Theorem mul0eqap 8588

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 7913	. . . . 5 |- ( ph -> 0 e. CC )
5		apcotr 8526	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ 0 e. CC ) -> ( A # B -> ( A # 0 \/ B # 0 ) ) )
6	2, 3, 4, 5	syl3anc 1233	. . . 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 7913	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> 0 e. CC )
12	2, 3	mulcld 7940	. . . . . . . 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 8575	. . . . . . . 8 |- ( ph -> ( ( A # 0 /\ B # 0 ) <-> ( A x. B ) # 0 ) )
16	14, 15	sylan9bbr 460	. . . . . . 7 |- ( ( ph /\ A # 0 ) -> ( B # 0 <-> ( A x. B ) # 0 ) )
17	10, 11, 13, 11, 16	apcon4bid 8543	. . . . . 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 7913	. . . . . . 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 460	. . . . . . 7 |- ( ( ph /\ B # 0 ) -> ( A # 0 <-> ( A x. B ) # 0 ) )
26	21, 22, 23, 22, 25	apcon4bid 8543	. . . . . 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 781	. . 3 |- ( ph -> ( ( A # 0 \/ B # 0 ) -> ( B = 0 \/ A = 0 ) ) )
30	7, 29	mpd 13	. 2 |- ( ph -> ( B = 0 \/ A = 0 ) )
31	30	orcomd 724	1 |- ( ph -> ( A = 0 \/ B = 0 ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   e. wcel 2141   class class class wbr 3989  (class class class)co 5853   CCcc 7772   0cc0 7774   x. cmul 7779   # cap 8500

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501

This theorem is referenced by: (None)