Theorem irrmul 9606

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 3130	. . 3 |- ( A e. ( RR \ QQ ) <-> ( A e. RR /\ -. A e. QQ ) )
2		qre 9584	. . . . . . 7 |- ( B e. QQ -> B e. RR )
3		remulcl 7902	. . . . . . 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 506	. . . . 5 |- ( ( ( A e. RR /\ -. A e. QQ ) /\ ( B e. QQ /\ B =/= 0 ) ) -> ( A x. B ) e. RR )
6		qdivcl 9602	. . . . . . . . . . . . 13 |- ( ( ( A x. B ) e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) e. QQ )
7	6	3expb 1199	. . . . . . . . . . . 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 7907	. . . . . . . . . . . . . 14 |- ( A e. RR -> A e. CC )
11	10	3ad2ant1 1013	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> A e. CC )
12		qcn 9593	. . . . . . . . . . . . . 14 |- ( B e. QQ -> B e. CC )
13	12	3ad2ant2 1014	. . . . . . . . . . . . 13 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B e. CC )
14		simp3 994	. . . . . . . . . . . . . 14 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> B =/= 0 )
15		0z 9223	. . . . . . . . . . . . . . . . 17 |- 0 e. ZZ
16		zq 9585	. . . . . . . . . . . . . . . . 17 |- ( 0 e. ZZ -> 0 e. QQ )
17	15, 16	ax-mp 5	. . . . . . . . . . . . . . . 16 |- 0 e. QQ
18		qapne 9598	. . . . . . . . . . . . . . . 16 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
19	17, 18	mpan2 423	. . . . . . . . . . . . . . 15 |- ( B e. QQ -> ( B # 0 <-> B =/= 0 ) )
20	19	3ad2ant2 1014	. . . . . . . . . . . . . 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 8713	. . . . . . . . . . . 12 |- ( ( A e. RR /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) / B ) = A )
23	22	3expb 1199	. . . . . . . . . . 11 |- ( ( A e. RR /\ ( B e. QQ /\ B =/= 0 ) ) -> ( ( A x. B ) / B ) = A )
24	23	eleq1d 2239	. . . . . . . . . 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 626	. . . . . . . 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 1194	. . 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 1269	. 2 |- ( ( A e. ( RR \ QQ ) /\ B e. QQ /\ B =/= 0 ) -> ( ( A x. B ) e. RR /\ -. ( A x. B ) e. QQ ) )
33		eldif 3130	. 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 973   = wceq 1348   e. wcel 2141   =/= wne 2340   \ cdif 3118   class class class wbr 3989  (class class class)co 5853   CCcc 7772   RRcr 7773   0cc0 7774   x. cmul 7779   # cap 8500   / cdiv 8589   ZZcz 9212   QQcq 9578

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-n0 9136  df-z 9213  df-q 9579

This theorem is referenced by:  2logb9irrALT  13686