Theorem nnm00 6697

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 723	. . . . . 6 ⊢ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
4	3	orcd 740	. . . . 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 741	. . . . . 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 6683	. . . . . . . . . . . . 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 6642	. . . . . . . . . . . . 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 3500	. . . . . . . 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 724	. . . 4 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((∅ ∈ 𝐴 ∧ 𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24		0elnn 4717	. . . . . . 7 ⊢ (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25		0elnn 4717	. . . . . . 7 ⊢ (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
26	24, 25	anim12i 338	. . . . . 6 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27		anddi 828	. . . . . 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 725	. . 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 6024	. . . . . 6 ⊢ (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
33		nnm0r 6646	. . . . . 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 6025	. . . . . 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 724	. 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 715   = wceq 1397   ∈ wcel 2202  ∅c0 3494  ωcom 4688  (class class class)co 6017   ·_o comu 6579

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-oadd 6585  df-omul 6586

This theorem is referenced by:  enq0tr  7653  nqnq0pi  7657