Theorem irrmulap 9860

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 9859. (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 4087	. . . 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 9855	. . . . 5 |- ( ( Q e. QQ /\ B e. QQ /\ B =/= 0 ) -> ( Q / B ) e. QQ )
7	3, 4, 5, 6	syl3anc 1271	. . . 4 |- ( ph -> ( Q / B ) e. QQ )
8	1, 2, 7	rspcdva 2912	. . 3 |- ( ph -> A # ( Q / B ) )
9		qcn 9846	. . . . 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 8191	. . . 4 |- ( ph -> A e. CC )
13		apsym 8769	. . . 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 9846	. . . . 5 |- ( Q e. QQ -> Q e. CC )
17	3, 16	syl 14	. . . 4 |- ( ph -> Q e. CC )
18		qcn 9846	. . . . 5 |- ( B e. QQ -> B e. CC )
19	4, 18	syl 14	. . . 4 |- ( ph -> B e. CC )
20		0z 9473	. . . . . . 7 |- 0 e. ZZ
21		zq 9838	. . . . . . 7 |- ( 0 e. ZZ -> 0 e. QQ )
22	20, 21	ax-mp 5	. . . . . 6 |- 0 e. QQ
23		qapne 9851	. . . . . 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 8976	. . 3 |- ( ph -> ( ( Q / B ) # A <-> ( B x. A ) # Q ) )
27	19, 12	mulcomd 8184	. . . 4 |- ( ph -> ( B x. A ) = ( A x. B ) )
28	27	breq1d 4093	. . 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 2200   =/= wne 2400   A.wral 2508   class class class wbr 4083  (class class class)co 6010   CCcc 8013   RRcr 8014   0cc0 8015   x. cmul 8020   # cap 8744   / cdiv 8835   ZZcz 9462   QQcq 9831

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4259  ax-pr 4294  ax-un 4525  ax-setind 4630  ax-cnex 8106  ax-resscn 8107  ax-1cn 8108  ax-1re 8109  ax-icn 8110  ax-addcl 8111  ax-addrcl 8112  ax-mulcl 8113  ax-mulrcl 8114  ax-addcom 8115  ax-mulcom 8116  ax-addass 8117  ax-mulass 8118  ax-distr 8119  ax-i2m1 8120  ax-0lt1 8121  ax-1rid 8122  ax-0id 8123  ax-rnegex 8124  ax-precex 8125  ax-cnre 8126  ax-pre-ltirr 8127  ax-pre-ltwlin 8128  ax-pre-lttrn 8129  ax-pre-apti 8130  ax-pre-ltadd 8131  ax-pre-mulgt0 8132  ax-pre-mulext 8133

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4385  df-po 4388  df-iso 4389  df-xp 4726  df-rel 4727  df-cnv 4728  df-co 4729  df-dm 4730  df-rn 4731  df-res 4732  df-ima 4733  df-iota 5281  df-fun 5323  df-fn 5324  df-f 5325  df-fv 5329  df-riota 5963  df-ov 6013  df-oprab 6014  df-mpo 6015  df-1st 6295  df-2nd 6296  df-pnf 8199  df-mnf 8200  df-xr 8201  df-ltxr 8202  df-le 8203  df-sub 8335  df-neg 8336  df-reap 8738  df-ap 8745  df-div 8836  df-inn 9127  df-n0 9386  df-z 9463  df-q 9832

This theorem is referenced by: (None)