Theorem ulmuni 24146

Description: An sequence of functions uniformly converges to at most one limit. (Contributed by Mario Carneiro, 5-Jul-2017.)

Assertion

Ref	Expression
ulmuni	|- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> G = H )

Proof of Theorem ulmuni

Dummy variables i k n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ulmcl 24135	. . . 4 |- ( F ( ~~> u ` S ) G -> G : S --> CC )
2	1	adantr 481	. . 3 |- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> G : S --> CC )
3		ffn 6045	. . 3 |- ( G : S --> CC -> G Fn S )
4	2, 3	syl 17	. 2 |- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> G Fn S )
5		ulmcl 24135	. . . 4 |- ( F ( ~~> u ` S ) H -> H : S --> CC )
6	5	adantl 482	. . 3 |- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> H : S --> CC )
7		ffn 6045	. . 3 |- ( H : S --> CC -> H Fn S )
8	6, 7	syl 17	. 2 |- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> H Fn S )
9		eqid 2622	. . . . 5 |- ( ZZ>= ` n ) = ( ZZ>= ` n )
10		simplr 792	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> n e. ZZ )
11		simpr 477	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F : ( ZZ>= ` n ) --> ( CC ^m S ) )
12		simpllr 799	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> x e. S )
13		fvex 6201	. . . . . . 7 |- ( ZZ>= ` n ) e. _V
14	13	mptex 6486	. . . . . 6 |- ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) e. _V
15	14	a1i 11	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) e. _V )
16		fveq2 6191	. . . . . . . . 9 |- ( i = k -> ( F ` i ) = ( F ` k ) )
17	16	fveq1d 6193	. . . . . . . 8 |- ( i = k -> ( ( F ` i ) ` x ) = ( ( F ` k ) ` x ) )
18		eqid 2622	. . . . . . . 8 |- ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) = ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) )
19		fvex 6201	. . . . . . . 8 |- ( ( F ` k ) ` x ) e. _V
20	17, 18, 19	fvmpt 6282	. . . . . . 7 |- ( k e. ( ZZ>= ` n ) -> ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) = ( ( F ` k ) ` x ) )
21	20	eqcomd 2628	. . . . . 6 |- ( k e. ( ZZ>= ` n ) -> ( ( F ` k ) ` x ) = ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) )
22	21	adantl 482	. . . . 5 |- ( ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) /\ k e. ( ZZ>= ` n ) ) -> ( ( F ` k ) ` x ) = ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ` k ) )
23		simp-4l 806	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F ( ~~> u ` S ) G )
24	9, 10, 11, 12, 15, 22, 23	ulmclm 24141	. . . 4 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( G ` x ) )
25		simp-4r 807	. . . . 5 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> F ( ~~> u ` S ) H )
26	9, 10, 11, 12, 15, 22, 25	ulmclm 24141	. . . 4 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( H ` x ) )
27		climuni 14283	. . . 4 |- ( ( ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( G ` x ) /\ ( i e. ( ZZ>= ` n ) |-> ( ( F ` i ) ` x ) ) ~~> ( H ` x ) ) -> ( G ` x ) = ( H ` x ) )
28	24, 26, 27	syl2anc 693	. . 3 |- ( ( ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) /\ n e. ZZ ) /\ F : ( ZZ>= ` n ) --> ( CC ^m S ) ) -> ( G ` x ) = ( H ` x ) )
29		ulmf 24136	. . . 4 |- ( F ( ~~> u ` S ) G -> E. n e. ZZ F : ( ZZ>= ` n ) --> ( CC ^m S ) )
30	29	ad2antrr 762	. . 3 |- ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) -> E. n e. ZZ F : ( ZZ>= ` n ) --> ( CC ^m S ) )
31	28, 30	r19.29a 3078	. 2 |- ( ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) /\ x e. S ) -> ( G ` x ) = ( H ` x ) )
32	4, 8, 31	eqfnfvd 6314	1 |- ( ( F ( ~~> u ` S ) G /\ F ( ~~> u ` S ) H ) -> G = H )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   _Vcvv 3200   class class class wbr 4653   |-> cmpt 4729   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   ZZcz 11377   ZZ>=cuz 11687   ~~> cli 14215   ~~> uculm 24130

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  ax-pre-sup 10014

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-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-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  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-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-ulm 24131

This theorem is referenced by:  ulmdm  24147