Theorem irradd 9879

Description: The sum of an irrational number and a rational number is irrational. (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 3209	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9858	. . . . . 6 |- ( B e. QQ -> B e. RR )
3		readdcl 8157	. . . . . 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 477	. . . 4 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ B e. QQ ) -> ( A + B ) e. RR )
6		qsubcl 9871	. . . . . . . . . . 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 8164	. . . . . . . . . . 11 |- ( A e. RR -> A e. CC )
10		qcn 9867	. . . . . . . . . . 11 |- ( B e. QQ -> B e. CC )
11		pncan 8384	. . . . . . . . . . 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 2300	. . . . . . . . 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 636	. . . . . . 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 3209	. 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 ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   \ cdif 3197  (class class class)co 6017   CCcc 8029   RRcr 8030   + caddc 8034   - cmin 8349   QQcq 9852

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-po 4393  df-iso 4394  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-n0 9402  df-z 9479  df-q 9853

This theorem is referenced by: (None)