ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nnm00 GIF version

Theorem nnm00 6393
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 108 . . . . . . 7 ((𝐴 = ∅ ∧ 𝐵 = ∅) → 𝐴 = ∅)
2 simpl 108 . . . . . . 7 ((𝐴 = ∅ ∧ ∅ ∈ 𝐵) → 𝐴 = ∅)
31, 2jaoi 690 . . . . . 6 (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) → 𝐴 = ∅)
43orcd 707 . . . . 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 109 . . . . . . 7 ((∅ ∈ 𝐴𝐵 = ∅) → 𝐵 = ∅)
76olcd 708 . . . . . 6 ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
87a1i 9 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴𝐵 = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
9 simplr 504 . . . . . . 7 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 ·o 𝐵) = ∅)
10 nnmordi 6380 . . . . . . . . . . . . 13 (((𝐵 ∈ ω ∧ 𝐴 ∈ ω) ∧ ∅ ∈ 𝐴) → (∅ ∈ 𝐵 → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
1110expimpd 360 . . . . . . . . . . . 12 ((𝐵 ∈ ω ∧ 𝐴 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
1211ancoms 266 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵)))
13 nnm0 6339 . . . . . . . . . . . . 13 (𝐴 ∈ ω → (𝐴 ·o ∅) = ∅)
1413adantr 274 . . . . . . . . . . . 12 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ·o ∅) = ∅)
1514eleq1d 2186 . . . . . . . . . . 11 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o ∅) ∈ (𝐴 ·o 𝐵) ↔ ∅ ∈ (𝐴 ·o 𝐵)))
1612, 15sylibd 148 . . . . . . . . . 10 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
1716adantr 274 . . . . . . . . 9 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → ∅ ∈ (𝐴 ·o 𝐵)))
1817imp 123 . . . . . . . 8 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → ∅ ∈ (𝐴 ·o 𝐵))
19 n0i 3338 . . . . . . . 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 594 . . . . . 6 ((((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) ∧ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅))
2221ex 114 . . . . 5 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → ((∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
238, 22jaod 691 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵)) → (𝐴 = ∅ ∨ 𝐵 = ∅)))
24 0elnn 4502 . . . . . . 7 (𝐴 ∈ ω → (𝐴 = ∅ ∨ ∅ ∈ 𝐴))
25 0elnn 4502 . . . . . . 7 (𝐵 ∈ ω → (𝐵 = ∅ ∨ ∅ ∈ 𝐵))
2624, 25anim12i 336 . . . . . 6 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)))
27 anddi 795 . . . . . 6 (((𝐴 = ∅ ∨ ∅ ∈ 𝐴) ∧ (𝐵 = ∅ ∨ ∅ ∈ 𝐵)) ↔ (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2826, 27sylib 121 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
2928adantr 274 . . . 4 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (((𝐴 = ∅ ∧ 𝐵 = ∅) ∨ (𝐴 = ∅ ∧ ∅ ∈ 𝐵)) ∨ ((∅ ∈ 𝐴𝐵 = ∅) ∨ (∅ ∈ 𝐴 ∧ ∅ ∈ 𝐵))))
305, 23, 29mpjaod 692 . . 3 (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐴 ·o 𝐵) = ∅) → (𝐴 = ∅ ∨ 𝐵 = ∅))
3130ex 114 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ → (𝐴 = ∅ ∨ 𝐵 = ∅)))
32 oveq1 5749 . . . . . 6 (𝐴 = ∅ → (𝐴 ·o 𝐵) = (∅ ·o 𝐵))
33 nnm0r 6343 . . . . . 6 (𝐵 ∈ ω → (∅ ·o 𝐵) = ∅)
3432, 33sylan9eqr 2172 . . . . 5 ((𝐵 ∈ ω ∧ 𝐴 = ∅) → (𝐴 ·o 𝐵) = ∅)
3534ex 114 . . . 4 (𝐵 ∈ ω → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
3635adantl 275 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 = ∅ → (𝐴 ·o 𝐵) = ∅))
37 oveq2 5750 . . . . . 6 (𝐵 = ∅ → (𝐴 ·o 𝐵) = (𝐴 ·o ∅))
3837, 13sylan9eqr 2172 . . . . 5 ((𝐴 ∈ ω ∧ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅)
3938ex 114 . . . 4 (𝐴 ∈ ω → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
4039adantr 274 . . 3 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 = ∅ → (𝐴 ·o 𝐵) = ∅))
4136, 40jaod 691 . 2 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 = ∅ ∨ 𝐵 = ∅) → (𝐴 ·o 𝐵) = ∅))
4231, 41impbid 128 1 ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ·o 𝐵) = ∅ ↔ (𝐴 = ∅ ∨ 𝐵 = ∅)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 682   = wceq 1316  wcel 1465  c0 3333  ωcom 4474  (class class class)co 5742   ·o comu 6279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-iord 4258  df-on 4260  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-irdg 6235  df-oadd 6285  df-omul 6286
This theorem is referenced by:  enq0tr  7210  nqnq0pi  7214
  Copyright terms: Public domain W3C validator