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Theorem psrbagcon 17546
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 994 . . . . . . . 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 17537 . . . . . . . . 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 5657 . . . . . 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 995 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G : I --> NN0 )
10 ffn 5657 . . . . . 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 3657 . . . . 5  |-  ( I  i^i  I )  =  I
148, 11, 12, 12, 13offn 6431 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
)  Fn  I )
15 eqidd 2452 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2452 . . . . . . 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 6429 . . . . . 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 996 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G  oR  <_  F )
1911, 8, 12, 12, 13, 16, 15ofrfval 6428 . . . . . . . . 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 2910 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  <_  ( F `  x
) )
229ffvelrnda 5942 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( G `  x )  e.  NN0 )
236ffvelrnda 5942 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  e.  NN0 )
24 nn0sub 10731 . . . . . . . 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 2539 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F  oF  -  G ) `  x )  e.  NN0 )
2827ralrimiva 2822 . . . 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 5968 . . . 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 10740 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  0  <_  ( G `  x
) )
33 nn0re 10689 . . . . . . . . . 10  |-  ( ( F `  x )  e.  NN0  ->  ( F `
 x )  e.  RR )
34 nn0re 10689 . . . . . . . . . 10  |-  ( ( G `  x )  e.  NN0  ->  ( G `
 x )  e.  RR )
35 subge02 9956 . . . . . . . . . 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 2822 . . . . . 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 6428 . . . . . 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 17541 . . . . 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 1221 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( `' ( F  oF  -  G ) " NN )  C_  ( `' F " NN ) )
44 ssfi 7634 . . . 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 17537 . . . 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 913 . 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 965    = wceq 1370    e. wcel 1758   A.wral 2795   {crab 2799    C_ wss 3426   class class class wbr 4390   `'ccnv 4937   "cima 4941    Fn wfn 5511   -->wf 5512   ` cfv 5516  (class class class)co 6190    oFcof 6418    oRcofr 6419    ^m cmap 7314   Fincfn 7410   RRcr 9382   0cc0 9383    <_ cle 9520    - cmin 9696   NNcn 10423   NN0cn0 10680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4501  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440  ax-1cn 9441  ax-icn 9442  ax-addcl 9443  ax-addrcl 9444  ax-mulcl 9445  ax-mulrcl 9446  ax-mulcom 9447  ax-addass 9448  ax-mulass 9449  ax-distr 9450  ax-i2m1 9451  ax-1ne0 9452  ax-1rid 9453  ax-rnegex 9454  ax-rrecex 9455  ax-cnre 9456  ax-pre-lttri 9457  ax-pre-lttrn 9458  ax-pre-ltadd 9459  ax-pre-mulgt0 9460
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-tp 3980  df-op 3982  df-uni 4190  df-iun 4271  df-br 4391  df-opab 4449  df-mpt 4450  df-tr 4484  df-eprel 4730  df-id 4734  df-po 4739  df-so 4740  df-fr 4777  df-we 4779  df-ord 4820  df-on 4821  df-lim 4822  df-suc 4823  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-riota 6151  df-ov 6193  df-oprab 6194  df-mpt2 6195  df-of 6420  df-ofr 6421  df-om 6577  df-supp 6791  df-recs 6932  df-rdg 6966  df-er 7201  df-map 7316  df-en 7411  df-dom 7412  df-sdom 7413  df-fin 7414  df-pnf 9521  df-mnf 9522  df-xr 9523  df-ltxr 9524  df-le 9525  df-sub 9698  df-neg 9699  df-nn 10424  df-n0 10681
This theorem is referenced by:  psrbagconcl  17549  psrbagconf1o  17550  gsumbagdiaglem  17551  psrmulcllem  17564  psrlidm  17580  psrlidmOLD  17581  psrridm  17582  psrridmOLD  17583  psrass1  17584  psrcom  17588
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