Theorem nnm00 6038

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		simpl 102	. . . . . . 7 ⊢ ((A = ∅ ∧ B = ∅) → A = ∅)
2		simpl 102	. . . . . . 7 ⊢ ((A = ∅ ∧ ∅ ∈ B) → A = ∅)
3	1, 2	jaoi 635	. . . . . 6 ⊢ (((A = ∅ ∧ B = ∅) ∨ (A = ∅ ∧ ∅ ∈ B)) → A = ∅)
4	3	orcd 651	. . . . 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		simpr 103	. . . . . . 7 ⊢ ((∅ ∈ A ∧ B = ∅) → B = ∅)
7	6	olcd 652	. . . . . 6 ⊢ ((∅ ∈ A ∧ B = ∅) → (A = ∅ ∨ B = ∅))
8	7	a1i 9	. . . . 5 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → ((∅ ∈ A ∧ B = ∅) → (A = ∅ ∨ B = ∅)))
9		simplr 482	. . . . . . 7 ⊢ ((((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) ∧ (∅ ∈ A ∧ ∅ ∈ B)) → (A ·𝑜 B) = ∅)
10		nnmordi 6025	. . . . . . . . . . . . 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 5993	. . . . . . . . . . . . 13 ⊢ (A ∈ 𝜔 → (A ·𝑜 ∅) = ∅)
14	13	adantr 261	. . . . . . . . . . . 12 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → (A ·𝑜 ∅) = ∅)
15	14	eleq1d 2103	. . . . . . . . . . 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 3223	. . . . . . . 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 550	. . . . . 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 636	. . . 4 ⊢ (((A ∈ 𝜔 ∧ B ∈ 𝜔) ∧ (A ·𝑜 B) = ∅) → (((∅ ∈ A ∧ B = ∅) ∨ (∅ ∈ A ∧ ∅ ∈ B)) → (A = ∅ ∨ B = ∅)))
24		0elnn 4283	. . . . . . 7 ⊢ (A ∈ 𝜔 → (A = ∅ ∨ ∅ ∈ A))
25		0elnn 4283	. . . . . . 7 ⊢ (B ∈ 𝜔 → (B = ∅ ∨ ∅ ∈ B))
26	24, 25	anim12i 321	. . . . . 6 ⊢ ((A ∈ 𝜔 ∧ B ∈ 𝜔) → ((A = ∅ ∨ ∅ ∈ A) ∧ (B = ∅ ∨ ∅ ∈ B)))
27		anddi 733	. . . . . 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 637	. . 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 5462	. . . . . 6 ⊢ (A = ∅ → (A ·𝑜 B) = (∅ ·𝑜 B))
33		nnm0r 5997	. . . . . 6 ⊢ (B ∈ 𝜔 → (∅ ·𝑜 B) = ∅)
34	32, 33	sylan9eqr 2091	. . . . 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 5463	. . . . . 6 ⊢ (B = ∅ → (A ·𝑜 B) = (A ·𝑜 ∅))
38	37, 13	sylan9eqr 2091	. . . . 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 636	. 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 628   = wceq 1242   ∈ wcel 1390  ∅c0 3218  𝜔com 4256  (class class class)co 5455   ·_𝑜 comu 5938

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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945

This theorem is referenced by:  enq0tr  6416  nqnq0pi  6420