Theorem mptnn0fsuppr 12242

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 7914	. . . 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 2813	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. NN0 C e. B )
5		eqid 2461	. . . . . . . . . . . . . . 15 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
6	5	fnmpt 5725	. . . . . . . . . . . . . 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 10903	. . . . . . . . . . . . . 14 |- NN0 e. _V
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> NN0 e. _V )
10		mptnn0fsupp.0	. . . . . . . . . . . . . 14 |- ( ph -> .0. e. V )
11		elex 3065	. . . . . . . . . . . . . 14 |- ( .0. e. V -> .0. e. _V )
12	10, 11	syl 17	. . . . . . . . . . . . 13 |- ( ph -> .0. e. _V )
13	7, 9, 12	3jca 1194	. . . . . . . . . . . 12 |- ( ph -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
14	13	adantr 471	. . . . . . . . . . 11 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
15		suppvalfn 6947	. . . . . . . . . . 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 467	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> x e. NN0 )
18	4	adantr 471	. . . . . . . . . . . . . . 15 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> A. k e. NN0 C e. B )
19	18	adantr 471	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> A. k e. NN0 C e. B )
20		rspcsbela 3806	. . . . . . . . . . . . . 14 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
21	17, 19, 20	syl2anc 671	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
22	5	fvmpts 5973	. . . . . . . . . . . . 13 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
23	17, 21, 22	syl2anc 671	. . . . . . . . . . . 12 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
24	23	neeq1d 2694	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( ( k e. NN0 |-> C ) ` x ) =/= .0. <-> [_ x / k ]_ C =/= .0. ) )
25	24	rabbidva 3046	. . . . . . . . . 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 2495	. . . . . . . . 9 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | [_ x / k ]_ C =/= .0. } )
27	26	eleq1d 2523	. . . . . . . 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 212	. . . . . . 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 441	. . . . . 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 81	. . . . 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 435	. . . 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 37	. 2 |- ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin )
34		rabssnn0fi 12229	. . 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 2638	. . . . . 6 |- ( -. [_ x / k ]_ C =/= .0. <-> [_ x / k ]_ C = .0. )
36	35	imbi2i 318	. . . . 5 |- ( ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> ( s < x -> [_ x / k ]_ C = .0. ) )
37	36	ralbii 2830	. . . 4 |- ( A. x e. NN0 ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
38	37	rexbii 2900	. . 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 257	. 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 201	1 |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 375   /\ w3a 991   = wceq 1454   e. wcel 1897   =/= wne 2632   A.wral 2748   E.wrex 2749   {crab 2752   _Vcvv 3056   [_csb 3374   class class class wbr 4415   |-> cmpt 4474   Fun wfun 5594   Fn wfn 5595   ` cfv 5600  (class class class)co 6314   supp csupp 6940   Fincfn 7594   finSupp cfsupp 7908   < clt 9700   NN0cn0 10897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621  ax-1cn 9622  ax-icn 9623  ax-addcl 9624  ax-addrcl 9625  ax-mulcl 9626  ax-mulrcl 9627  ax-mulcom 9628  ax-addass 9629  ax-mulass 9630  ax-distr 9631  ax-i2m1 9632  ax-1ne0 9633  ax-1rid 9634  ax-rnegex 9635  ax-rrecex 9636  ax-cnre 9637  ax-pre-lttri 9638  ax-pre-lttrn 9639  ax-pre-ltadd 9640  ax-pre-mulgt0 9641

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  df-3an 993  df-tru 1457  df-fal 1460  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-nel 2635  df-ral 2753  df-rex 2754  df-reu 2755  df-rmo 2756  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-riota 6276  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-1st 6819  df-2nd 6820  df-supp 6941  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-er 7388  df-en 7595  df-dom 7596  df-sdom 7597  df-fin 7598  df-fsupp 7909  df-pnf 9702  df-mnf 9703  df-xr 9704  df-ltxr 9705  df-le 9706  df-sub 9887  df-neg 9888  df-nn 10637  df-n0 10898  df-z 10966  df-uz 11188  df-fz 11813

This theorem is referenced by:  cpmidpmatlem3  19944