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Theorem psrbagcon 17786
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 997 . . . . . . . 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 17777 . . . . . . . . 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 5722 . . . . . 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 998 . . . . . 6  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G : I --> NN0 )
10 ffn 5722 . . . . . 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 3700 . . . . 5  |-  ( I  i^i  I )  =  I
148, 11, 12, 12, 13offn 6526 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( F  oF  -  G
)  Fn  I )
15 eqidd 2461 . . . . . . 7  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2461 . . . . . . 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 6524 . . . . . 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 999 . . . . . . . . 9  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  G  oR  <_  F )
1911, 8, 12, 12, 13, 16, 15ofrfval 6523 . . . . . . . . 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 2826 . . . . . . 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 10835 . . . . . . . 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 2548 . . . . 5  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  (
( F  oF  -  G ) `  x )  e.  NN0 )
2827ralrimiva 2871 . . . 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 10844 . . . . . . . 8  |-  ( ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  /\  x  e.  I )  ->  0  <_  ( G `  x
) )
33 nn0re 10793 . . . . . . . . . 10  |-  ( ( F `  x )  e.  NN0  ->  ( F `
 x )  e.  RR )
34 nn0re 10793 . . . . . . . . . 10  |-  ( ( G `  x )  e.  NN0  ->  ( G `
 x )  e.  RR )
35 subge02 10057 . . . . . . . . . 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 2871 . . . . . 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 6523 . . . . . 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 17781 . . . . 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 1225 . . . 4  |-  ( ( I  e.  V  /\  ( F  e.  D  /\  G : I --> NN0  /\  G  oR  <_  F
) )  ->  ( `' ( F  oF  -  G ) " NN )  C_  ( `' F " NN ) )
44 ssfi 7730 . . . 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 17777 . . . 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 915 . 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 968    = wceq 1374    e. wcel 1762   A.wral 2807   {crab 2811    C_ wss 3469   class class class wbr 4440   `'ccnv 4991   "cima 4995    Fn wfn 5574   -->wf 5575   ` cfv 5579  (class class class)co 6275    oFcof 6513    oRcofr 6514    ^m cmap 7410   Fincfn 7506   RRcr 9480   0cc0 9481    <_ cle 9618    - cmin 9794   NNcn 10525   NN0cn0 10784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-supp 6892  df-recs 7032  df-rdg 7066  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-n0 10785
This theorem is referenced by:  psrbagconcl  17789  psrbagconf1o  17790  gsumbagdiaglem  17791  psrmulcllem  17804  psrlidm  17820  psrlidmOLD  17821  psrridm  17822  psrridmOLD  17823  psrass1  17824  psrcom  17828
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