Theorem nzadd 9243

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 3125	. . 3 |- ( A e. ( RR \ ZZ ) <-> ( A e. RR /\ -. A e. ZZ ) )
2		zre 9195	. . . . . 6 |- ( B e. ZZ -> B e. RR )
3		readdcl 7879	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	2, 3	sylan2 284	. . . . 5 |- ( ( A e. RR /\ B e. ZZ ) -> ( A + B ) e. RR )
5	4	adantlr 469	. . . 4 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( A + B ) e. RR )
6		zsubcl 9232	. . . . . . . . . . 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 275	. . . . . . . . 9 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) e. ZZ -> ( ( A + B ) - B ) e. ZZ ) )
9		recn 7886	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		zcn 9196	. . . . . . . . . . 11 |- ( B e. ZZ -> B e. CC )
11		pncan 8104	. . . . . . . . . . 11 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
12	9, 10, 11	syl2an 287	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. ZZ ) -> ( ( A + B ) - B ) = A )
13	12	eleq1d 2235	. . . . . . . . 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 621	. . . . . . 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 304	. . 3 |- ( ( ( A e. RR /\ -. A e. ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
20	1, 19	sylanb 282	. 2 |- ( ( A e. ( RR \ ZZ ) /\ B e. ZZ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. ZZ ) )
21		eldif 3125	. 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 1343   e. wcel 2136   \ cdif 3113  (class class class)co 5842   CCcc 7751   RRcr 7752   + caddc 7756   - cmin 8069   ZZcz 9191

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192

This theorem is referenced by: (None)