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Theorem dvgt0lem2 19366
Description: Lemma for dvgt0 19367 and dvlt0 19368. (Contributed by Mario Carneiro, 19-Feb-2015.)
Hypotheses
Ref Expression
dvgt0.a  |-  ( ph  ->  A  e.  RR )
dvgt0.b  |-  ( ph  ->  B  e.  RR )
dvgt0.f  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
dvgt0lem.d  |-  ( ph  ->  ( RR  _D  F
) : ( A (,) B ) --> S )
dvgt0lem.o  |-  O  Or  RR
dvgt0lem.i  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x ) O ( F `  y ) )
Assertion
Ref Expression
dvgt0lem2  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
Distinct variable groups:    x, y, A    x, O, y    ph, x, y    x, B, y    x, F, y
Allowed substitution hints:    S( x, y)

Proof of Theorem dvgt0lem2
StepHypRef Expression
1 dvgt0lem.i . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  ( A [,] B )  /\  y  e.  ( A [,] B ) ) )  /\  x  <  y
)  ->  ( F `  x ) O ( F `  y ) )
21ex 423 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( F `  x ) O ( F `  y ) ) )
32ralrimivva 2648 . . . 4  |-  ( ph  ->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
) O ( F `
 y ) ) )
4 dvgt0.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
5 dvgt0.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
6 iccssre 10747 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ltso 8919 . . . . . 6  |-  <  Or  RR
9 soss 4348 . . . . . 6  |-  ( ( A [,] B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A [,] B ) ) )
107, 8, 9ee10 1366 . . . . 5  |-  ( ph  ->  <  Or  ( A [,] B ) )
11 dvgt0lem.o . . . . . 6  |-  O  Or  RR
1211a1i 10 . . . . 5  |-  ( ph  ->  O  Or  RR )
13 dvgt0.f . . . . . 6  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
14 cncff 18413 . . . . . 6  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
1513, 14syl 15 . . . . 5  |-  ( ph  ->  F : ( A [,] B ) --> RR )
16 ssid 3210 . . . . . 6  |-  ( A [,] B )  C_  ( A [,] B )
1716a1i 10 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  ( A [,] B ) )
18 soisores 5840 . . . . 5  |-  ( ( (  <  Or  ( A [,] B )  /\  O  Or  RR )  /\  ( F : ( A [,] B ) --> RR  /\  ( A [,] B )  C_  ( A [,] B ) ) )  ->  (
( F  |`  ( A [,] B ) ) 
Isom  <  ,  O  ( ( A [,] B
) ,  ( F
" ( A [,] B ) ) )  <->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x ) O ( F `  y ) ) ) )
1910, 12, 15, 17, 18syl22anc 1183 . . . 4  |-  ( ph  ->  ( ( F  |`  ( A [,] B ) )  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  A. x  e.  ( A [,] B ) A. y  e.  ( A [,] B ) ( x  <  y  ->  ( F `  x
) O ( F `
 y ) ) ) )
203, 19mpbird 223 . . 3  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) 
Isom  <  ,  O  ( ( A [,] B
) ,  ( F
" ( A [,] B ) ) ) )
21 ffn 5405 . . . . 5  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2213, 14, 213syl 18 . . . 4  |-  ( ph  ->  F  Fn  ( A [,] B ) )
23 fnresdm 5369 . . . 4  |-  ( F  Fn  ( A [,] B )  ->  ( F  |`  ( A [,] B ) )  =  F )
24 isoeq1 5832 . . . 4  |-  ( ( F  |`  ( A [,] B ) )  =  F  ->  ( ( F  |`  ( A [,] B ) )  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) ) )
2522, 23, 243syl 18 . . 3  |-  ( ph  ->  ( ( F  |`  ( A [,] B ) )  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) ) )
2620, 25mpbid 201 . 2  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) )
27 fnima 5378 . . 3  |-  ( F  Fn  ( A [,] B )  ->  ( F " ( A [,] B ) )  =  ran  F )
28 isoeq5 5836 . . 3  |-  ( ( F " ( A [,] B ) )  =  ran  F  -> 
( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
2922, 27, 283syl 18 . 2  |-  ( ph  ->  ( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
3026, 29mpbid 201 1  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556    C_ wss 3165   class class class wbr 4039    Or wor 4329   ran crn 4706    |` cres 4707   "cima 4708    Fn wfn 5266   -->wf 5267   ` cfv 5271    Isom wiso 5272  (class class class)co 5874   RRcr 8752    < clt 8883   (,)cioo 10672   [,]cicc 10675   -cn->ccncf 18396    _D cdv 19229
This theorem is referenced by:  dvgt0  19367  dvlt0  19368
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-pre-lttri 8827  ax-pre-lttrn 8828
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-er 6676  df-map 6790  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-icc 10679  df-cncf 18398
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