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Theorem psrbagcon 19371
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 1067 . . . . . . . 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 19364 . . . . . . . . 9 |- ( I e. V -> ( F e. D <-> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) ) )
43adantr 481 . . . . . . . 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 222 . . . . . . 7 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( F : I --> NN0 /\ ( `' F " NN ) e. Fin ) )
65simpld 475 . . . . . 6 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> F : I --> NN0 )
7 ffn 6045 . . . . . 6 |- ( F : I --> NN0 -> F Fn I )
86, 7syl 17 . . . . 5 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> F Fn I )
9 simpr2 1068 . . . . . 6 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> G : I --> NN0 )
10 ffn 6045 . . . . . 6 |- ( G : I --> NN0 -> G Fn I )
119, 10syl 17 . . . . 5 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> G Fn I )
12 simpl 473 . . . . 5 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> I e. V )
13 inidm 3822 . . . . 5 |- ( I i^i I ) = I
148, 11, 12, 12, 13offn 6908 . . . 4 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( F oF - G ) Fn I )
15 eqidd 2623 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( F ` x ) = ( F ` x ) )
16 eqidd 2623 . . . . . . 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 6906 . . . . . 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 1069 . . . . . . . . 9 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> G oR <_ F )
1911, 8, 12, 12, 13, 16, 15ofrfval 6905 . . . . . . . . 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 222 . . . . . . . 8 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> A. x e. I ( G ` x ) <_ ( F ` x ) )
2120r19.21bi 2932 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( G ` x ) <_ ( F ` x ) )
229ffvelrnda 6359 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( G ` x ) e. NN0 )
236ffvelrnda 6359 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( F ` x ) e. NN0 )
24 nn0sub 11343 . . . . . . . 8 |- ( ( ( G ` x ) e. NN0 /\ ( F ` x ) e. NN0 ) -> ( ( G ` x ) <_ ( F ` x ) <-> ( ( F ` x ) - ( G ` x ) ) e. NN0 ) )
2522, 23, 24syl2anc 693 . . . . . . 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 222 . . . . . 6 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( ( F ` x ) - ( G ` x ) ) e. NN0 )
2717, 26eqeltrd 2701 . . . . 5 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( ( F oF - G ) ` x ) e. NN0 )
2827ralrimiva 2966 . . . 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 6388 . . . 4 |- ( ( F oF - G ) : I --> NN0 <-> ( ( F oF - G ) Fn I /\ A. x e. I ( ( F oF - G ) ` x ) e. NN0 ) )
3014, 28, 29sylanbrc 698 . . 3 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( F oF - G ) : I --> NN0 )
315simprd 479 . . . 4 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' F " NN ) e. Fin )
3222nn0ge0d 11354 . . . . . . . 8 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> 0 <_ ( G ` x ) )
33 nn0re 11301 . . . . . . . . . 10 |- ( ( F ` x ) e. NN0 -> ( F ` x ) e. RR )
34 nn0re 11301 . . . . . . . . . 10 |- ( ( G ` x ) e. NN0 -> ( G ` x ) e. RR )
35 subge02 10544 . . . . . . . . . 10 |- ( ( ( F ` x ) e. RR /\ ( G ` x ) e. RR ) -> ( 0 <_ ( G ` x ) <-> ( ( F ` x ) - ( G ` x ) ) <_ ( F ` x ) ) )
3633, 34, 35syl2an 494 . . . . . . . . 9 |- ( ( ( F ` x ) e. NN0 /\ ( G ` x ) e. NN0 ) -> ( 0 <_ ( G ` x ) <-> ( ( F ` x ) - ( G ` x ) ) <_ ( F ` x ) ) )
3723, 22, 36syl2anc 693 . . . . . . . 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 222 . . . . . . 7 |- ( ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) /\ x e. I ) -> ( ( F ` x ) - ( G ` x ) ) <_ ( F ` x ) )
3938ralrimiva 2966 . . . . . 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 6905 . . . . . 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 247 . . . . 5 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( F oF - G ) oR <_ F )
422psrbaglesupp 19368 . . . . 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 1328 . . . 4 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' ( F oF - G ) " NN ) C_ ( `' F " NN ) )
44 ssfi 8180 . . . 4 |- ( ( ( `' F " NN ) e. Fin /\ ( `' ( F oF - G ) " NN ) C_ ( `' F " NN ) ) -> ( `' ( F oF - G ) " NN ) e. Fin )
4531, 43, 44syl2anc 693 . . 3 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( `' ( F oF - G ) " NN ) e. Fin )
462psrbag 19364 . . . 4 |- ( I e. V -> ( ( F oF - G ) e. D <-> ( ( F oF - G ) : I --> NN0 /\ ( `' ( F oF - G ) " NN ) e. Fin ) ) )
4746adantr 481 . . 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 957 . 2 |- ( ( I e. V /\ ( F e. D /\ G : I --> NN0 /\ G oR <_ F ) ) -> ( F oF - G ) e. D )
4948, 41jca 554 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 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   class class class wbr 4653   `'ccnv 5113   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   oRcofr 6896   ^m cmap 7857   Fincfn 7955   RRcr 9935   0cc0 9936   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293
This theorem is referenced by:  psrbagconcl  19373  psrbagconf1o  19374  gsumbagdiaglem  19375  psrmulcllem  19387  psrlidm  19403  psrridm  19404  psrass1  19405  psrcom  19409
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