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Theorem dvgt0lem2 19887
Description: Lemma for dvgt0 19888 and dvlt0 19889. (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 424 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( F `  x ) O ( F `  y ) ) )
32ralrimivva 2798 . . . 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 10992 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 643 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ltso 9156 . . . . . 6  |-  <  Or  RR
9 soss 4521 . . . . . 6  |-  ( ( A [,] B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A [,] B ) ) )
107, 8, 9ee10 1385 . . . . 5  |-  ( ph  ->  <  Or  ( A [,] B ) )
11 dvgt0lem.o . . . . . 6  |-  O  Or  RR
1211a1i 11 . . . . 5  |-  ( ph  ->  O  Or  RR )
13 dvgt0.f . . . . . 6  |-  ( ph  ->  F  e.  ( ( A [,] B )
-cn-> RR ) )
14 cncff 18923 . . . . . 6  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  F : ( A [,] B ) --> RR )
16 ssid 3367 . . . . . 6  |-  ( A [,] B )  C_  ( A [,] B )
1716a1i 11 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  ( A [,] B ) )
18 soisores 6047 . . . . 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 1185 . . . 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 224 . . 3  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) 
Isom  <  ,  O  ( ( A [,] B
) ,  ( F
" ( A [,] B ) ) ) )
21 ffn 5591 . . . . 5  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2213, 14, 213syl 19 . . . 4  |-  ( ph  ->  F  Fn  ( A [,] B ) )
23 fnresdm 5554 . . . 4  |-  ( F  Fn  ( A [,] B )  ->  ( F  |`  ( A [,] B ) )  =  F )
24 isoeq1 6039 . . . 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 19 . . 3  |-  ( ph  ->  ( ( F  |`  ( A [,] B ) )  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) ) )
2620, 25mpbid 202 . 2  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) )
27 fnima 5563 . . 3  |-  ( F  Fn  ( A [,] B )  ->  ( F " ( A [,] B ) )  =  ran  F )
28 isoeq5 6043 . . 3  |-  ( ( F " ( A [,] B ) )  =  ran  F  -> 
( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
2922, 27, 283syl 19 . 2  |-  ( ph  ->  ( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
3026, 29mpbid 202 1  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705    C_ wss 3320   class class class wbr 4212    Or wor 4502   ran crn 4879    |` cres 4880   "cima 4881    Fn wfn 5449   -->wf 5450   ` cfv 5454    Isom wiso 5455  (class class class)co 6081   RRcr 8989    < clt 9120   (,)cioo 10916   [,]cicc 10919   -cn->ccncf 18906    _D cdv 19750
This theorem is referenced by:  dvgt0  19888  dvlt0  19889
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-icc 10923  df-cncf 18908
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