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Theorem nzadd 9264
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 3130 . . 3  |-  ( A  e.  ( RR  \  ZZ )  <->  ( A  e.  RR  /\  -.  A  e.  ZZ ) )
2 zre 9216 . . . . . 6  |-  ( B  e.  ZZ  ->  B  e.  RR )
3 readdcl 7900 . . . . . 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 474 . . . 4  |-  ( ( ( A  e.  RR  /\ 
-.  A  e.  ZZ )  /\  B  e.  ZZ )  ->  ( A  +  B )  e.  RR )
6 zsubcl 9253 . . . . . . . . . . 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 7907 . . . . . . . . . . 11  |-  ( A  e.  RR  ->  A  e.  CC )
10 zcn 9217 . . . . . . . . . . 11  |-  ( B  e.  ZZ  ->  B  e.  CC )
11 pncan 8125 . . . . . . . . . . 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 2239 . . . . . . . . 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 626 . . . . . . 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 3130 . 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 1348    e. wcel 2141    \ cdif 3118  (class class class)co 5853   CCcc 7772   RRcr 7773    + caddc 7777    - cmin 8090   ZZcz 9212
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-addcom 7874  ax-addass 7876  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-0id 7882  ax-rnegex 7883  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-ltadd 7890
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-inn 8879  df-n0 9136  df-z 9213
This theorem is referenced by: (None)
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