Theorem fsuppmapnn0fiub0 12793

Description: If all functions of a finite set of functions over the nonnegative integers are finitely supported, then all these functions are zero for all integers greater than a fixed integer. (Contributed by AV, 3-Oct-2019.)

Assertion

Ref	Expression
fsuppmapnn0fiub0	|- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( A. f e. M f finSupp Z -> E. m e. NN0 A. f e. M A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )

Distinct variable groups:   f, M, m   R, f, m   f, V, m   f, Z, m   x, M   x, R   x, V   x, Z, f, m

Proof of Theorem fsuppmapnn0fiub0

Step	Hyp	Ref	Expression
1		fsuppmapnn0fiubex 12792	. 2 |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( A. f e. M f finSupp Z -> E. m e. NN0 A. f e. M ( f supp Z ) C_ ( 0 ... m ) ) )
2		ssel2 3598	. . . . . . . . . . . . . 14 |- ( ( M C_ ( R ^m NN0 ) /\ f e. M ) -> f e. ( R ^m NN0 ) )
3	2	ancoms 469	. . . . . . . . . . . . 13 |- ( ( f e. M /\ M C_ ( R ^m NN0 ) ) -> f e. ( R ^m NN0 ) )
4		elmapfn 7880	. . . . . . . . . . . . 13 |- ( f e. ( R ^m NN0 ) -> f Fn NN0 )
5	3, 4	syl 17	. . . . . . . . . . . 12 |- ( ( f e. M /\ M C_ ( R ^m NN0 ) ) -> f Fn NN0 )
6	5	expcom 451	. . . . . . . . . . 11 |- ( M C_ ( R ^m NN0 ) -> ( f e. M -> f Fn NN0 ) )
7	6	3ad2ant1 1082	. . . . . . . . . 10 |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( f e. M -> f Fn NN0 ) )
8	7	adantr 481	. . . . . . . . 9 |- ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) -> ( f e. M -> f Fn NN0 ) )
9	8	imp 445	. . . . . . . 8 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> f Fn NN0 )
10		nn0ex 11298	. . . . . . . . 9 |- NN0 e. _V
11	10	a1i 11	. . . . . . . 8 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> NN0 e. _V )
12		simpll3 1102	. . . . . . . 8 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> Z e. V )
13		suppvalfn 7302	. . . . . . . 8 |- ( ( f Fn NN0 /\ NN0 e. _V /\ Z e. V ) -> ( f supp Z ) = { x e. NN0 | ( f ` x ) =/= Z } )
14	9, 11, 12, 13	syl3anc 1326	. . . . . . 7 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> ( f supp Z ) = { x e. NN0 | ( f ` x ) =/= Z } )
15	14	sseq1d 3632	. . . . . 6 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> ( ( f supp Z ) C_ ( 0 ... m ) <-> { x e. NN0 | ( f ` x ) =/= Z } C_ ( 0 ... m ) ) )
16		rabss 3679	. . . . . 6 |- ( { x e. NN0 | ( f ` x ) =/= Z } C_ ( 0 ... m ) <-> A. x e. NN0 ( ( f ` x ) =/= Z -> x e. ( 0 ... m ) ) )
17	15, 16	syl6bb 276	. . . . 5 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> ( ( f supp Z ) C_ ( 0 ... m ) <-> A. x e. NN0 ( ( f ` x ) =/= Z -> x e. ( 0 ... m ) ) ) )
18		nne 2798	. . . . . . . . . 10 |- ( -. ( f ` x ) =/= Z <-> ( f ` x ) = Z )
19	18	biimpi 206	. . . . . . . . 9 |- ( -. ( f ` x ) =/= Z -> ( f ` x ) = Z )
20	19	2a1d 26	. . . . . . . 8 |- ( -. ( f ` x ) =/= Z -> ( ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) /\ x e. NN0 ) -> ( m < x -> ( f ` x ) = Z ) ) )
21		elfz2nn0 12431	. . . . . . . . 9 |- ( x e. ( 0 ... m ) <-> ( x e. NN0 /\ m e. NN0 /\ x <_ m ) )
22		nn0re 11301	. . . . . . . . . . . . 13 |- ( x e. NN0 -> x e. RR )
23		nn0re 11301	. . . . . . . . . . . . 13 |- ( m e. NN0 -> m e. RR )
24		lenlt 10116	. . . . . . . . . . . . 13 |- ( ( x e. RR /\ m e. RR ) -> ( x <_ m <-> -. m < x ) )
25	22, 23, 24	syl2an 494	. . . . . . . . . . . 12 |- ( ( x e. NN0 /\ m e. NN0 ) -> ( x <_ m <-> -. m < x ) )
26		pm2.21 120	. . . . . . . . . . . 12 |- ( -. m < x -> ( m < x -> ( f ` x ) = Z ) )
27	25, 26	syl6bi 243	. . . . . . . . . . 11 |- ( ( x e. NN0 /\ m e. NN0 ) -> ( x <_ m -> ( m < x -> ( f ` x ) = Z ) ) )
28	27	3impia 1261	. . . . . . . . . 10 |- ( ( x e. NN0 /\ m e. NN0 /\ x <_ m ) -> ( m < x -> ( f ` x ) = Z ) )
29	28	a1d 25	. . . . . . . . 9 |- ( ( x e. NN0 /\ m e. NN0 /\ x <_ m ) -> ( ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) /\ x e. NN0 ) -> ( m < x -> ( f ` x ) = Z ) ) )
30	21, 29	sylbi 207	. . . . . . . 8 |- ( x e. ( 0 ... m ) -> ( ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) /\ x e. NN0 ) -> ( m < x -> ( f ` x ) = Z ) ) )
31	20, 30	ja 173	. . . . . . 7 |- ( ( ( f ` x ) =/= Z -> x e. ( 0 ... m ) ) -> ( ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) /\ x e. NN0 ) -> ( m < x -> ( f ` x ) = Z ) ) )
32	31	com12 32	. . . . . 6 |- ( ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) /\ x e. NN0 ) -> ( ( ( f ` x ) =/= Z -> x e. ( 0 ... m ) ) -> ( m < x -> ( f ` x ) = Z ) ) )
33	32	ralimdva 2962	. . . . 5 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> ( A. x e. NN0 ( ( f ` x ) =/= Z -> x e. ( 0 ... m ) ) -> A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )
34	17, 33	sylbid 230	. . . 4 |- ( ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) /\ f e. M ) -> ( ( f supp Z ) C_ ( 0 ... m ) -> A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )
35	34	ralimdva 2962	. . 3 |- ( ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) /\ m e. NN0 ) -> ( A. f e. M ( f supp Z ) C_ ( 0 ... m ) -> A. f e. M A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )
36	35	reximdva 3017	. 2 |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( E. m e. NN0 A. f e. M ( f supp Z ) C_ ( 0 ... m ) -> E. m e. NN0 A. f e. M A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )
37	1, 36	syld 47	1 |- ( ( M C_ ( R ^m NN0 ) /\ M e. Fin /\ Z e. V ) -> ( A. f e. M f finSupp Z -> E. m e. NN0 A. f e. M A. x e. NN0 ( m < x -> ( f ` x ) = Z ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   ^m cmap 7857   Fincfn 7955   finSupp cfsupp 8275   RRcr 9935   0cc0 9936   < clt 10074   <_ cle 10075   NN0cn0 11292   ...cfz 12326

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-rmo 2920  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-int 4476  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-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  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  df-z 11378  df-uz 11688  df-fz 12327

This theorem is referenced by:  pmatcoe1fsupp  20506