Theorem irradd 10062

Description: The sum of a real which is not rational and a rational number is not rational. (Contributed by NM, 7-Nov-2008.)

Assertion

Ref	Expression
irradd	|- ( ( A e. ( RR \ QQ ) /\ B e. QQ ) -> ( A + B ) e. ( RR \ QQ ) )

Proof of Theorem irradd

Step	Hyp	Ref	Expression
1		eldif 3229	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 10041	. . . . . 6 |- ( B e. QQ -> B e. RR )
3		readdcl 8311	. . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR )
4	2, 3	sylan2 286	. . . . 5 |- ( ( A e. RR /\ B e. QQ ) -> ( A + B ) e. RR )
5	4	adantlr 481	. . . 4 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ B e. QQ ) -> ( A + B ) e. RR )
6		qsubcl 10054	. . . . . . . . . . 11 |- ( ( ( A + B ) e. QQ /\ B e. QQ ) -> ( ( A + B ) - B ) e. QQ )
7	6	expcom 116	. . . . . . . . . 10 |- ( B e. QQ -> ( ( A + B ) e. QQ -> ( ( A + B ) - B ) e. QQ ) )
8	7	adantl 277	. . . . . . . . 9 |- ( ( A e. RR /\ B e. QQ ) -> ( ( A + B ) e. QQ -> ( ( A + B ) - B ) e. QQ ) )
9		recn 8318	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		qcn 10050	. . . . . . . . . . 11 |- ( B e. QQ -> B e. CC )
11		pncan 8539	. . . . . . . . . . 11 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) - B ) = A )
12	9, 10, 11	syl2an 289	. . . . . . . . . 10 |- ( ( A e. RR /\ B e. QQ ) -> ( ( A + B ) - B ) = A )
13	12	eleq1d 2307	. . . . . . . . 9 |- ( ( A e. RR /\ B e. QQ ) -> ( ( ( A + B ) - B ) e. QQ <-> A e. QQ ) )
14	8, 13	sylibd 149	. . . . . . . 8 |- ( ( A e. RR /\ B e. QQ ) -> ( ( A + B ) e. QQ -> A e. QQ ) )
15	14	con3d 640	. . . . . . 7 |- ( ( A e. RR /\ B e. QQ ) -> ( -. A e. QQ -> -. ( A + B ) e. QQ ) )
16	15	ex 115	. . . . . 6 |- ( A e. RR -> ( B e. QQ -> ( -. A e. QQ -> -. ( A + B ) e. QQ ) ) )
17	16	com23 78	. . . . 5 |- ( A e. RR -> ( -. A e. QQ -> ( B e. QQ -> -. ( A + B ) e. QQ ) ) )
18	17	imp31 256	. . . 4 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ B e. QQ ) -> -. ( A + B ) e. QQ )
19	5, 18	jca 306	. . 3 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ B e. QQ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. QQ ) )
20	1, 19	sylanb 284	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ ) -> ( ( A + B ) e. RR /\ -. ( A + B ) e. QQ ) )
21		eldif 3229	. 2 |- ( ( A + B ) e. ( RR \ QQ ) <-> ( ( A + B ) e. RR /\ -. ( A + B ) e. QQ ) )
22	20, 21	sylibr 134	1 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ ) -> ( A + B ) e. ( RR \ QQ ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   \ cdif 3217  (class class class)co 6088   CCcc 8183   RRcr 8184   + caddc 8188   - cmin 8504   QQcq 10035

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-po 4443  df-iso 4444  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-n0 9575  df-z 9656  df-q 10036

This theorem is used by: (None)