Theorem irrmul 9439

Description: The product of a real which is not rational with a nonzero rational is not rational. Note that by "not rational" we mean the negation of "is rational" (whereas "irrational" is often defined to mean apart from any rational number - given excluded middle these two definitions would be equivalent). (Contributed by NM, 7-Nov-2008.)

Assertion

Ref	Expression
irrmul	|- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( A x. B ) e. ( RR \ QQ ) )

Proof of Theorem irrmul

Step	Hyp	Ref	Expression
1		eldif 3080	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9417	. . . . . . 7 |- ( B e. QQ -> B e. RR )
3		remulcl 7748	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4	2, 3	sylan2 284	. . . . . 6 |- ( ( A e. RR /\ B e. QQ ) -> ( A x. B ) e. RR )
5	4	ad2ant2r 500	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
6		qdivcl 9435	. . . . . . . . . . . . 13 |- ( ( ( A x. B ) e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) e. QQ )
7	6	3expb 1182	. . . . . . . . . . . 12 |- ( ( ( A x. B ) e. QQ /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) e. QQ )
8	7	expcom 115	. . . . . . . . . . 11 |- ( ( B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. QQ -> ( ( A x. B ) / B ) e. QQ ) )
9	8	adantl 275	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) e. QQ -> ( ( A x. B ) / B ) e. QQ ) )
10		recn 7753	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. CC )
11	10	3ad2ant1 1002	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> A e. CC )
12		qcn 9426	. . . . . . . . . . . . . 14 |- ( B e. QQ -> B e. CC )
13	12	3ad2ant2 1003	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B e. CC )
14		simp3 983	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B =/= 0 )
15		0z 9065	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
16		zq 9418	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ZZ -> 0 e. QQ )
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . . 16 |- 0 e. QQ
18		qapne 9431	. . . . . . . . . . . . . . . 16 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
19	17, 18	mpan2 421	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> ( B # 0 <-> B =/= 0 ) )
20	19	3ad2ant2 1003	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( B # 0 <-> B =/= 0 ) )
21	14, 20	mpbird 166	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B # 0 )
22	11, 13, 21	divcanap4d 8556	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) = A )
23	22	3expb 1182	. . . . . . . . . . 11 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) = A )
24	23	eleq1d 2208	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( ( A x. B ) / B ) e. QQ <-> A e. QQ ) )
25	9, 24	sylibd 148	. . . . . . . . 9 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) e. QQ -> A e. QQ ) )
26	25	con3d 620	. . . . . . . 8 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( -. A e. QQ -> -. ( A x. B ) e. QQ ) )
27	26	ex 114	. . . . . . 7 |- ( A e. RR -> ( ( B e. QQ /\ B =/= 0 ) -> ( -. A e. QQ -> -. ( A x. B ) e. QQ ) ) )
28	27	com23 78	. . . . . 6 |- ( A e. RR -> ( -. A e. QQ -> ( ( B e. QQ /\ B =/= 0 ) -> -. ( A x. B ) e. QQ ) ) )
29	28	imp31 254	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> -. ( A x. B ) e. QQ )
30	5, 29	jca 304	. . . 4 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
31	30	3impb 1177	. . 3 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
32	1, 31	syl3an1b 1252	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
33		eldif 3080	. 2 |- ( ( A x. B ) e. ( RR \ QQ ) <-> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
34	32, 33	sylibr 133	1 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( A x. B ) e. ( RR \ QQ ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 962   = wceq 1331   e. wcel 1480   =/= wne 2308   \ cdif 3068   class class class wbr 3929  (class class class)co 5774   CCcc 7618   RRcr 7619   0cc0 7620   x. cmul 7625   # cap 8343   / cdiv 8432   ZZcz 9054   QQcq 9411

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1cn 7713  ax-1re 7714  ax-icn 7715  ax-addcl 7716  ax-addrcl 7717  ax-mulcl 7718  ax-mulrcl 7719  ax-addcom 7720  ax-mulcom 7721  ax-addass 7722  ax-mulass 7723  ax-distr 7724  ax-i2m1 7725  ax-0lt1 7726  ax-1rid 7727  ax-0id 7728  ax-rnegex 7729  ax-precex 7730  ax-cnre 7731  ax-pre-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734  ax-pre-apti 7735  ax-pre-ltadd 7736  ax-pre-mulgt0 7737  ax-pre-mulext 7738

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-sub 7935  df-neg 7936  df-reap 8337  df-ap 8344  df-div 8433  df-inn 8721  df-n0 8978  df-z 9055  df-q 9412

This theorem is referenced by: (None)