Theorem mul0eqap 8714

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 8036	. . . . 5 |- ( ph -> 0 e. CC )
5		apcotr 8651	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ 0 e. CC ) -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
6	2, 3, 4, 5	syl3anc 1249	. . . 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 276	. . . . . 6 |- ( ( ph /\ A # 0 ) -> ( A x. B ) = 0 )
10	3	adantr 276	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> B e. CC )
11		0cnd 8036	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> 0 e. CC )
12	2, 3	mulcld 8064	. . . . . . . 8 |- ( ph -> ( A x. B ) e. CC )
13	12	adantr 276	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> ( A x. B ) e. CC )
14		ibar 301	. . . . . . . 8 |- ( A # 0 -> ( B # 0 <-> ( A # 0 /\ B # 0 ) ) )
15	2, 3	mulap0bd 8701	. . . . . . . 8 |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
16	14, 15	sylan9bbr 463	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> ( B # 0 <-> ( A x. B ) # 0 ) )
17	10, 11, 13, 11, 16	apcon4bid 8668	. . . . . 6 |- ( ( ph /\ A # 0 ) -> ( B = 0 <-> ( A x. B ) = 0 ) )
18	9, 17	mpbird 167	. . . . 5 |- ( ( ph /\ A # 0 ) -> B = 0 )
19	18	ex 115	. . . 4 |- ( ph -> ( A # 0 -> B = 0 ) )
20	8	adantr 276	. . . . . 6 |- ( ( ph /\ B # 0 ) -> ( A x. B ) = 0 )
21	2	adantr 276	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> A e. CC )
22		0cnd 8036	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> 0 e. CC )
23	12	adantr 276	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> ( A x. B ) e. CC )
24		iba 300	. . . . . . . 8 |- ( B # 0 -> ( A # 0 <-> ( A # 0 /\ B # 0 ) ) )
25	24, 15	sylan9bbr 463	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> ( A # 0 <-> ( A x. B ) # 0 ) )
26	21, 22, 23, 22, 25	apcon4bid 8668	. . . . . 6 |- ( ( ph /\ B # 0 ) -> ( A = 0 <-> ( A x. B ) = 0 ) )
27	20, 26	mpbird 167	. . . . 5 |- ( ( ph /\ B # 0 ) -> A = 0 )
28	27	ex 115	. . . 4 |- ( ph -> ( B # 0 -> A = 0 ) )
29	19, 28	orim12d 787	. . 3 |- ( ph -> ( ( A # 0 \/ B # 0 ) -> ( B = 0 \/ A = 0 ) ) )
30	7, 29	mpd 13	. 2 |- ( ph -> ( B = 0 \/ A = 0 ) )
31	30	orcomd 730	1 |- ( ph -> ( A = 0 \/ B = 0 ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   e. wcel 2167   class class class wbr 4034  (class class class)co 5925   CCcc 7894   0cc0 7896   x. cmul 7901   # cap 8625

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-br 4035  df-opab 4096  df-id 4329  df-po 4332  df-iso 4333  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-iota 5220  df-fun 5261  df-fv 5267  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626

This theorem is referenced by: (None)