Theorem nzadd 9303

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 3138	. . 3 |- ( A e. ( RR \ ZZ ) <-> ( A e. RR /\ -. A e. ZZ ) )
2		zre 9255	. . . . . 6 |- ( B e. ZZ -> B e. RR )
3		readdcl 7936	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	2, 3	sylan2 286	. . . . 5 |- ( ( A e. RR /\ B e. ZZ ) -> ( A + B ) e. RR )
5	4	adantlr 477	. . . 4 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( A + B ) e. RR )
6		zsubcl 9292	. . . . . . . . . . 11 |- ( ( ( A + B ) e. ZZ /\ B e. ZZ ) -> ( ( A + B ) - B ) e. ZZ )
7	6	expcom 116	. . . . . . . . . 10 |- ( B e. ZZ -> ( ( A + B ) e. ZZ -> ( ( A + B ) - B ) e. ZZ ) )
8	7	adantl 277	. . . . . . . . 9 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) e. ZZ -> ( ( A + B ) - B ) e. ZZ ) )
9		recn 7943	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		zcn 9256	. . . . . . . . . . 11 |- ( B e. ZZ -> B e. CC )
11		pncan 8161	. . . . . . . . . . 11 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
12	9, 10, 11	syl2an 289	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) - B ) = A )
13	12	eleq1d 2246	. . . . . . . . 9 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( ( A + B ) - B ) e. ZZ <-> A e. ZZ ) )
14	8, 13	sylibd 149	. . . . . . . 8 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) e. ZZ -> A e. ZZ ) )
15	14	con3d 631	. . . . . . 7 |- ( ( A e. RR /\ B e. ZZ ) -> ( -. A e. ZZ -> -. ( A + B ) e. ZZ ) )
16	15	ex 115	. . . . . 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 256	. . . 4 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> -. ( A + B ) e. ZZ )
19	5, 18	jca 306	. . 3 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
20	1, 19	sylanb 284	. 2 |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
21		eldif 3138	. 2 |- ( ( A + B ) e. ( RR \ ZZ ) <-> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
22	20, 21	sylibr 134	1 |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( A + B ) e. ( RR \ ZZ ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   \ cdif 3126  (class class class)co 5874   CCcc 7808   RRcr 7809   + caddc 7813   - cmin 8126   ZZcz 9251

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-addcom 7910  ax-addass 7912  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-0id 7918  ax-rnegex 7919  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-ltadd 7926

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-br 4004  df-opab 4065  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-iota 5178  df-fun 5218  df-fv 5224  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-inn 8918  df-n0 9175  df-z 9252

This theorem is referenced by: (None)