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.

Step	Hyp	Ref	Expression
1		smodm 7041	. . . . 5
2		ordtr1 4926	. . . . . . 7
3	2	ancomsd 454	. . . . . 6
4	3	expdimp 437	. . . . 5
5	1, 4	sylan 471	. . . 4
6		df-smo 7036	. . . . . 6
7		eleq1 2529	. . . . . . . . . . 11
8		fveq2 5871	. . . . . . . . . . . 12
9	8	eleq1d 2526	. . . . . . . . . . 11
10	7, 9	imbi12d 320	. . . . . . . . . 10
11		eleq2 2530	. . . . . . . . . . 11
12		fveq2 5871	. . . . . . . . . . . 12
13	12	eleq2d 2527	. . . . . . . . . . 11
14	11, 13	imbi12d 320	. . . . . . . . . 10
15	10, 14	rspc2v 3219	. . . . . . . . 9
16	15	ancoms 453	. . . . . . . 8
17	16	com12 31	. . . . . . 7
18	17	3ad2ant3 1019	. . . . . 6
19	6, 18	sylbi 195	. . . . 5
20	19	expdimp 437	. . . 4
21	5, 20	syld 44	. . 3
22	21	pm2.43d 48	. 2
23	22	3impia 1193	1

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  Smowsmo 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