
This theorem is referenced by: (None)
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