Theorem mul0eqap 8567

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 7892	. . . . 5 |- ( ph -> 0 e. CC )
5		apcotr 8505	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ 0 e. CC ) -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
6	2, 3, 4, 5	syl3anc 1228	. . . 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 7892	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> 0 e. CC )
12	2, 3	mulcld 7919	. . . . . . . 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 8554	. . . . . . . 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 8522	. . . . . 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 7892	. . . . . . 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 8522	. . . . . 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 1343   e. wcel 2136   class class class wbr 3982  (class class class)co 5842   CCcc 7751   0cc0 7753   x. cmul 7758   # cap 8479

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-iota 5153  df-fun 5190  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480

This theorem is referenced by: (None)