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Theorem psrbagcon 17890
Description: The analogue of the statement " 0  <_  G  <_  F implies  0  <_  F  -  G  <_  F " for finite bags. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypothesis
Ref Expression
psrbag.d  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
Assertion
Ref Expression
psrbagcon  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  (
( F  oF  -  G )  e.  D  /\  ( F  oF  -  G
)  oR  <_  F ) )
Distinct variable groups:    f, F    f, G    f, I
Allowed substitution hints:    D( f)    V( f)

Proof of Theorem psrbagcon
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpr1 1001 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  F  e.  D )
2 psrbag.d . . . . . . . . . 10  |-  D  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
32psrbag 17881 . . . . . . . . 9  |-  ( I  e.  V  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
43adantr 465 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  e.  D  <->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) ) )
51, 4mpbid 210 . . . . . . 7  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F : I --> NN0  /\  ( `' F " NN )  e.  Fin ) )
65simpld 459 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  F : I --> NN0 )
7 ffn 5717 . . . . . 6  |-  ( F : I --> NN0  ->  F  Fn  I )
86, 7syl 16 . . . . 5  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  F  Fn  I )
9 simpr2 1002 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G : I --> NN0 )
10 ffn 5717 . . . . . 6  |-  ( G : I --> NN0  ->  G  Fn  I )
119, 10syl 16 . . . . 5  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G  Fn  I )
12 simpl 457 . . . . 5  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  I  e.  V )
13 inidm 3689 . . . . 5  |-  ( I  i^i  I )  =  I
148, 11, 12, 12, 13offn 6532 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
)  Fn  I )
15 eqidd 2442 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2442 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  =  ( G `  x ) )
178, 11, 12, 12, 13, 15, 16ofval 6530 . . . . . 6  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F  oF  -  G ) `  x )  =  ( ( F `  x
)  -  ( G `
 x ) ) )
18 simpr3 1003 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G  oR  <_  F )
1911, 8, 12, 12, 13, 16, 15ofrfval 6529 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( G  oR  <_  F  <->  A. x  e.  I  ( G `  x )  <_  ( F `  x ) ) )
2018, 19mpbid 210 . . . . . . . 8  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  A. x  e.  I  ( G `  x )  <_  ( F `  x )
)
2120r19.21bi 2810 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  <_  ( F `  x
) )
229ffvelrnda 6012 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  e.  NN0 )
236ffvelrnda 6012 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  e.  NN0 )
24 nn0sub 10847 . . . . . . . 8  |-  ( ( ( G `  x
)  e.  NN0  /\  ( F `  x )  e.  NN0 )  -> 
( ( G `  x )  <_  ( F `  x )  <->  ( ( F `  x
)  -  ( G `
 x ) )  e.  NN0 ) )
2522, 23, 24syl2anc 661 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( G `  x
)  <_  ( F `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  e. 
