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Theorem nzadd 8772
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 3006 . . 3  |-  ( A  e.  ( RR  \  ZZ )  <->  ( A  e.  RR  /\  -.  A  e.  ZZ ) )
2 zre 8724 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  RR )
3 readdcl 7447 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B
)  e.  RR )
42, 3sylan2 280 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( A  +  B
)  e.  RR )
54adantlr 461 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B )  e.  RR )
6 zsubcl 8761 . . . . . . . . . . 11  |-  ( ( ( A  +  B
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A  +  B )  -  B
)  e.  ZZ )
76expcom 114 . . . . . . . . . 10  |-  ( B  e.  ZZ  ->  (
( A  +  B
)  e.  ZZ  ->  ( ( A  +  B
)  -  B )  e.  ZZ ) )
87adantl 271 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  ZZ  ->  ( ( A  +  B )  -  B
)  e.  ZZ ) )
9 recn 7454 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
10 zcn 8725 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  CC )
11 pncan 7667 . . . . . . . . . . 11  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  -  B
)  =  A )
129, 10, 11syl2an 283 . . . . . . . . . 10  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  -  B
)  =  A )
1312eleq1d 2156 . . . . . . . . 9  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( ( A  +  B )  -  B )  e.  ZZ  <->  A  e.  ZZ ) )
148, 13sylibd 147 . . . . . . . 8  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  ZZ  ->  A  e.  ZZ ) )
1514con3d 596 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  ZZ )  ->  ( -.  A  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) )
1615ex 113 . . . . . 6  |-  ( A  e.  RR  ->  ( B  e.  ZZ  ->  ( -.  A  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) ) )
1716com23 77 . . . . 5  |-  ( A  e.  RR  ->  ( -.  A  e.  ZZ  ->  ( B  e.  ZZ  ->  -.  ( A  +  B )  e.  ZZ ) ) )
1817imp31 252 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  -.  ( A  +  B )  e.  ZZ )
195, 18jca 300 . . 3  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  RR  /\  -.  ( A  +  B )  e.  ZZ ) )
201, 19sylanb 278 . 2  |-  ( ( A  e.  ( RR 
\  ZZ )  /\  B  e.  ZZ )  ->  ( ( A  +  B )  e.  RR  /\ 
-.  ( A  +  B )  e.  ZZ ) )
21 eldif 3006 . 2  |-  ( ( A  +  B )  e.  ( RR  \  ZZ )  <->  ( ( A  +  B )  e.  RR  /\  -.  ( A  +  B )  e.  ZZ ) )
2220, 21sylibr 132 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 102    = wceq 1289    e. wcel 1438    \ cdif 2994  (class class class)co 5634   CCcc 7327   RRcr 7328    + caddc 7332    - cmin 7632   ZZcz 8720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-addcom 7424  ax-addass 7426  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-0id 7432  ax-rnegex 7433  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-ltadd 7440
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-int 3684  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-inn 8395  df-n0 8644  df-z 8721
This theorem is referenced by: (None)
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