Theorem nnm00 6741

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 724	. . . . . 6 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
4	3	orcd 741	. . . . 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 742	. . . . . 6 ⊢ ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
8	7	a1i 9	. . . . 5 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ 𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9		simplr 529	. . . . . . 7 ⊢ ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·o 𝐵) = ∅)
10		nnmordi 6727	. . . . . . . . . . . . 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 6686	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
14	13	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
15	14	eleq1d 2300	. . . . . . . . . . 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 3502	. . . . . . . 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 625	. . . . . 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 725	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24		0elnn 4723	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25		0elnn 4723	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
26	24, 25	anim12i 338	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27		anddi 829	. . . . . 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 726	. . 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 6035	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
33		nnm0r 6690	. . . . . 6 ⊢ (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
34	32, 33	sylan9eqr 2286	. . . . 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 6036	. . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
38	37, 13	sylan9eqr 2286	. . . . 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 725	. 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 716   = wceq 1398   ∈ wcel 2202  ∅c0 3496  ωcom 4694  (class class class)co 6028   ·_o comu 6623

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-oadd 6629  df-omul 6630

This theorem is referenced by:  enq0tr  7697  nqnq0pi  7701