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Theorem dvgt0lem2 19350
Description: Lemma for dvgt0 19351 and dvlt0 19352. (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 2635 . . . 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 10731 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 642 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ltso 8903 . . . . . 6  |-  <  Or  RR
9 soss 4332 . . . . . 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 18397 . . . . . 6  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
1513, 14syl 15 . . . . 5  |-  ( ph  ->  F : ( A [,] B ) --> RR )
16 ssid 3197 . . . . . 6  |-  ( A [,] B )  C_  ( A [,] B )
1716a1i 10 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  ( A [,] B ) )
18 soisores 5824 . . . . 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 5389 . . . . 5  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2213, 14, 213syl 18 . . . 4  |-  ( ph  ->  F  Fn  ( A [,] B ) )
23 fnresdm 5353 . . . 4  |-  ( F  Fn  ( A [,] B )  ->  ( F  |`  ( A [,] B ) )  =  F )
24 isoeq1 5816 . . . 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 5362 . . 3  |-  ( F  Fn  ( A [,] B )  ->  ( F " ( A [,] B ) )  =  ran  F )
28 isoeq5 5820 . . 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 1623    e. wcel 1684   A.wral 2543    C_ wss 3152   class class class wbr 4023    Or wor 4313   ran crn 4690    |` cres 4691   "cima 4692    Fn wfn 5250   -->wf 5251   ` cfv 5255    Isom wiso 5256  (class class class)co 5858   RRcr 8736    < clt 8867   (,)cioo 10656   [,]cicc 10659   -cn->ccncf 18380    _D cdv 19213
This theorem is referenced by:  dvgt0  19351  dvlt0  19352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-icc 10663  df-cncf 18382
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