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Theorem nnm00 6013
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 ax-ia1 99 . . . . . . 7 ((A = ∅ B = ∅) → A = ∅)
2 ax-ia1 99 . . . . . . 7 ((A = ∅ B) → A = ∅)
31, 2jaoi 623 . . . . . 6 (((A = ∅ B = ∅) (A = ∅ B)) → A = ∅)
43orcd 639 . . . . 5 (((A = ∅ B = ∅) (A = ∅ B)) → (A = ∅ B = ∅))
54a1i 9 . . . 4 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (((A = ∅ B = ∅) (A = ∅ B)) → (A = ∅ B = ∅)))
6 ax-ia2 100 . . . . . . 7 ((∅ A B = ∅) → B = ∅)
76olcd 640 . . . . . 6 ((∅ A B = ∅) → (A = ∅ B = ∅))
87a1i 9 . . . . 5 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → ((∅ A B = ∅) → (A = ∅ B = ∅)))
9 simplr 470 . . . . . . 7 ((((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) (∅ A B)) → (A ·𝑜 B) = ∅)
10 nnmordi 6000 . . . . . . . . . . . . 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 5969 . . . . . . . . . . . . 13 (A 𝜔 → (A ·𝑜 ∅) = ∅)
1413adantr 261 . . . . . . . . . . . 12 ((A 𝜔 B 𝜔) → (A ·𝑜 ∅) = ∅)
1514eleq1d 2088 . . . . . . . . . . 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 3206 . . . . . . . 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 538 . . . . . 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 624 . . . 4 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (((∅ A B = ∅) (∅ A B)) → (A = ∅ B = ∅)))
24 0elnn 4267 . . . . . . 7 (A 𝜔 → (A = ∅ A))
25 0elnn 4267 . . . . . . 7 (B 𝜔 → (B = ∅ B))
2624, 25anim12i 321 . . . . . 6 ((A 𝜔 B 𝜔) → ((A = ∅ A) (B = ∅ B)))
27 anddi 722 . . . . . 6 (((A = ∅ A) (B = ∅ B)) ↔ (((A = ∅ B = ∅) (A = ∅ B)) ((∅ A B = ∅) (∅ A B))))
2826, 27sylib 127 . . . . 5 ((A 𝜔 B 𝜔) → (((A = ∅ B = ∅) (A = ∅ B)) ((∅ A B = ∅) (∅ A B))))
2928adantr 261 . . . 4 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (((A = ∅ B = ∅) (A = ∅ B)) ((∅ A B = ∅) (∅ A B))))
305, 23, 29mpjaod 625 . . 3 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (A = ∅ B = ∅))
3130ex 108 . 2 ((A 𝜔 B 𝜔) → ((A ·𝑜 B) = ∅ → (A = ∅ B = ∅)))
32 oveq1 5443 . . . . . 6 (A = ∅ → (A ·𝑜 B) = (∅ ·𝑜 B))
33 nnm0r 5973 . . . . . 6 (B 𝜔 → (∅ ·𝑜 B) = ∅)
3432, 33sylan9eqr 2076 . . . . 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 5444 . . . . . 6 (B = ∅ → (A ·𝑜 B) = (A ·𝑜 ∅))
3837, 13sylan9eqr 2076 . . . . 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 624 . 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 616   = wceq 1228   wcel 1374  c0 3201  𝜔com 4240  (class class class)co 5436   ·𝑜 comu 5914
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-oadd 5920  df-omul 5921
This theorem is referenced by:  enq0tr  6289  nqnq0pi  6293
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