Theorem nnm00 6693

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 109	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = ∅)
2		simpl 109	. . . . . . 7 ⊢ ((𝐴 = ∅ ∧ ∅ ∈ 𝐵) → 𝐴 = ∅)
3	1, 2	jaoi 721	. . . . . 6 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
4	3	orcd 738	. . . . 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 110	. . . . . . 7 ⊢ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → 𝐵 = ∅)
7	6	olcd 739	. . . . . 6 ⊢ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
8	7	a1i 9	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9		simplr 528	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·o 𝐵) = ∅)
10		nnmordi 6679	. . . . . . . . . . . . 13 ⊢ (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
11	10	expimpd 363	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
12	11	ancoms 268	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
13		nnm0 6638	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
14	13	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
15	14	eleq1d 2298	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵) ↔ ∅ ∈ (𝐴 ·o 𝐵)))
16	12, 15	sylibd 149	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
17	16	adantr 276	. . . . . . . . 9 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
18	17	imp 124	. . . . . . . 8 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ∅ ∈ (𝐴 ·o 𝐵))
19		n0i 3498	. . . . . . . 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 623	. . . . . 6 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
22	21	ex 115	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
23	8, 22	jaod 722	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24		0elnn 4715	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25		0elnn 4715	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
26	24, 25	anim12i 338	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27		anddi 826	. . . . . 6 ⊢ (((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
28	26, 27	sylib 122	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
29	28	adantr 276	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
30	5, 23, 29	mpjaod 723	. . 3 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
31	30	ex 115	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
32		oveq1 6020	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
33		nnm0r 6642	. . . . . 6 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
34	32, 33	sylan9eqr 2284	. . . . 5 ⊢ ((𝐵 ∈ ω ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
35	34	ex 115	. . . 4 ⊢ (𝐵 ∈ ω → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
36	35	adantl 277	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
37		oveq2 6021	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
38	37, 13	sylan9eqr 2284	. . . . 5 ⊢ ((𝐴 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
39	38	ex 115	. . . 4 ⊢ (𝐴 ∈ ω → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
40	39	adantr 276	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
41	36, 40	jaod 722	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
42	31, 41	impbid 129	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713   = wceq 1395   ∈ wcel 2200  ∅c0 3492  ωcom 4686  (class class class)co 6013   ·_o comu 6575

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-irdg 6531  df-oadd 6581  df-omul 6582

This theorem is referenced by:  enq0tr  7644  nqnq0pi  7648