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Theorem dvgt0lem2 22832
Description: Lemma for dvgt0 22833 and dvlt0 22834. (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 435 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( F `  x ) O ( F `  y ) ) )
32ralrimivva 2853 . . . 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 11716 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 665 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ltso 9713 . . . . . 6  |-  <  Or  RR
9 soss 4793 . . . . . 6  |-  ( ( A [,] B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A [,] B ) ) )
107, 8, 9mpisyl 22 . . . . 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 21821 . . . . . 6  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
1513, 14syl 17 . . . . 5  |-  ( ph  ->  F : ( A [,] B ) --> RR )
16 ssid 3489 . . . . . 6  |-  ( A [,] B )  C_  ( A [,] B )
1716a1i 11 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  ( A [,] B ) )
18 soisores 6233 . . . . 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 1265 . . . 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 235 . . 3  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) 
Isom  <  ,  O  ( ( A [,] B
) ,  ( F
" ( A [,] B ) ) ) )
21 ffn 5746 . . . . 5  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2213, 14, 213syl 18 . . . 4  |-  ( ph  ->  F  Fn  ( A [,] B ) )
23 fnresdm 5703 . . . 4  |-  ( F  Fn  ( A [,] B )  ->  ( F  |`  ( A [,] B ) )  =  F )
24 isoeq1 6225 . . . 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 213 . 2  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) )
27 fnima 5712 . . 3  |-  ( F  Fn  ( A [,] B )  ->  ( F " ( A [,] B ) )  =  ran  F )
28 isoeq5 6229 . . 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 213 1  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   A.wral 2782    C_ wss 3442   class class class wbr 4426    Or wor 4774   ran crn 4855    |` cres 4856   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601    Isom wiso 5602  (class class class)co 6305   RRcr 9537    < clt 9674   (,)cioo 11635   [,]cicc 11638   -cn->ccncf 21804    _D cdv 22695
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 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-pre-lttri 9612  ax-pre-lttrn 9613
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 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-po 4775  df-so 4776  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-icc 11642  df-cncf 21806
This theorem is referenced by:  dvgt0  22833  dvlt0  22834
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