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Theorem nnm00 6102
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 102 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = ∅)
2 simpl 102 . . . . . . 7 ((𝐴 = ∅ ∧ ∅ ∈ 𝐵) → 𝐴 = ∅)
31, 2jaoi 636 . . . . . 6 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
43orcd 652 . . . . 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 103 . . . . . . 7 ((∅ ∈ 𝐴𝐵 = ∅) → 𝐵 = ∅)
76olcd 653 . . . . . 6 ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
87a1i 9 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9 simplr 482 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·𝑜 𝐵) = ∅)
10 nnmordi 6089 . . . . . . . . . . . . 13 (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
1110expimpd 345 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
1211ancoms 255 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵)))
13 nnm0 6054 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (𝐴 ·𝑜 ∅) = ∅)
1413adantr 261 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·𝑜 ∅) = ∅)
1514eleq1d 2106 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 ∅) ∈ (𝐴 ·𝑜 𝐵) ↔ ∅ ∈ (𝐴 ·𝑜 𝐵)))
1612, 15sylibd 138 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·𝑜 𝐵)))
1716adantr 261 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·𝑜 𝐵)))
1817imp 115 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ∅ ∈ (𝐴 ·𝑜 𝐵))
19 n0i 3229 . . . . . . . 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 550 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
2221ex 108 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
238, 22jaod 637 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24 0elnn 4340 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25 0elnn 4340 . . . . . . 7 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
2624, 25anim12i 321 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27 anddi 734 . . . . . 6 (((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2826, 27sylib 127 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2928adantr 261 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
305, 23, 29mpjaod 638 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·𝑜 𝐵) = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
3130ex 108 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
32 oveq1 5519 . . . . . 6 (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = (∅ ·𝑜 𝐵))
33 nnm0r 6058 . . . . . 6 (𝐵 ∈ ω → (∅ ·𝑜 𝐵) = ∅)
3432, 33sylan9eqr 2094 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3534ex 108 . . . 4 (𝐵 ∈ ω → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
3635adantl 262 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
37 oveq2 5520 . . . . . 6 (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = (𝐴 ·𝑜 ∅))
3837, 13sylan9eqr 2094 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅)
3938ex 108 . . . 4 (𝐴 ∈ ω → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
4039adantr 261 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 = ∅ → (𝐴 ·𝑜 𝐵) = ∅))
4136, 40jaod 637 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·𝑜 𝐵) = ∅))
4231, 41impbid 120 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·𝑜 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 97  wb 98  wo 629   = wceq 1243  wcel 1393  c0 3224  ωcom 4313  (class class class)co 5512   ·𝑜 comu 5999
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-1st 5767  df-2nd 5768  df-recs 5920  df-irdg 5957  df-oadd 6005  df-omul 6006
This theorem is referenced by:  enq0tr  6530  nqnq0pi  6534
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