Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvds1lem Structured version   Unicode version

Theorem dvds1lem 13873
 Description: A lemma to assist theorems of with one antecedent. (Contributed by Paul Chapman, 21-Mar-2011.)
Hypotheses
Ref Expression
dvds1lem.1
dvds1lem.2
dvds1lem.3
dvds1lem.4
Assertion
Ref Expression
dvds1lem
Distinct variable groups:   ,   ,   ,   ,   ,
Allowed substitution hint:   ()

Proof of Theorem dvds1lem
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 dvds1lem.3 . . . 4
2 dvds1lem.4 . . . 4
3 oveq1 6302 . . . . . 6
43eqeq1d 2469 . . . . 5
54rspcev 3219 . . . 4
61, 2, 5syl6an 545 . . 3
76rexlimdva 2959 . 2
8 dvds1lem.1 . . 3
9 divides 13866 . . 3
108, 9syl 16 . 2
11 dvds1lem.2 . . 3
12 divides 13866 . . 3
1311, 12syl 16 . 2
147, 10, 133imtr4d 268 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  wrex 2818   class class class wbr 4453  (class class class)co 6295   cmul 9509  cz 10876   cdivides 13864 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-iota 5557  df-fv 5602  df-ov 6298  df-dvds 13865 This theorem is referenced by:  negdvdsb  13878  dvdsnegb  13879  muldvds1  13886  muldvds2  13887  dvdscmul  13888  dvdsmulc  13889  dvdscmulr  13890  dvdsmulcr  13891
 Copyright terms: Public domain W3C validator