Theorem mul0eqap 8544

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 7871	. . . . 5 |- ( ph -> 0 e. CC )
5		apcotr 8482	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ 0 e. CC ) -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
6	2, 3, 4, 5	syl3anc 1220	. . . 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 7871	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> 0 e. CC )
12	2, 3	mulcld 7898	. . . . . . . 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 8531	. . . . . . . 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 8499	. . . . . 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 7871	. . . . . . 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 8499	. . . . . 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 1335   e. wcel 2128   class class class wbr 3965  (class class class)co 5824   CCcc 7730   0cc0 7732   x. cmul 7737   # cap 8456

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-1cn 7825  ax-1re 7826  ax-icn 7827  ax-addcl 7828  ax-addrcl 7829  ax-mulcl 7830  ax-mulrcl 7831  ax-addcom 7832  ax-mulcom 7833  ax-addass 7834  ax-mulass 7835  ax-distr 7836  ax-i2m1 7837  ax-0lt1 7838  ax-1rid 7839  ax-0id 7840  ax-rnegex 7841  ax-precex 7842  ax-cnre 7843  ax-pre-ltirr 7844  ax-pre-ltwlin 7845  ax-pre-lttrn 7846  ax-pre-apti 7847  ax-pre-ltadd 7848  ax-pre-mulgt0 7849  ax-pre-mulext 7850

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-id 4253  df-po 4256  df-iso 4257  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-iota 5135  df-fun 5172  df-fv 5178  df-riota 5780  df-ov 5827  df-oprab 5828  df-mpo 5829  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917  df-le 7918  df-sub 8048  df-neg 8049  df-reap 8450  df-ap 8457

This theorem is referenced by: (None)