Theorem nzadd 9004

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

Step	Hyp	Ref	Expression
1		eldif 3044	. . 3 |- ( A e. ( RR \ ZZ ) <-> ( A e. RR /\ -. A e. ZZ ) )
2		zre 8956	. . . . . 6 |- ( B e. ZZ -> B e. RR )
3		readdcl 7664	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	2, 3	sylan2 282	. . . . 5 |- ( ( A e. RR /\ B e. ZZ ) -> ( A + B ) e. RR )
5	4	adantlr 466	. . . 4 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( A + B ) e. RR )
6		zsubcl 8993	. . . . . . . . . . 11 |- ( ( ( A + B ) e. ZZ /\ B e. ZZ ) -> ( ( A + B ) - B ) e. ZZ )
7	6	expcom 115	. . . . . . . . . 10 |- ( B e. ZZ -> ( ( A + B ) e. ZZ -> ( ( A + B ) - B ) e. ZZ ) )
8	7	adantl 273	. . . . . . . . 9 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) e. ZZ -> ( ( A + B ) - B ) e. ZZ ) )
9		recn 7671	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		zcn 8957	. . . . . . . . . . 11 |- ( B e. ZZ -> B e. CC )
11		pncan 7885	. . . . . . . . . . 11 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
12	9, 10, 11	syl2an 285	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) - B ) = A )
13	12	eleq1d 2181	. . . . . . . . 9 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( ( A + B ) - B ) e. ZZ <-> A e. ZZ ) )
14	8, 13	sylibd 148	. . . . . . . 8 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) e. ZZ -> A e. ZZ ) )
15	14	con3d 603	. . . . . . 7 |- ( ( A e. RR /\ B e. ZZ ) -> ( -. A e. ZZ -> -. ( A + B ) e. ZZ ) )
16	15	ex 114	. . . . . 6 |- ( A e. RR -> ( B e. ZZ -> ( -. A e. ZZ -> -. ( A + B ) e. ZZ ) ) )
17	16	com23 78	. . . . 5 |- ( A e. RR -> ( -. A e. ZZ -> ( B e. ZZ -> -. ( A + B ) e. ZZ ) ) )
18	17	imp31 254	. . . 4 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> -. ( A + B ) e. ZZ )
19	5, 18	jca 302	. . 3 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
20	1, 19	sylanb 280	. 2 |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
21		eldif 3044	. 2 |- ( ( A + B ) e. ( RR \ ZZ ) <-> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
22	20, 21	sylibr 133	1 |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( A + B ) e. ( RR \ ZZ ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   = wceq 1312   e. wcel 1461   \ cdif 3032  (class class class)co 5726   CCcc 7539   RRcr 7540   + caddc 7544   - cmin 7850   ZZcz 8952

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-addcom 7639  ax-addass 7641  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-0id 7647  ax-rnegex 7648  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-ltadd 7655

This theorem depends on definitions:  df-bi 116  df-3or 944  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-int 3736  df-br 3894  df-opab 3948  df-id 4173  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-inn 8625  df-n0 8876  df-z 8953

This theorem is referenced by: (None)