Theorem irrmul 9803

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 9804. (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 3183	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9781	. . . . . . 7 |- ( B e. QQ -> B e. RR )
3		remulcl 8088	. . . . . . 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 9799	. . . . . . . . . . . . 13 |- ( ( ( A x. B ) e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) e. QQ )
7	6	3expb 1207	. . . . . . . . . . . 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 8093	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. CC )
11	10	3ad2ant1 1021	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> A e. CC )
12		qcn 9790	. . . . . . . . . . . . . 14 |- ( B e. QQ -> B e. CC )
13	12	3ad2ant2 1022	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B e. CC )
14		simp3 1002	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B =/= 0 )
15		0z 9418	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
16		zq 9782	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ZZ -> 0 e. QQ )
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . . 16 |- 0 e. QQ
18		qapne 9795	. . . . . . . . . . . . . . . 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 1022	. . . . . . . . . . . . . 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 8904	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) = A )
23	22	3expb 1207	. . . . . . . . . . 11 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) = A )
24	23	eleq1d 2276	. . . . . . . . . 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 632	. . . . . . . 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 1202	. . 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 1286	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
33		eldif 3183	. 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 981   = wceq 1373   e. wcel 2178   =/= wne 2378   \ cdif 3171   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   x. cmul 7965   # cap 8689   / cdiv 8780   ZZcz 9407   QQcq 9775

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-po 4361  df-iso 4362  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-q 9776

This theorem is referenced by:  2logb9irrALT  15561