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Theorem dvgt0lem2 22167
Description: Lemma for dvgt0 22168 and dvlt0 22169. (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 434 . . . . 5  |-  ( (
ph  /\  ( x  e.  ( A [,] B
)  /\  y  e.  ( A [,] B ) ) )  ->  (
x  <  y  ->  ( F `  x ) O ( F `  y ) ) )
32ralrimivva 2885 . . . 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 11606 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
74, 5, 6syl2anc 661 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
8 ltso 9665 . . . . . 6  |-  <  Or  RR
9 soss 4818 . . . . . 6  |-  ( ( A [,] B ) 
C_  RR  ->  (  < 
Or  RR  ->  <  Or  ( A [,] B ) ) )
107, 8, 9mpisyl 18 . . . . 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 21160 . . . . . 6  |-  ( F  e.  ( ( A [,] B ) -cn-> RR )  ->  F :
( A [,] B
) --> RR )
1513, 14syl 16 . . . . 5  |-  ( ph  ->  F : ( A [,] B ) --> RR )
16 ssid 3523 . . . . . 6  |-  ( A [,] B )  C_  ( A [,] B )
1716a1i 11 . . . . 5  |-  ( ph  ->  ( A [,] B
)  C_  ( A [,] B ) )
18 soisores 6211 . . . . 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 1229 . . . 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 232 . . 3  |-  ( ph  ->  ( F  |`  ( A [,] B ) ) 
Isom  <  ,  O  ( ( A [,] B
) ,  ( F
" ( A [,] B ) ) ) )
21 ffn 5731 . . . . 5  |-  ( F : ( A [,] B ) --> RR  ->  F  Fn  ( A [,] B ) )
2213, 14, 213syl 20 . . . 4  |-  ( ph  ->  F  Fn  ( A [,] B ) )
23 fnresdm 5690 . . . 4  |-  ( F  Fn  ( A [,] B )  ->  ( F  |`  ( A [,] B ) )  =  F )
24 isoeq1 6203 . . . 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 20 . . 3  |-  ( ph  ->  ( ( F  |`  ( A [,] B ) )  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) ) )
2620, 25mpbid 210 . 2  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) ) )
27 fnima 5699 . . 3  |-  ( F  Fn  ( A [,] B )  ->  ( F " ( A [,] B ) )  =  ran  F )
28 isoeq5 6207 . . 3  |-  ( ( F " ( A [,] B ) )  =  ran  F  -> 
( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
2922, 27, 283syl 20 . 2  |-  ( ph  ->  ( F  Isom  <  ,  O  ( ( A [,] B ) ,  ( F " ( A [,] B ) ) )  <->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) ) )
3026, 29mpbid 210 1  |-  ( ph  ->  F  Isom  <  ,  O  ( ( A [,] B ) ,  ran  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476   class class class wbr 4447    Or wor 4799   ran crn 5000    |` cres 5001   "cima 5002    Fn wfn 5583   -->wf 5584   ` cfv 5588    Isom wiso 5589  (class class class)co 6284   RRcr 9491    < clt 9628   (,)cioo 11529   [,]cicc 11532   -cn->ccncf 21143    _D cdv 22030
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-isom 5597  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-icc 11536  df-cncf 21145
This theorem is referenced by:  dvgt0  22168  dvlt0  22169
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