Theorem irrmul 9854

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). For a similar theorem with irrational in place of not rational, see irrmulap 9855. (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 3206	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9832	. . . . . . 7 |- ( B e. QQ -> B e. RR )
3		remulcl 8138	. . . . . . 7 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4	2, 3	sylan2 286	. . . . . 6 |- ( ( A e. RR /\ B e. QQ ) -> ( A x. B ) e. RR )
5	4	ad2ant2r 509	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
6		qdivcl 9850	. . . . . . . . . . . . 13 |- ( ( ( A x. B ) e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) e. QQ )
7	6	3expb 1228	. . . . . . . . . . . 12 |- ( ( ( A x. B ) e. QQ /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) e. QQ )
8	7	expcom 116	. . . . . . . . . . 11 |- ( ( B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. QQ -> ( ( A x. B ) / B ) e. QQ ) )
9	8	adantl 277	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) e. QQ -> ( ( A x. B ) / B ) e. QQ ) )
10		recn 8143	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. CC )
11	10	3ad2ant1 1042	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> A e. CC )
12		qcn 9841	. . . . . . . . . . . . . 14 |- ( B e. QQ -> B e. CC )
13	12	3ad2ant2 1043	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B e. CC )
14		simp3 1023	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B =/= 0 )
15		0z 9468	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
16		zq 9833	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ZZ -> 0 e. QQ )
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . . 16 |- 0 e. QQ
18		qapne 9846	. . . . . . . . . . . . . . . 16 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
19	17, 18	mpan2 425	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> ( B # 0 <-> B =/= 0 ) )
20	19	3ad2ant2 1043	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( B # 0 <-> B =/= 0 ) )
21	14, 20	mpbird 167	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B # 0 )
22	11, 13, 21	divcanap4d 8954	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) = A )
23	22	3expb 1228	. . . . . . . . . . 11 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) = A )
24	23	eleq1d 2298	. . . . . . . . . 10 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( ( A x. B ) / B ) e. QQ <-> A e. QQ ) )
25	9, 24	sylibd 149	. . . . . . . . 9 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) e. QQ -> A e. QQ ) )
26	25	con3d 634	. . . . . . . 8 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( -. A e. QQ -> -. ( A x. B ) e. QQ ) )
27	26	ex 115	. . . . . . 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 256	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> -. ( A x. B ) e. QQ )
30	5, 29	jca 306	. . . 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 1223	. . 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 1307	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
33		eldif 3206	. 2 |- ( ( A x. B ) e. ( RR \ QQ ) <-> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
34	32, 33	sylibr 134	1 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( A x. B ) e. ( RR \ QQ ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   = wceq 1395   e. wcel 2200   =/= wne 2400   \ cdif 3194   class class class wbr 4083  (class class class)co 6007   CCcc 8008   RRcr 8009   0cc0 8010   x. cmul 8015   # cap 8739   / cdiv 8830   ZZcz 9457   QQcq 9826

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-mulrcl 8109  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-0lt1 8116  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-precex 8120  ax-cnre 8121  ax-pre-ltirr 8122  ax-pre-ltwlin 8123  ax-pre-lttrn 8124  ax-pre-apti 8125  ax-pre-ltadd 8126  ax-pre-mulgt0 8127  ax-pre-mulext 8128

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-po 4387  df-iso 4388  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-pnf 8194  df-mnf 8195  df-xr 8196  df-ltxr 8197  df-le 8198  df-sub 8330  df-neg 8331  df-reap 8733  df-ap 8740  df-div 8831  df-inn 9122  df-n0 9381  df-z 9458  df-q 9827

This theorem is referenced by:  2logb9irrALT  15663