Theorem irrmulap 9713

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 9712. (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 4033	. . . 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 9708	. . . . 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 2869	. . 3 |- ( ph -> A # ( Q / B ) )
9		qcn 9699	. . . . 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 8048	. . . 4 |- ( ph -> A e. CC )
13		apsym 8625	. . . 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 9699	. . . . 5 |- ( Q e. QQ -> Q e. CC )
17	3, 16	syl 14	. . . 4 |- ( ph -> Q e. CC )
18		qcn 9699	. . . . 5 |- ( B e. QQ -> B e. CC )
19	4, 18	syl 14	. . . 4 |- ( ph -> B e. CC )
20		0z 9328	. . . . . . 7 |- 0 e. ZZ
21		zq 9691	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
22	20, 21	ax-mp 5	. . . . . 6 |- 0 e. QQ
23		qapne 9704	. . . . . 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 8832	. . 3 |- ( ph -> ( ( Q / B ) # A <-> ( B x. A ) # Q ) )
27	19, 12	mulcomd 8041	. . . 4 |- ( ph -> ( B x. A ) = ( A x. B ) )
28	27	breq1d 4039	. . 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 4029  (class class class)co 5918   CCcc 7870   RRcr 7871   0cc0 7872   x. cmul 7877   # cap 8600   / cdiv 8691   ZZcz 9317   QQcq 9684

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 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-1cn 7965  ax-1re 7966  ax-icn 7967  ax-addcl 7968  ax-addrcl 7969  ax-mulcl 7970  ax-mulrcl 7971  ax-addcom 7972  ax-mulcom 7973  ax-addass 7974  ax-mulass 7975  ax-distr 7976  ax-i2m1 7977  ax-0lt1 7978  ax-1rid 7979  ax-0id 7980  ax-rnegex 7981  ax-precex 7982  ax-cnre 7983  ax-pre-ltirr 7984  ax-pre-ltwlin 7985  ax-pre-lttrn 7986  ax-pre-apti 7987  ax-pre-ltadd 7988  ax-pre-mulgt0 7989  ax-pre-mulext 7990

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 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-po 4327  df-iso 4328  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-fv 5262  df-riota 5873  df-ov 5921  df-oprab 5922  df-mpo 5923  df-1st 6193  df-2nd 6194  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059  df-le 8060  df-sub 8192  df-neg 8193  df-reap 8594  df-ap 8601  df-div 8692  df-inn 8983  df-n0 9241  df-z 9318  df-q 9685

This theorem is referenced by: (None)