Theorem mptnn0fsupp 12797

Description: A mapping from the nonnegative integers is finitely supported under certain conditions. (Contributed by AV, 5-Oct-2019.) (Revised by AV, 23-Dec-2019.)

Hypotheses

Ref	Expression
mptnn0fsupp.0	|- ( ph -> .0. e. V )
mptnn0fsupp.c	|- ( ( ph /\ k e. NN0 ) -> C e. B )
mptnn0fsupp.s	|- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )

Assertion

Ref	Expression
mptnn0fsupp	|- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )

Distinct variable groups:   B, k   C, s, x   ph, k, s, x   .0. , s, x

Allowed substitution hints:   B( x, s)   C( k)   V( x, k, s)   .0. ( k)

Proof of Theorem mptnn0fsupp

Step	Hyp	Ref	Expression
1		mptnn0fsupp.c	. . . . . 6 |- ( ( ph /\ k e. NN0 ) -> C e. B )
2	1	ralrimiva 2966	. . . . 5 |- ( ph -> A. k e. NN0 C e. B )
3		eqid 2622	. . . . . 6 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
4	3	fnmpt 6020	. . . . 5 |- ( A. k e. NN0 C e. B -> ( k e. NN0 |-> C ) Fn NN0 )
5	2, 4	syl 17	. . . 4 |- ( ph -> ( k e. NN0 |-> C ) Fn NN0 )
6		nn0ex 11298	. . . . 5 |- NN0 e. _V
7	6	a1i 11	. . . 4 |- ( ph -> NN0 e. _V )
8		mptnn0fsupp.0	. . . . 5 |- ( ph -> .0. e. V )
9		elex 3212	. . . . 5 |- ( .0. e. V -> .0. e. _V )
10	8, 9	syl 17	. . . 4 |- ( ph -> .0. e. _V )
11		suppvalfn 7302	. . . 4 |- ( ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } )
12	5, 7, 10, 11	syl3anc 1326	. . 3 |- ( ph -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } )
13		mptnn0fsupp.s	. . . . 5 |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
14		nne 2798	. . . . . . . . 9 |- ( -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. <-> ( ( k e. NN0 |-> C ) ` x ) = .0. )
15		simpr 477	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> x e. NN0 )
16	2	ad2antrr 762	. . . . . . . . . . . 12 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> A. k e. NN0 C e. B )
17		rspcsbela 4006	. . . . . . . . . . . 12 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
18	15, 16, 17	syl2anc 693	. . . . . . . . . . 11 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
19	3	fvmpts 6285	. . . . . . . . . . 11 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
20	15, 18, 19	syl2anc 693	. . . . . . . . . 10 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
21	20	eqeq1d 2624	. . . . . . . . 9 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> ( ( ( k e. NN0 |-> C ) ` x ) = .0. <-> [_ x / k ]_ C = .0. ) )
22	14, 21	syl5bb 272	. . . . . . . 8 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> ( -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. <-> [_ x / k ]_ C = .0. ) )
23	22	imbi2d 330	. . . . . . 7 |- ( ( ( ph /\ s e. NN0 ) /\ x e. NN0 ) -> ( ( s < x -> -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. ) <-> ( s < x -> [_ x / k ]_ C = .0. ) ) )
24	23	ralbidva 2985	. . . . . 6 |- ( ( ph /\ s e. NN0 ) -> ( A. x e. NN0 ( s < x -> -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. ) <-> A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) ) )
25	24	rexbidva 3049	. . . . 5 |- ( ph -> ( E. s e. NN0 A. x e. NN0 ( s < x -> -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. ) <-> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) ) )
26	13, 25	mpbird 247	. . . 4 |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. ) )
27		rabssnn0fi 12785	. . . 4 |- ( { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } e. Fin <-> E. s e. NN0 A. x e. NN0 ( s < x -> -. ( ( k e. NN0 |-> C ) ` x ) =/= .0. ) )
28	26, 27	sylibr 224	. . 3 |- ( ph -> { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } e. Fin )
29	12, 28	eqeltrd 2701	. 2 |- ( ph -> ( ( k e. NN0 |-> C ) supp .0. ) e. Fin )
30		funmpt 5926	. . . 4 |- Fun ( k e. NN0 |-> C )
31	30	a1i 11	. . 3 |- ( ph -> Fun ( k e. NN0 |-> C ) )
32	6	mptex 6486	. . . 4 |- ( k e. NN0 |-> C ) e. _V
33	32	a1i 11	. . 3 |- ( ph -> ( k e. NN0 |-> C ) e. _V )
34		funisfsupp 8280	. . 3 |- ( ( Fun ( k e. NN0 |-> C ) /\ ( k e. NN0 |-> C ) e. _V /\ .0. e. _V ) -> ( ( k e. NN0 |-> C ) finSupp .0. <-> ( ( k e. NN0 |-> C ) supp .0. ) e. Fin ) )
35	31, 33, 10, 34	syl3anc 1326	. 2 |- ( ph -> ( ( k e. NN0 |-> C ) finSupp .0. <-> ( ( k e. NN0 |-> C ) supp .0. ) e. Fin ) )
36	29, 35	mpbird 247	1 |- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   {crab 2916   _Vcvv 3200   [_csb 3533   class class class wbr 4653   |-> cmpt 4729   Fun wfun 5882   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   supp csupp 7295   Fincfn 7955   finSupp cfsupp 8275   < clt 10074   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-fal 1489  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-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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  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:  mptnn0fsuppd  12798  mptcoe1fsupp  19585  mptcoe1matfsupp  20607  pm2mp  20630  chfacffsupp  20661  chfacfscmulfsupp  20664  chfacfpmmulfsupp  20668  cayhamlem4  20693  ply1mulgsumlem3  42176  ply1mulgsumlem4  42177