Theorem mptnn0fsuppr 12799

Description: A finitely supported mapping from the nonnegative integers fulfills certain conditions. (Contributed by AV, 3-Nov-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 )
mptnn0fsuppr.s	|- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )

Assertion

Ref	Expression
mptnn0fsuppr	|- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .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 mptnn0fsuppr

Step	Hyp	Ref	Expression
1		mptnn0fsuppr.s	. . 3 |- ( ph -> ( k e. NN0 |-> C ) finSupp .0. )
2		fsuppimp 8281	. . . 4 |- ( ( k e. NN0 |-> C ) finSupp .0. -> ( Fun ( k e. NN0 |-> C ) /\ ( ( k e. NN0 |-> C ) supp .0. ) e. Fin ) )
3		mptnn0fsupp.c	. . . . . . . . . . . . . . 15 |- ( ( ph /\ k e. NN0 ) -> C e. B )
4	3	ralrimiva 2966	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. NN0 C e. B )
5		eqid 2622	. . . . . . . . . . . . . . 15 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
6	5	fnmpt 6020	. . . . . . . . . . . . . 14 |- ( A. k e. NN0 C e. B -> ( k e. NN0 |-> C ) Fn NN0 )
7	4, 6	syl 17	. . . . . . . . . . . . 13 |- ( ph -> ( k e. NN0 |-> C ) Fn NN0 )
8		nn0ex 11298	. . . . . . . . . . . . . 14 |- NN0 e. _V
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> NN0 e. _V )
10		mptnn0fsupp.0	. . . . . . . . . . . . . 14 |- ( ph -> .0. e. V )
11		elex 3212	. . . . . . . . . . . . . 14 |- ( .0. e. V -> .0. e. _V )
12	10, 11	syl 17	. . . . . . . . . . . . 13 |- ( ph -> .0. e. _V )
13	7, 9, 12	3jca 1242	. . . . . . . . . . . 12 |- ( ph -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
14	13	adantr 481	. . . . . . . . . . 11 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
15		suppvalfn 7302	. . . . . . . . . . 11 |- ( ( ( 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. } )
16	14, 15	syl 17	. . . . . . . . . 10 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } )
17		simpr 477	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> x e. NN0 )
18	4	adantr 481	. . . . . . . . . . . . . . 15 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> A. k e. NN0 C e. B )
19	18	adantr 481	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> A. k e. NN0 C e. B )
20		rspcsbela 4006	. . . . . . . . . . . . . 14 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
21	17, 19, 20	syl2anc 693	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
22	5	fvmpts 6285	. . . . . . . . . . . . 13 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
23	17, 21, 22	syl2anc 693	. . . . . . . . . . . 12 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
24	23	neeq1d 2853	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( ( k e. NN0 |-> C ) ` x ) =/= .0. <-> [_ x / k ]_ C =/= .0. ) )
25	24	rabbidva 3188	. . . . . . . . . 10 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } = { x e. NN0 | [_ x / k ]_ C =/= .0. } )
26	16, 25	eqtrd 2656	. . . . . . . . 9 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | [_ x / k ]_ C =/= .0. } )
27	26	eleq1d 2686	. . . . . . . 8 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( ( k e. NN0 |-> C ) supp .0. ) e. Fin <-> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) )
28	27	biimpd 219	. . . . . . 7 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( ( k e. NN0 |-> C ) supp .0. ) e. Fin -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) )
29	28	expcom 451	. . . . . 6 |- ( Fun ( k e. NN0 |-> C ) -> ( ph -> ( ( ( k e. NN0 |-> C ) supp .0. ) e. Fin -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) ) )
30	29	com23 86	. . . . 5 |- ( Fun ( k e. NN0 |-> C ) -> ( ( ( k e. NN0 |-> C ) supp .0. ) e. Fin -> ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) ) )
31	30	imp 445	. . . 4 |- ( ( Fun ( k e. NN0 |-> C ) /\ ( ( k e. NN0 |-> C ) supp .0. ) e. Fin ) -> ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) )
32	2, 31	syl 17	. . 3 |- ( ( k e. NN0 |-> C ) finSupp .0. -> ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) )
33	1, 32	mpcom 38	. 2 |- ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin )
34		rabssnn0fi 12785	. . 3 |- ( { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin <-> E. s e. NN0 A. x e. NN0 ( s < x -> -. [_ x / k ]_ C =/= .0. ) )
35		nne 2798	. . . . . 6 |- ( -. [_ x / k ]_ C =/= .0. <-> [_ x / k ]_ C = .0. )
36	35	imbi2i 326	. . . . 5 |- ( ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> ( s < x -> [_ x / k ]_ C = .0. ) )
37	36	ralbii 2980	. . . 4 |- ( A. x e. NN0 ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
38	37	rexbii 3041	. . 3 |- ( E. s e. NN0 A. x e. NN0 ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
39	34, 38	bitri 264	. 2 |- ( { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin <-> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
40	33, 39	sylib 208	1 |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  cpmidpmatlem3  20677