Theorem nzadd 9499

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 3206	. . 3 |- ( A e. ( RR \ ZZ ) <-> ( A e. RR /\ -. A e. ZZ ) )
2		zre 9450	. . . . . 6 |- ( B e. ZZ -> B e. RR )
3		readdcl 8125	. . . . . 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 9487	. . . . . . . . . . 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 8132	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		zcn 9451	. . . . . . . . . . 11 |- ( B e. ZZ -> B e. CC )
11		pncan 8352	. . . . . . . . . . 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 2298	. . . . . . . . 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 634	. . . . . . 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 3206	. 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 1395   e. wcel 2200   \ cdif 3194  (class class class)co 6001   CCcc 7997   RRcr 7998   + caddc 8002   - cmin 8317   ZZcz 9446

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8090  ax-resscn 8091  ax-1cn 8092  ax-1re 8093  ax-icn 8094  ax-addcl 8095  ax-addrcl 8096  ax-mulcl 8097  ax-addcom 8099  ax-addass 8101  ax-distr 8103  ax-i2m1 8104  ax-0lt1 8105  ax-0id 8107  ax-rnegex 8108  ax-cnre 8110  ax-pre-ltirr 8111  ax-pre-ltwlin 8112  ax-pre-lttrn 8113  ax-pre-ltadd 8115

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-iota 5278  df-fun 5320  df-fv 5326  df-riota 5954  df-ov 6004  df-oprab 6005  df-mpo 6006  df-pnf 8183  df-mnf 8184  df-xr 8185  df-ltxr 8186  df-le 8187  df-sub 8319  df-neg 8320  df-inn 9111  df-n0 9370  df-z 9447

This theorem is referenced by: (None)