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

Theorem wemappo 8067
Description: Construct lexicographic order on a function space based on a well-ordering of the indexes and a total ordering of the values.

Without totality on the values or least differing indexes, the best we can prove here is a partial order. (Contributed by Stefan O'Rear, 18-Jan-2015.)

Hypothesis
Ref Expression
wemapso.t  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
Assertion
Ref Expression
wemappo  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
Distinct variable groups:    x, B    x, w, y, z, A   
w, R, x, y, z    w, S, x, y, z
Allowed substitution hints:    B( y, z, w)    T( x, y, z, w)    V( x, y, z, w)

Proof of Theorem wemappo
Dummy variables  a 
b  c are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 3090 . 2  |-  ( A  e.  V  ->  A  e.  _V )
2 simpll3 1046 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  S  Po  B )
3 elmapi 7498 . . . . . . . . 9  |-  ( a  e.  ( B  ^m  A )  ->  a : A --> B )
43adantl 467 . . . . . . . 8  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  a : A --> B )
54ffvelrnda 6034 . . . . . . 7  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  (
a `  b )  e.  B )
6 poirr 4782 . . . . . . 7  |-  ( ( S  Po  B  /\  ( a `  b
)  e.  B )  ->  -.  ( a `  b ) S ( a `  b ) )
72, 5, 6syl2anc 665 . . . . . 6  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  -.  ( a `  b
) S ( a `
 b ) )
87intnanrd 925 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  a  e.  ( B  ^m  A ) )  /\  b  e.  A )  ->  -.  ( ( a `  b ) S ( a `  b )  /\  A. c  e.  A  ( c R b  ->  ( a `  c )  =  ( a `  c ) ) ) )
98nrexdv 2881 . . . 4  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  -.  E. b  e.  A  ( (
a `  b ) S ( a `  b )  /\  A. c  e.  A  (
c R b  -> 
( a `  c
)  =  ( a `
 c ) ) ) )
10 vex 3084 . . . . 5  |-  a  e. 
_V
11 wemapso.t . . . . . 6  |-  T  =  { <. x ,  y
>.  |  E. z  e.  A  ( (
x `  z ) S ( y `  z )  /\  A. w  e.  A  (
w R z  -> 
( x `  w
)  =  ( y `
 w ) ) ) }
1211wemaplem1 8064 . . . . 5  |-  ( ( a  e.  _V  /\  a  e.  _V )  ->  ( a T a  <->  E. b  e.  A  ( ( a `  b ) S ( a `  b )  /\  A. c  e.  A  ( c R b  ->  ( a `  c )  =  ( a `  c ) ) ) ) )
1310, 10, 12mp2an 676 . . . 4  |-  ( a T a  <->  E. b  e.  A  ( (
a `  b ) S ( a `  b )  /\  A. c  e.  A  (
c R b  -> 
( a `  c
)  =  ( a `
 c ) ) ) )
149, 13sylnibr 306 . . 3  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  a  e.  ( B  ^m  A ) )  ->  -.  a T
a )
15 simpll1 1044 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  A  e.  _V )
16 simplr1 1047 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a  e.  ( B  ^m  A ) )
17 simplr2 1048 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  b  e.  ( B  ^m  A ) )
18 simplr3 1049 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  c  e.  ( B  ^m  A ) )
19 simpll2 1045 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  R  Or  A
)
20 simpll3 1046 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  S  Po  B
)
21 simprl 762 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a T b )
22 simprr 764 . . . . 5  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  b T c )
2311, 15, 16, 17, 18, 19, 20, 21, 22wemaplem3 8066 . . . 4  |-  ( ( ( ( A  e. 
_V  /\  R  Or  A  /\  S  Po  B
)  /\  ( a  e.  ( B  ^m  A
)  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  /\  (
a T b  /\  b T c ) )  ->  a T c )
2423ex 435 . . 3  |-  ( ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  /\  ( a  e.  ( B  ^m  A )  /\  b  e.  ( B  ^m  A )  /\  c  e.  ( B  ^m  A ) ) )  ->  (
( a T b  /\  b T c )  ->  a T
c ) )
2514, 24ispod 4779 . 2  |-  ( ( A  e.  _V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
261, 25syl3an1 1297 1  |-  ( ( A  e.  V  /\  R  Or  A  /\  S  Po  B )  ->  T  Po  ( B  ^m  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   _Vcvv 3081   class class class wbr 4420   {copab 4478    Po wpo 4769    Or wor 4770   -->wf 5594   ` cfv 5598  (class class class)co 6302    ^m cmap 7477
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-1st 6804  df-2nd 6805  df-map 7479
This theorem is referenced by:  wemapsolem  8068
  Copyright terms: Public domain W3C validator