Theorem nnm00 6013

Description: The product of two natural numbers is zero iff at least one of them is zero. (Contributed by Jim Kingdon, 11-Nov-2004.)

Assertion

Ref	Expression
nnm00	⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A ·𝑜 B) = ∅ ↔ (A = ∅ ∨ B = ∅)))

Proof of Theorem nnm00

Step	Hyp	Ref	Expression
1		ax-ia1 99	. . . . . . 7 ⊢ ((A = ∅ ∧ B = ∅) → A = ∅)
2		ax-ia1 99	. . . . . . 7 ⊢ ((A = ∅ ∧ ∅ ∈ B) → A = ∅)
3	1, 2	jaoi 623	. . . . . 6 ⊢ (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) → A = ∅)
4	3	orcd 639	. . . . 5 ⊢ (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) → (A = ∅ ∨ B = ∅))
5	4	a1i 9	. . . 4 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) → (A = ∅ ∨ B = ∅)))
6		ax-ia2 100	. . . . . . 7 ⊢ ((∅ ∈ A ∧ B = ∅) → B = ∅)
7	6	olcd 640	. . . . . 6 ⊢ ((∅ ∈ A ∧ B = ∅) → (A = ∅ ∨ B = ∅))
8	7	a1i 9	. . . . 5 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → ((∅ ∈ A ∧ B = ∅) → (A = ∅ ∨ B = ∅)))
9		simplr 470	. . . . . . 7 ⊢ ((((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) ∧ (∅ ∈ A ∧ ∅ ∈ B)) → (A ·𝑜 B) = ∅)
10		nnmordi 6000	. . . . . . . . . . . . 13 ⊢ (((B ∈ 𝜔 ∧ A ∈ 𝜔) ∧ ∅ ∈ A) → (∅ ∈ B → (A ·𝑜 ∅) ∈ (A ·𝑜 B)))
11	10	expimpd 345	. . . . . . . . . . . 12 ⊢ ((B ∈ 𝜔 ∧ A ∈ 𝜔) → ((∅ ∈ A ∧ ∅ ∈ B) → (A ·𝑜 ∅) ∈ (A ·𝑜 B)))
12	11	ancoms 255	. . . . . . . . . . 11 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((∅ ∈ A ∧ ∅ ∈ B) → (A ·𝑜 ∅) ∈ (A ·𝑜 B)))
13		nnm0 5969	. . . . . . . . . . . . 13 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) = ∅)
14	13	adantr 261	. . . . . . . . . . . 12 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ·𝑜 ∅) = ∅)
15	14	eleq1d 2088	. . . . . . . . . . 11 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A ·𝑜 ∅) ∈ (A ·𝑜 B) ↔ ∅ ∈ (A ·𝑜 B)))
16	12, 15	sylibd 138	. . . . . . . . . 10 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((∅ ∈ A ∧ ∅ ∈ B) → ∅ ∈ (A ·𝑜 B)))
17	16	adantr 261	. . . . . . . . 9 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → ((∅ ∈ A ∧ ∅ ∈ B) → ∅ ∈ (A ·𝑜 B)))
18	17	imp 115	. . . . . . . 8 ⊢ ((((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) ∧ (∅ ∈ A ∧ ∅ ∈ B)) → ∅ ∈ (A ·𝑜 B))
19		n0i 3206	. . . . . . . 8 ⊢ (∅ ∈ (A ·𝑜 B) → ¬ (A ·𝑜 B) = ∅)
20	18, 19	syl 14	. . . . . . 7 ⊢ ((((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) ∧ (∅ ∈ A ∧ ∅ ∈ B)) → ¬ (A ·𝑜 B) = ∅)
21	9, 20	pm2.21dd 538	. . . . . 6 ⊢ ((((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) ∧ (∅ ∈ A ∧ ∅ ∈ B)) → (A = ∅ ∨ B = ∅))
22	21	ex 108	. . . . 5 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → ((∅ ∈ A ∧ ∅ ∈ B) → (A = ∅ ∨ B = ∅)))
23	8, 22	jaod 624	. . . 4 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → (((∅ ∈ A ∧ B = ∅) ∨ (∅ ∈ A ∧ ∅ ∈ B)) → (A = ∅ ∨ B = ∅)))
24		0elnn 4267	. . . . . . 7 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))
25		0elnn 4267	. . . . . . 7 ⊢ (B ∈ 𝜔 → (B = ∅ ∨ ∅ ∈ B))
26	24, 25	anim12i 321	. . . . . 6 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A = ∅ ∨ ∅ ∈ A) ∧ (B = ∅ ∨ ∅ ∈ B)))
27		anddi 722	. . . . . 6 ⊢ (((A = ∅ ∨ ∅ ∈ A) ∧ (B = ∅ ∨ ∅ ∈ B)) ↔ (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) ∨ ((∅ ∈ A ∧ B = ∅) ∨ (∅ ∈ A ∧ ∅ ∈ B))))
28	26, 27	sylib 127	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) ∨ ((∅ ∈ A ∧ B = ∅) ∨ (∅ ∈ A ∧ ∅ ∈ B))))
29	28	adantr 261	. . . 4 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) ∨ ((∅ ∈ A ∧ B = ∅) ∨ (∅ ∈ A ∧ ∅ ∈ B))))
30	5, 23, 29	mpjaod 625	. . 3 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → (A = ∅ ∨ B = ∅))
31	30	ex 108	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A ·𝑜 B) = ∅ → (A = ∅ ∨ B = ∅)))
32		oveq1 5443	. . . . . 6 ⊢ (A = ∅ → (A ·𝑜 B) = (∅ ·𝑜 B))
33		nnm0r 5973	. . . . . 6 ⊢ (B ∈ 𝜔 → (∅ ·𝑜 B) = ∅)
34	32, 33	sylan9eqr 2076	. . . . 5 ⊢ ((B ∈ 𝜔 ∧ A = ∅) → (A ·𝑜 B) = ∅)
35	34	ex 108	. . . 4 ⊢ (B ∈ 𝜔 → (A = ∅ → (A ·𝑜 B) = ∅))
36	35	adantl 262	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A = ∅ → (A ·𝑜 B) = ∅))
37		oveq2 5444	. . . . . 6 ⊢ (B = ∅ → (A ·𝑜 B) = (A ·𝑜 ∅))
38	37, 13	sylan9eqr 2076	. . . . 5 ⊢ ((A ∈ 𝜔 ∧ B = ∅) → (A ·𝑜 B) = ∅)
39	38	ex 108	. . . 4 ⊢ (A ∈ 𝜔 → (B = ∅ → (A ·𝑜 B) = ∅))
40	39	adantr 261	. . 3 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (B = ∅ → (A ·𝑜 B) = ∅))
41	36, 40	jaod 624	. 2 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A = ∅ ∨ B = ∅) → (A ·𝑜 B) = ∅))
42	31, 41	impbid 120	1 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A ·𝑜 B) = ∅ ↔ (A = ∅ ∨ B = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 616   = wceq 1228   ∈ wcel 1374  ∅c0 3201  𝜔com 4240  (class class class)co 5436   ·_𝑜 comu 5914

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238

This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921

This theorem is referenced by:  enq0tr  6289  nqnq0pi  6293