Theorem mptnn0fsuppr 12090

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 7827	. . . 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 2868	. . . . . . . . . . . . . 14 |- ( ph -> A. k e. NN0 C e. B )
5		eqid 2454	. . . . . . . . . . . . . . 15 |- ( k e. NN0 |-> C ) = ( k e. NN0 |-> C )
6	5	fnmpt 5689	. . . . . . . . . . . . . 14 |- ( A. k e. NN0 C e. B -> ( k e. NN0 |-> C ) Fn NN0 )
7	4, 6	syl 16	. . . . . . . . . . . . 13 |- ( ph -> ( k e. NN0 |-> C ) Fn NN0 )
8		nn0ex 10797	. . . . . . . . . . . . . 14 |- NN0 e. _V
9	8	a1i 11	. . . . . . . . . . . . 13 |- ( ph -> NN0 e. _V )
10		mptnn0fsupp.0	. . . . . . . . . . . . . 14 |- ( ph -> .0. e. V )
11		elex 3115	. . . . . . . . . . . . . 14 |- ( .0. e. V -> .0. e. _V )
12	10, 11	syl 16	. . . . . . . . . . . . 13 |- ( ph -> .0. e. _V )
13	7, 9, 12	3jca 1174	. . . . . . . . . . . 12 |- ( ph -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
14	13	adantr 463	. . . . . . . . . . 11 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) Fn NN0 /\ NN0 e. _V /\ .0. e. _V ) )
15		suppvalfn 6898	. . . . . . . . . . 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 16	. . . . . . . . . 10 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> ( ( k e. NN0 |-> C ) supp .0. ) = { x e. NN0 | ( ( k e. NN0 |-> C ) ` x ) =/= .0. } )
17		simpr 459	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> x e. NN0 )
18	4	adantr 463	. . . . . . . . . . . . . . 15 |- ( ( ph /\ Fun ( k e. NN0 |-> C ) ) -> A. k e. NN0 C e. B )
19	18	adantr 463	. . . . . . . . . . . . . 14 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> A. k e. NN0 C e. B )
20		rspcsbela 3845	. . . . . . . . . . . . . 14 |- ( ( x e. NN0 /\ A. k e. NN0 C e. B ) -> [_ x / k ]_ C e. B )
21	17, 19, 20	syl2anc 659	. . . . . . . . . . . . 13 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> [_ x / k ]_ C e. B )
22	5	fvmpts 5933	. . . . . . . . . . . . 13 |- ( ( x e. NN0 /\ [_ x / k ]_ C e. B ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
23	17, 21, 22	syl2anc 659	. . . . . . . . . . . 12 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( k e. NN0 |-> C ) ` x ) = [_ x / k ]_ C )
24	23	neeq1d 2731	. . . . . . . . . . 11 |- ( ( ( ph /\ Fun ( k e. NN0 |-> C ) ) /\ x e. NN0 ) -> ( ( ( k e. NN0 |-> C ) ` x ) =/= .0. <-> [_ x / k ]_ C =/= .0. ) )
25	24	rabbidva 3097	. . . . . . . . . 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 207	. . . . . . 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 433	. . . . . 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 78	. . . . 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 427	. . . 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 16	. . 3 |- ( ( k e. NN0 |-> C ) finSupp .0. -> ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin ) )
33	1, 32	mpcom 36	. 2 |- ( ph -> { x e. NN0 | [_ x / k ]_ C =/= .0. } e. Fin )
34		rabssnn0fi 12080	. . 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 2655	. . . . . 6 |- ( -. [_ x / k ]_ C =/= .0. <-> [_ x / k ]_ C = .0. )
36	35	imbi2i 310	. . . . 5 |- ( ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> ( s < x -> [_ x / k ]_ C = .0. ) )
37	36	ralbii 2885	. . . 4 |- ( A. x e. NN0 ( s < x -> -. [_ x / k ]_ C =/= .0. ) <-> A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )
38	37	rexbii 2956	. . 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 249	. 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 196	1 |- ( ph -> E. s e. NN0 A. x e. NN0 ( s < x -> [_ x / k ]_ C = .0. ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 367   /\ w3a 971   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   {crab 2808   _Vcvv 3106   [_csb 3420   class class class wbr 4439   |-> cmpt 4497   Fun wfun 5564   Fn wfn 5565   ` cfv 5570  (class class class)co 6270   supp csupp 6891   Fincfn 7509   finSupp cfsupp 7821   < clt 9617   NN0cn0 10791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676

This theorem is referenced by:  cpmidpmatlem3  19543