Theorem mul0eqap 8455

Description: If two numbers are apart from each other and their product is zero, one of them must be zero. (Contributed by Jim Kingdon, 31-Jul-2023.)

Hypotheses

Ref	Expression
mul0eqap.a	|- ( ph -> A e. CC )
mul0eqap.b	|- ( ph -> B e. CC )
mul0eqap.ab	|- ( ph -> A # B )
mul0eqap.0	|- ( ph -> ( A x. B ) = 0 )

Assertion

Ref	Expression
mul0eqap	|- ( ph -> ( A = 0 \/ B = 0 ) )

Proof of Theorem mul0eqap

Step	Hyp	Ref	Expression
1		mul0eqap.ab	. . . 4 |- ( ph -> A # B )
2		mul0eqap.a	. . . . 5 |- ( ph -> A e. CC )
3		mul0eqap.b	. . . . 5 |- ( ph -> B e. CC )
4		0cnd 7783	. . . . 5 |- ( ph -> 0 e. CC )
5		apcotr 8393	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ 0 e. CC ) -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
6	2, 3, 4, 5	syl3anc 1217	. . . 4 |- ( ph -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
7	1, 6	mpd 13	. . 3 |- ( ph -> ( A # 0 \/ B # 0 ) )
8		mul0eqap.0	. . . . . . 7 |- ( ph -> ( A x. B ) = 0 )
9	8	adantr 274	. . . . . 6 |- ( ( ph /\ A # 0 ) -> ( A x. B ) = 0 )
10	3	adantr 274	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> B e. CC )
11		0cnd 7783	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> 0 e. CC )
12	2, 3	mulcld 7810	. . . . . . . 8 |- ( ph -> ( A x. B ) e. CC )
13	12	adantr 274	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> ( A x. B ) e. CC )
14		ibar 299	. . . . . . . 8 |- ( A # 0 -> ( B # 0 <-> ( A # 0 /\ B # 0 ) ) )
15	2, 3	mulap0bd 8442	. . . . . . . 8 |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
16	14, 15	sylan9bbr 459	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> ( B # 0 <-> ( A x. B ) # 0 ) )
17	10, 11, 13, 11, 16	apcon4bid 8410	. . . . . 6 |- ( ( ph /\ A # 0 ) -> ( B = 0 <-> ( A x. B ) = 0 ) )
18	9, 17	mpbird 166	. . . . 5 |- ( ( ph /\ A # 0 ) -> B = 0 )
19	18	ex 114	. . . 4 |- ( ph -> ( A # 0 -> B = 0 ) )
20	8	adantr 274	. . . . . 6 |- ( ( ph /\ B # 0 ) -> ( A x. B ) = 0 )
21	2	adantr 274	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> A e. CC )
22		0cnd 7783	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> 0 e. CC )
23	12	adantr 274	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> ( A x. B ) e. CC )
24		iba 298	. . . . . . . 8 |- ( B # 0 -> ( A # 0 <-> ( A # 0 /\ B # 0 ) ) )
25	24, 15	sylan9bbr 459	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> ( A # 0 <-> ( A x. B ) # 0 ) )
26	21, 22, 23, 22, 25	apcon4bid 8410	. . . . . 6 |- ( ( ph /\ B # 0 ) -> ( A = 0 <-> ( A x. B ) = 0 ) )
27	20, 26	mpbird 166	. . . . 5 |- ( ( ph /\ B # 0 ) -> A = 0 )
28	27	ex 114	. . . 4 |- ( ph -> ( B # 0 -> A = 0 ) )
29	19, 28	orim12d 776	. . 3 |- ( ph -> ( ( A # 0 \/ B # 0 ) -> ( B = 0 \/ A = 0 ) ) )
30	7, 29	mpd 13	. 2 |- ( ph -> ( B = 0 \/ A = 0 ) )
31	30	orcomd 719	1 |- ( ph -> ( A = 0 \/ B = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1332   e. wcel 1481   class class class wbr 3937  (class class class)co 5782   CCcc 7642   0cc0 7644   x. cmul 7649   # cap 8367

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-id 4223  df-po 4226  df-iso 4227  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-iota 5096  df-fun 5133  df-fv 5139  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368

This theorem is referenced by: (None)