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

Theorem nnm00 6038
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 635 . . . . . 6 (((A = ∅ B = ∅) (A = ∅ B)) → A = ∅)
43orcd 651 . . . . 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 ((∅ A B = ∅) → B = ∅)
76olcd 652 . . . . . 6 ((∅ A B = ∅) → (A = ∅ B = ∅))
87a1i 9 . . . . 5 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → ((∅ A B = ∅) → (A = ∅ B = ∅)))
9 simplr 482 . . . . . . 7 ((((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) (∅ A B)) → (A ·𝑜 B) = ∅)
10 nnmordi 6025 . . . . . . . . . . . . 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 5993 . . . . . . . . . . . . 13 (A 𝜔 → (A ·𝑜 ∅) = ∅)
1413adantr 261 . . . . . . . . . . . 12 ((A 𝜔 B 𝜔) → (A ·𝑜 ∅) = ∅)
1514eleq1d 2103 . . . . . . . . . . 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 3223 . . . . . . . 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 636 . . . 4 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (((∅ A B = ∅) (∅ A B)) → (A = ∅ B = ∅)))
24 0elnn 4283 . . . . . . 7 (A 𝜔 → (A = ∅ A))
25 0elnn 4283 . . . . . . 7 (B 𝜔 → (B = ∅ B))
2624, 25anim12i 321 . . . . . 6 ((A 𝜔 B 𝜔) → ((A = ∅ A) (B = ∅ B)))
27 anddi 733 . . . . . 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 637 . . 3 (((A 𝜔 B 𝜔) (A ·𝑜 B) = ∅) → (A = ∅ B = ∅))
3130ex 108 . 2 ((A 𝜔 B 𝜔) → ((A ·𝑜 B) = ∅ → (A = ∅ B = ∅)))
32 oveq1 5462 . . . . . 6 (A = ∅ → (A ·𝑜 B) = (∅ ·𝑜 B))
33 nnm0r 5997 . . . . . 6 (B 𝜔 → (∅ ·𝑜 B) = ∅)
3432, 33sylan9eqr 2091 . . . . 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 5463 . . . . . 6 (B = ∅ → (A ·𝑜 B) = (A ·𝑜 ∅))
3837, 13sylan9eqr 2091 . . . . 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 636 . 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 628   = wceq 1242   wcel 1390  c0 3218  𝜔com 4256  (class class class)co 5455   ·𝑜 comu 5938
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-oadd 5944  df-omul 5945
This theorem is referenced by:  enq0tr  6416  nqnq0pi  6420
  Copyright terms: Public domain W3C validator