Theorem irrmulap 9986

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 9985. (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 4115	. . . 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 9981	. . . . 5 |- ( ( Q e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( Q / B ) e. QQ )
7	3, 4, 5, 6	syl3anc 1274	. . . 4 |- ( ph -> ( Q / B ) e. QQ )
8	1, 2, 7	rspcdva 2928	. . 3 |- ( ph -> A # ( Q / B ) )
9		qcn 9972	. . . . 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 8307	. . . 4 |- ( ph -> A e. CC )
13		apsym 8885	. . . 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 9972	. . . . 5 |- ( Q e. QQ -> Q e. CC )
17	3, 16	syl 14	. . . 4 |- ( ph -> Q e. CC )
18		qcn 9972	. . . . 5 |- ( B e. QQ -> B e. CC )
19	4, 18	syl 14	. . . 4 |- ( ph -> B e. CC )
20		0z 9593	. . . . . . 7 |- 0 e. ZZ
21		zq 9964	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
22	20, 21	ax-mp 5	. . . . . 6 |- 0 e. QQ
23		qapne 9977	. . . . . 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 9092	. . 3 |- ( ph -> ( ( Q / B ) # A <-> ( B x. A ) # Q ) )
27	19, 12	mulcomd 8300	. . . 4 |- ( ph -> ( B x. A ) = ( A x. B ) )
28	27	breq1d 4121	. . 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 2205   =/= wne 2414   A.wral 2522   class class class wbr 4111  (class class class)co 6052   CCcc 8130   RRcr 8131   0cc0 8132   x. cmul 8137   # cap 8860   / cdiv 8951   ZZcz 9582   QQcq 9957

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-1cn 8225  ax-1re 8226  ax-icn 8227  ax-addcl 8228  ax-addrcl 8229  ax-mulcl 8230  ax-mulrcl 8231  ax-addcom 8232  ax-mulcom 8233  ax-addass 8234  ax-mulass 8235  ax-distr 8236  ax-i2m1 8237  ax-0lt1 8238  ax-1rid 8239  ax-0id 8240  ax-rnegex 8241  ax-precex 8242  ax-cnre 8243  ax-pre-ltirr 8244  ax-pre-ltwlin 8245  ax-pre-lttrn 8246  ax-pre-apti 8247  ax-pre-ltadd 8248  ax-pre-mulgt0 8249  ax-pre-mulext 8250

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-id 4416  df-po 4419  df-iso 4420  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318  df-le 8319  df-sub 8451  df-neg 8452  df-reap 8854  df-ap 8861  df-div 8952  df-inn 9243  df-n0 9502  df-z 9583  df-q 9958

This theorem is referenced by: (None)