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Theorem dmmulpi 7178
 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 4849 . . 3 dom ( ·o ↾ (N × N)) = ((N × N) ∩ dom ·o )
2 fnom 6355 . . . . 5 ·o Fn (On × On)
3 fndm 5231 . . . . 5 ( ·o Fn (On × On) → dom ·o = (On × On))
42, 3ax-mp 5 . . . 4 dom ·o = (On × On)
54ineq2i 3280 . . 3 ((N × N) ∩ dom ·o ) = ((N × N) ∩ (On × On))
61, 5eqtri 2161 . 2 dom ( ·o ↾ (N × N)) = ((N × N) ∩ (On × On))
7 df-mi 7158 . . 3 ·N = ( ·o ↾ (N × N))
87dmeqi 4749 . 2 dom ·N = dom ( ·o ↾ (N × N))
9 df-ni 7156 . . . . . . 7 N = (ω ∖ {∅})
10 difss 3208 . . . . . . 7 (ω ∖ {∅}) ⊆ ω
119, 10eqsstri 3135 . . . . . 6 N ⊆ ω
12 omsson 4535 . . . . . 6 ω ⊆ On
1311, 12sstri 3112 . . . . 5 N ⊆ On
14 anidm 394 . . . . 5 ((N ⊆ On ∧ N ⊆ On) ↔ N ⊆ On)
1513, 14mpbir 145 . . . 4 (N ⊆ On ∧ N ⊆ On)
16 xpss12 4655 . . . 4 ((N ⊆ On ∧ N ⊆ On) → (N × N) ⊆ (On × On))
1715, 16ax-mp 5 . . 3 (N × N) ⊆ (On × On)
18 dfss 3091 . . 3 ((N × N) ⊆ (On × On) ↔ (N × N) = ((N × N) ∩ (On × On)))
1917, 18mpbi 144 . 2 (N × N) = ((N × N) ∩ (On × On))
206, 8, 193eqtr4i 2171 1 dom ·N = (N × N)
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   = wceq 1332   ∖ cdif 3074   ∩ cin 3076   ⊆ wss 3077  ∅c0 3369  {csn 3533  Oncon0 4294  ωcom 4513   × cxp 4546  dom cdm 4548   ↾ cres 4550   Fn wfn 5127   ·o comu 6320  Ncnpi 7124   ·N cmi 7126 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4052  ax-sep 4055  ax-nul 4063  ax-pow 4107  ax-pr 4140  ax-un 4364  ax-setind 4461  ax-iinf 4511 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-dif 3079  df-un 3081  df-in 3083  df-ss 3090  df-nul 3370  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-int 3781  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-tr 4036  df-id 4224  df-iord 4297  df-on 4299  df-suc 4302  df-iom 4514  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-f1 5137  df-fo 5138  df-f1o 5139  df-fv 5140  df-ov 5786  df-oprab 5787  df-mpo 5788  df-1st 6047  df-2nd 6048  df-recs 6211  df-irdg 6276  df-oadd 6326  df-omul 6327  df-ni 7156  df-mi 7158 This theorem is referenced by: (None)
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