Theorem irrmul 9585

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 3125	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9563	. . . . . . 7 |- ( B e. QQ -> B e. RR )
3		remulcl 7881	. . . . . . 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 501	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
6		qdivcl 9581	. . . . . . . . . . . . 13 |- ( ( ( A x. B ) e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) e. QQ )
7	6	3expb 1194	. . . . . . . . . . . 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 7886	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. CC )
11	10	3ad2ant1 1008	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> A e. CC )
12		qcn 9572	. . . . . . . . . . . . . 14 |- ( B e. QQ -> B e. CC )
13	12	3ad2ant2 1009	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B e. CC )
14		simp3 989	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B =/= 0 )
15		0z 9202	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
16		zq 9564	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ZZ -> 0 e. QQ )
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . . 16 |- 0 e. QQ
18		qapne 9577	. . . . . . . . . . . . . . . 16 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
19	17, 18	mpan2 422	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> ( B # 0 <-> B =/= 0 ) )
20	19	3ad2ant2 1009	. . . . . . . . . . . . . 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 8692	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) = A )
23	22	3expb 1194	. . . . . . . . . . 11 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) = A )
24	23	eleq1d 2235	. . . . . . . . . 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 621	. . . . . . . 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 1189	. . 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 1264	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
33		eldif 3125	. 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 968   = wceq 1343   e. wcel 2136   =/= wne 2336   \ cdif 3113   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   x. cmul 7758   # cap 8479   / cdiv 8568   ZZcz 9191   QQcq 9557

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-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

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-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  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-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-n0 9115  df-z 9192  df-q 9558

This theorem is referenced by:  2logb9irrALT  13532