NN0 ) )
2621, 25mpbid 210 . . . . . 6  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F `  x
)  -  ( G `
 x ) )  e.  NN0 )
2717, 26eqeltrd 2529 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F  oF  -  G ) `  x )  e.  NN0 )
2827ralrimiva 2855 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  A. x  e.  I  ( ( F  oF  -  G
) `  x )  e.  NN0 )
29 ffnfv 6038 . . . 4  |-  ( ( F  oF  -  G ) : I --> NN0  <->  ( ( F  oF  -  G
)  Fn  I  /\  A. x  e.  I  ( ( F  oF  -  G ) `  x )  e.  NN0 ) )
3014, 28, 29sylanbrc 664 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
) : I --> NN0 )
315simprd 463 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( `' F " NN )  e.  Fin )
3222nn0ge0d 10856 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  0  <_  ( G `  x
) )
33 nn0re 10805 . . . . . . . . . 10  |-  ( ( F `  x )  e.  NN0  ->  ( F `
 x )  e.  RR )
34 nn0re 10805 . . . . . . . . . 10  |-  ( ( G `  x )  e.  NN0  ->  ( G `
 x )  e.  RR )
35 subge02 10069 . . . . . . . . . 10  |-  ( ( ( F `  x
)  e.  RR  /\  ( G `  x )  e.  RR )  -> 
( 0  <_  ( G `  x )  <->  ( ( F `  x
)  -  ( G `
 x ) )  <_  ( F `  x ) ) )
3633, 34, 35syl2an 477 . . . . . . . . 9  |-  ( ( ( F `  x
)  e.  NN0  /\  ( G `  x )  e.  NN0 )  -> 
( 0  <_  ( G `  x )  <->  ( ( F `  x
)  -  ( G `
 x ) )  <_  ( F `  x ) ) )
3723, 22, 36syl2anc 661 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
0  <_  ( G `  x )  <->  ( ( F `  x )  -  ( G `  x ) )  <_ 
( F `  x
) ) )
3832, 37mpbid 210 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F `  x
)  -  ( G `
 x ) )  <_  ( F `  x ) )
3938ralrimiva 2855 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  A. x  e.  I  ( ( F `  x )  -  ( G `  x ) )  <_ 
( F `  x
) )
4014, 8, 12, 12, 13, 17, 15ofrfval 6529 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  (
( F  oF  -  G )  oR  <_  F  <->  A. x  e.  I  ( ( F `  x )  -  ( G `  x ) )  <_ 
( F `  x
) ) )
4139, 40mpbird 232 . . . . 5  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
)  oR  <_  F )
422psrbaglesupp 17885 . . . . 5  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  ( F  oF  -  G ) : I --> NN0  /\  ( F  oF  -  G
)  oR  <_  F ) )  -> 
( `' ( F  oF  -  G
) " NN ) 
C_  ( `' F " NN ) )
4312, 1, 30, 41, 42syl13anc 1229 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( `' ( F  oF  -  G ) " NN )  C_  ( `' F " NN ) )
44 ssfi 7738 . . . 4  |-  ( ( ( `' F " NN )  e.  Fin  /\  ( `' ( F  oF  -  G
) " NN ) 
C_  ( `' F " NN ) )  -> 
( `' ( F  oF  -  G
) " NN )  e.  Fin )
4531, 43, 44syl2anc 661 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( `' ( F  oF  -  G ) " NN )  e.  Fin )
462psrbag 17881 . . . 4  |-  ( I  e.  V  ->  (
( F  oF  -  G )  e.  D  <->  ( ( F  oF  -  G
) : I --> NN0  /\  ( `' ( F  oF  -  G ) " NN )  e.  Fin ) ) )
4746adantr 465 . . 3  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  (
( F  oF  -  G )  e.  D  <->  ( ( F  oF  -  G
) : I --> NN0  /\  ( `' ( F  oF  -  G ) " NN )  e.  Fin ) ) )
4830, 45, 47mpbir2and 920 . 2  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
)  e.  D )
4948, 41jca 532 1  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  (
( F  oF  -  G )  e.  D  /\  ( F  oF  -  G
)  oR  <_  F ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   {crab 2795    C_ wss 3458   class class class wbr 4433   `'ccnv 4984   "cima 4988    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277    oFcof 6519    oRcofr 6520    ^m cmap 7418   Fincfn 7514   RRcr 9489   0cc0 9490    <_ cle 9627    - cmin 9805   NNcn 10537   NN0cn0 10796
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-of 6521  df-ofr 6522  df-om 6682  df-supp 6900  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-fin 7518  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-nn 10538  df-n0 10797
This theorem is referenced by:  psrbagconcl  17893  psrbagconf1o  17894  gsumbagdiaglem  17895  psrmulcllem  17908  psrlidm  17924  psrlidmOLD  17925  psrridm  17926  psrridmOLD  17927  psrass1  17928  psrcom  17932
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