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

Theorem smoel 7050
 Description: If is less than then a strictly monotone function's value will be strictly less at than at . (Contributed by Andrew Salmon, 22-Nov-2011.)
Assertion
Ref Expression
smoel

Proof of Theorem smoel
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 smodm 7041 . . . . 5
2 ordtr1 4926 . . . . . . 7
32ancomsd 454 . . . . . 6
43expdimp 437 . . . . 5
51, 4sylan 471 . . . 4
6 df-smo 7036 . . . . . 6
7 eleq1 2529 . . . . . . . . . . 11
8 fveq2 5871 . . . . . . . . . . . 12
98eleq1d 2526 . . . . . . . . . . 11
107, 9imbi12d 320 . . . . . . . . . 10
11 eleq2 2530 . . . . . . . . . . 11
12 fveq2 5871 . . . . . . . . . . . 12
1312eleq2d 2527 . . . . . . . . . . 11
1411, 13imbi12d 320 . . . . . . . . . 10
1510, 14rspc2v 3219 . . . . . . . . 9
1615ancoms 453 . . . . . . . 8
1716com12 31 . . . . . . 7
18173ad2ant3 1019 . . . . . 6
196, 18sylbi 195 . . . . 5
2019expdimp 437 . . . 4
215, 20syld 44 . . 3
2221pm2.43d 48 . 2
23223impia 1193 1
 Colors of variables: wff setvar class Syntax hints:  ->wi 4  /\wa 369  /\w3a 973  =wceq 1395  e.wcel 1818  A.wral 2807  Ordword 4882   con0 4883  domcdm 5004  -->wf 5589  cfv 5593  Smo`wsmo 7035 This theorem is referenced by:  smoiun  7051  smoel2  7053 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-tr 4546  df-ord 4886  df-iota 5556  df-fv 5601  df-smo 7036
 Copyright terms: Public domain W3C validator