Theorem irrmulap 9716

Description: The product of an irrational with a nonzero rational is irrational. By irrational we mean apart from any rational number. For a similar theorem with not rational in place of irrational, see irrmul 9715. (Contributed by Jim Kingdon, 25-Aug-2025.)

Hypotheses

Ref	Expression
irrmulap.a	|- ( ph -> A e. RR )
irrmulap.aq	|- ( ph -> A. q e. QQ A # q )
irrmulap.b	|- ( ph -> B e. QQ )
irrmulap.b0	|- ( ph -> B =/= 0 )
irrmulap.q	|- ( ph -> Q e. QQ )

Assertion

Ref	Expression
irrmulap	|- ( ph -> ( A x. B ) # Q )

Distinct variable groups:   A, q   B, q   Q, q

Allowed substitution hint:   ph( q)

Proof of Theorem irrmulap

Step	Hyp	Ref	Expression
1		breq2 4034	. . . 4 |- ( q = ( Q / B ) -> ( A # q <-> A # ( Q / B ) ) )
2		irrmulap.aq	. . . 4 |- ( ph -> A. q e. QQ A # q )
3		irrmulap.q	. . . . 5 |- ( ph -> Q e. QQ )
4		irrmulap.b	. . . . 5 |- ( ph -> B e. QQ )
5		irrmulap.b0	. . . . 5 |- ( ph -> B =/= 0 )
6		qdivcl 9711	. . . . 5 |- ( ( Q e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( Q / B ) e. QQ )
7	3, 4, 5, 6	syl3anc 1249	. . . 4 |- ( ph -> ( Q / B ) e. QQ )
8	1, 2, 7	rspcdva 2870	. . 3 |- ( ph -> A # ( Q / B ) )
9		qcn 9702	. . . . 5 |- ( ( Q / B ) e. QQ -> ( Q / B ) e. CC )
10	7, 9	syl 14	. . . 4 |- ( ph -> ( Q / B ) e. CC )
11		irrmulap.a	. . . . 5 |- ( ph -> A e. RR )
12	11	recnd 8050	. . . 4 |- ( ph -> A e. CC )
13		apsym 8627	. . . 4 |- ( ( ( Q / B ) e. CC /\ A e. CC ) -> ( ( Q / B ) # A <-> A # ( Q / B ) ) )
14	10, 12, 13	syl2anc 411	. . 3 |- ( ph -> ( ( Q / B ) # A <-> A # ( Q / B ) ) )
15	8, 14	mpbird 167	. 2 |- ( ph -> ( Q / B ) # A )
16		qcn 9702	. . . . 5 |- ( Q e. QQ -> Q e. CC )
17	3, 16	syl 14	. . . 4 |- ( ph -> Q e. CC )
18		qcn 9702	. . . . 5 |- ( B e. QQ -> B e. CC )
19	4, 18	syl 14	. . . 4 |- ( ph -> B e. CC )
20		0z 9331	. . . . . . 7 |- 0 e. ZZ
21		zq 9694	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
22	20, 21	ax-mp 5	. . . . . 6 |- 0 e. QQ
23		qapne 9707	. . . . . 6 |- ( ( B e. QQ /\ 0 e. QQ ) -> ( B # 0 <-> B =/= 0 ) )
24	4, 22, 23	sylancl 413	. . . . 5 |- ( ph -> ( B # 0 <-> B =/= 0 ) )
25	5, 24	mpbird 167	. . . 4 |- ( ph -> B # 0 )
26	17, 19, 12, 25	apdivmuld 8834	. . 3 |- ( ph -> ( ( Q / B ) # A <-> ( B x. A ) # Q ) )
27	19, 12	mulcomd 8043	. . . 4 |- ( ph -> ( B x. A ) = ( A x. B ) )
28	27	breq1d 4040	. . 3 |- ( ph -> ( ( B x. A ) # Q <-> ( A x. B ) # Q ) )
29	26, 28	bitrd 188	. 2 |- ( ph -> ( ( Q / B ) # A <-> ( A x. B ) # Q ) )
30	15, 29	mpbid 147	1 |- ( ph -> ( A x. B ) # Q )

Syntax hints:   -> wi 4   <-> wb 105   e. wcel 2164   =/= wne 2364   A.wral 2472   class class class wbr 4030  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   x. cmul 7879   # cap 8602   / cdiv 8693   ZZcz 9320   QQcq 9687

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-po 4328  df-iso 4329  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-q 9688

This theorem is referenced by: (None)