Theorem nnm00 6433

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 108	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = ∅)
2		simpl 108	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ ∅ ∈ 𝐵) → 𝐴 = ∅)
3	1, 2	jaoi 706	. . . . . 6 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
4	3	orcd 723	. . . . 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 109	. . . . . . 7 ⊢ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → 𝐵 = ∅)
7	6	olcd 724	. . . . . 6 ⊢ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
8	7	a1i 9	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9		simplr 520	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·o 𝐵) = ∅)
10		nnmordi 6420	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
11	10	expimpd 361	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
12	11	ancoms 266	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
13		nnm0 6379	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
14	13	adantr 274	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
15	14	eleq1d 2209	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵) ↔ ∅ ∈ (𝐴 ·o 𝐵)))
16	12, 15	sylibd 148	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
17	16	adantr 274	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
18	17	imp 123	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ∅ ∈ (𝐴 ·o 𝐵))
19		n0i 3373	. . . . . . . 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 610	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
22	21	ex 114	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
23	8, 22	jaod 707	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24		0elnn 4540	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25		0elnn 4540	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
26	24, 25	anim12i 336	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27		anddi 811	. . . . . 6 ⊢ (((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
28	26, 27	sylib 121	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
29	28	adantr 274	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
30	5, 23, 29	mpjaod 708	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
31	30	ex 114	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
32		oveq1 5789	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
33		nnm0r 6383	. . . . . 6 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
34	32, 33	sylan9eqr 2195	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
35	34	ex 114	. . . 4 ⊢ (𝐵 ∈ ω → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
36	35	adantl 275	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
37		oveq2 5790	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
38	37, 13	sylan9eqr 2195	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
39	38	ex 114	. . . 4 ⊢ (𝐴 ∈ ω → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
40	39	adantr 274	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
41	36, 40	jaod 707	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
42	31, 41	impbid 128	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   = wceq 1332   ∈ wcel 1481  ∅c0 3368  ωcom 4512  (class class class)co 5782   ·_o comu 6319

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-iord 4296  df-on 4298  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-oadd 6325  df-omul 6326

This theorem is referenced by:  enq0tr  7266  nqnq0pi  7270