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Theorem nnm00 6218
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
StepHypRef Expression
1 simpl 107 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = ∅)
2 simpl 107 . . . . . . 7 ((𝐴 = ∅ ∧ ∅ ∈ 𝐵) → 𝐴 = ∅)
31, 2jaoi 669 . . . . . 6 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
43orcd 685 . . . . 5 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
54a1i 9 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
6 simpr 108 . . . . . . 7 ((∅ ∈ 𝐴𝐵 = ∅) → 𝐵 = ∅)
76olcd 686 . . . . . 6 ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
87a1i 9 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9 simplr 497 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 𝐵) = ∅)
10 nnmordi 6205 . . . . . . . . . . . . 13 (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
1110expimpd 355 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
1211ancoms 264 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
13 nnm0 6168 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
1413adantr 270 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ∅) = ∅)
1514eleq1d 2151 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵) ↔ ∅ ∈ (𝐴 ·𝑜 𝐵)))
1612, 15sylibd 147 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·𝑜 𝐵)))
1716adantr 270 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·𝑜 𝐵)))
1817imp 122 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ∅ ∈ (𝐴 ·𝑜 𝐵))
19 n0i 3274 . . . . . . . 8 (∅ ∈ (𝐴 ·𝑜 𝐵) → ¬ (𝐴 ·𝑜 𝐵) = ∅)
2018, 19syl 14 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ¬ (𝐴 ·𝑜 𝐵) = ∅)
219, 20pm2.21dd 583 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
2221ex 113 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
238, 22jaod 670 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24 0elnn 4395 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25 0elnn 4395 . . . . . . 7 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
2624, 25anim12i 331 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27 anddi 768 . . . . . 6 (((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2826, 27sylib 120 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2928adantr 270 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
305, 23, 29mpjaod 671 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
3130ex 113 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
32 oveq1 5598 . . . . . 6 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
33 nnm0r 6172 . . . . . 6 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
3432, 33sylan9eqr 2137 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3534ex 113 . . . 4 (𝐵 ∈ ω → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3635adantl 271 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
37 oveq2 5599 . . . . . 6 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
3837, 13sylan9eqr 2137 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3938ex 113 . . . 4 (𝐴 ∈ ω → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
4039adantr 270 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
4136, 40jaod 670 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅))
4231, 41impbid 127 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662   = wceq 1285  wcel 1434  c0 3269  ωcom 4368  (class class class)co 5591   ·𝑜 comu 6111
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3919  ax-sep 3922  ax-nul 3930  ax-pow 3974  ax-pr 4000  ax-un 4224  ax-setind 4316  ax-iinf 4366
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-dif 2986  df-un 2988  df-in 2990  df-ss 2997  df-nul 3270  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-int 3663  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-tr 3902  df-id 4084  df-iord 4157  df-on 4159  df-suc 4162  df-iom 4369  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-f1 4974  df-fo 4975  df-f1o 4976  df-fv 4977  df-ov 5594  df-oprab 5595  df-mpt2 5596  df-1st 5846  df-2nd 5847  df-recs 6002  df-irdg 6067  df-oadd 6117  df-omul 6118
This theorem is referenced by:  enq0tr  6896  nqnq0pi  6900
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