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Theorem dmmulpi 6310
 Description: Domain of multiplication on positive integers. (Contributed by NM, 26-Aug-1995.)
Assertion
Ref Expression
dmmulpi dom ·N = (N × N)

Proof of Theorem dmmulpi
StepHypRef Expression
1 dmres 4575 . . 3 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ dom ·𝑜 )
2 fnom 5969 . . . . 5 ·𝑜 Fn (On × On)
3 fndm 4941 . . . . 5 ( ·𝑜 Fn (On × On) → dom ·𝑜 = (On × On))
42, 3ax-mp 7 . . . 4 dom ·𝑜 = (On × On)
54ineq2i 3129 . . 3 ((N × N) ∩ dom ·𝑜 ) = ((N × N) ∩ (On × On))
61, 5eqtri 2057 . 2 dom ( ·𝑜 ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 6290 . . 3 ·N = ( ·𝑜 ↾ (N × N))
87dmeqi 4479 . 2 dom ·N = dom ( ·𝑜 ↾ (N × N))
9 df-ni 6288 . . . . . . 7 N = (𝜔 ∖ {∅})
10 difss 3064 . . . . . . 7 (𝜔 ∖ {∅}) ⊆ 𝜔
119, 10eqsstri 2969 . . . . . 6 N ⊆ 𝜔
12 omsson 4278 . . . . . 6 𝜔 ⊆ On
1311, 12sstri 2948 . . . . 5 N ⊆ On
14 anidm 376 . . . . 5 ((N ⊆ On N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 134 . . . 4 (N ⊆ On N ⊆ On)
16 xpss12 4388 . . . 4 ((N ⊆ On N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 7 . . 3 (N × N) ⊆ (On × On)
18 dfss 2926 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 133 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2067 1 dom ·N = (N × N)
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∖ cdif 2908   ∩ cin 2910   ⊆ wss 2911  ∅c0 3218  {csn 3367  Oncon0 4066  𝜔com 4256   × cxp 4286  dom cdm 4288   ↾ cres 4290   Fn wfn 4840   ·𝑜 comu 5938  Ncnpi 6256   ·N cmi 6258 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  df-ni 6288  df-mi 6290 This theorem is referenced by: (None)
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