Theorem irrmulap 9699

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 9698. (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 4029	. . . 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 9694	. . . . 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 2865	. . 3 |- ( ph -> A # ( Q / B ) )
9		qcn 9685	. . . . 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 8034	. . . 4 |- ( ph -> A e. CC )
13		apsym 8611	. . . 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 9685	. . . . 5 |- ( Q e. QQ -> Q e. CC )
17	3, 16	syl 14	. . . 4 |- ( ph -> Q e. CC )
18		qcn 9685	. . . . 5 |- ( B e. QQ -> B e. CC )
19	4, 18	syl 14	. . . 4 |- ( ph -> B e. CC )
20		0z 9314	. . . . . . 7 |- 0 e. ZZ
21		zq 9677	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
22	20, 21	ax-mp 5	. . . . . 6 |- 0 e. QQ
23		qapne 9690	. . . . . 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 8818	. . 3 |- ( ph -> ( ( Q / B ) # A <-> ( B x. A ) # Q ) )
27	19, 12	mulcomd 8027	. . . 4 |- ( ph -> ( B x. A ) = ( A x. B ) )
28	27	breq1d 4035	. . 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 2160   =/= wne 2360   A.wral 2468   class class class wbr 4025  (class class class)co 5906   CCcc 7856   RRcr 7857   0cc0 7858   x. cmul 7863   # cap 8586   / cdiv 8677   ZZcz 9303   QQcq 9670

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 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-cnex 7949  ax-resscn 7950  ax-1cn 7951  ax-1re 7952  ax-icn 7953  ax-addcl 7954  ax-addrcl 7955  ax-mulcl 7956  ax-mulrcl 7957  ax-addcom 7958  ax-mulcom 7959  ax-addass 7960  ax-mulass 7961  ax-distr 7962  ax-i2m1 7963  ax-0lt1 7964  ax-1rid 7965  ax-0id 7966  ax-rnegex 7967  ax-precex 7968  ax-cnre 7969  ax-pre-ltirr 7970  ax-pre-ltwlin 7971  ax-pre-lttrn 7972  ax-pre-apti 7973  ax-pre-ltadd 7974  ax-pre-mulgt0 7975  ax-pre-mulext 7976

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 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-id 4318  df-po 4321  df-iso 4322  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-fv 5250  df-riota 5861  df-ov 5909  df-oprab 5910  df-mpo 5911  df-1st 6180  df-2nd 6181  df-pnf 8042  df-mnf 8043  df-xr 8044  df-ltxr 8045  df-le 8046  df-sub 8178  df-neg 8179  df-reap 8580  df-ap 8587  df-div 8678  df-inn 8969  df-n0 9227  df-z 9304  df-q 9671

This theorem is referenced by: (None)