Theorem pl1cn 30001

Description: A univariate polynomial is continuous. (Contributed by Thierry Arnoux, 17-Sep-2018.)

Hypotheses

Ref	Expression
pl1cn.p	|- P = (Poly1 ` R )
pl1cn.e	|- E = (eval1 ` R )
pl1cn.b	|- B = ( Base ` P )
pl1cn.k	|- K = ( Base ` R )
pl1cn.j	|- J = ( TopOpen ` R )
pl1cn.1	|- ( ph -> R e. CRing )
pl1cn.2	|- ( ph -> R e. TopRing )
pl1cn.3	|- ( ph -> F e. B )

Assertion

Ref	Expression
pl1cn	|- ( ph -> ( E ` F ) e. ( J Cn J ) )

Proof of Theorem pl1cn

Dummy variables h f g x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pl1cn.k	. 2 |- K = ( Base ` R )
2		eqid 2622	. 2 |- ( +g ` R ) = ( +g ` R )
3		eqid 2622	. 2 |- ( .r ` R ) = ( .r ` R )
4		eqid 2622	. 2 |- ran (eval1 ` R ) = ran (eval1 ` R )
5		fvex 6201	. . . . . . . . 9 |- ( Base ` R ) e. _V
6	1, 5	eqeltri 2697	. . . . . . . 8 |- K e. _V
7	6	a1i 11	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> K e. _V )
8		fvexd 6203	. . . . . . 7 |- ( ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) /\ x e. K ) -> ( f ` x ) e. _V )
9		fvexd 6203	. . . . . . 7 |- ( ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) /\ x e. K ) -> ( g ` x ) e. _V )
10		simp1 1061	. . . . . . . 8 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ph )
11		eqid 2622	. . . . . . . . . . 11 |- U. J = U. J
12	11, 11	cnf 21050	. . . . . . . . . 10 |- ( f e. ( J Cn J ) -> f : U. J --> U. J )
13		ffn 6045	. . . . . . . . . 10 |- ( f : U. J --> U. J -> f Fn U. J )
14	12, 13	syl 17	. . . . . . . . 9 |- ( f e. ( J Cn J ) -> f Fn U. J )
15	14	3ad2ant2 1083	. . . . . . . 8 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> f Fn U. J )
16		dffn5 6241	. . . . . . . . . 10 |- ( f Fn K <-> f = ( x e. K |-> ( f ` x ) ) )
17		pl1cn.2	. . . . . . . . . . . . 13 |- ( ph -> R e. TopRing )
18		trgtgp 21971	. . . . . . . . . . . . 13 |- ( R e. TopRing -> R e. TopGrp )
19		pl1cn.j	. . . . . . . . . . . . . 14 |- J = ( TopOpen ` R )
20	19, 1	tgptopon 21886	. . . . . . . . . . . . 13 |- ( R e. TopGrp -> J e. (TopOn ` K ) )
21	17, 18, 20	3syl 18	. . . . . . . . . . . 12 |- ( ph -> J e. (TopOn ` K ) )
22		toponuni 20719	. . . . . . . . . . . 12 |- ( J e. (TopOn ` K ) -> K = U. J )
23	21, 22	syl 17	. . . . . . . . . . 11 |- ( ph -> K = U. J )
24	23	fneq2d 5982	. . . . . . . . . 10 |- ( ph -> ( f Fn K <-> f Fn U. J ) )
25	16, 24	syl5rbbr 275	. . . . . . . . 9 |- ( ph -> ( f Fn U. J <-> f = ( x e. K |-> ( f ` x ) ) ) )
26	25	biimpa 501	. . . . . . . 8 |- ( ( ph /\ f Fn U. J ) -> f = ( x e. K |-> ( f ` x ) ) )
27	10, 15, 26	syl2anc 693	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> f = ( x e. K |-> ( f ` x ) ) )
28	11, 11	cnf 21050	. . . . . . . . . 10 |- ( g e. ( J Cn J ) -> g : U. J --> U. J )
29		ffn 6045	. . . . . . . . . 10 |- ( g : U. J --> U. J -> g Fn U. J )
30	28, 29	syl 17	. . . . . . . . 9 |- ( g e. ( J Cn J ) -> g Fn U. J )
31	30	3ad2ant3 1084	. . . . . . . 8 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> g Fn U. J )
32		dffn5 6241	. . . . . . . . . 10 |- ( g Fn K <-> g = ( x e. K |-> ( g ` x ) ) )
33	23	fneq2d 5982	. . . . . . . . . 10 |- ( ph -> ( g Fn K <-> g Fn U. J ) )
34	32, 33	syl5rbbr 275	. . . . . . . . 9 |- ( ph -> ( g Fn U. J <-> g = ( x e. K |-> ( g ` x ) ) ) )
35	34	biimpa 501	. . . . . . . 8 |- ( ( ph /\ g Fn U. J ) -> g = ( x e. K |-> ( g ` x ) ) )
36	10, 31, 35	syl2anc 693	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> g = ( x e. K |-> ( g ` x ) ) )
37	7, 8, 9, 27, 36	offval2 6914	. . . . . 6 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( f oF ( +g ` R ) g ) = ( x e. K |-> ( ( f ` x ) ( +g ` R ) ( g ` x ) ) ) )
38	21	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> J e. (TopOn ` K ) )
39		simp2 1062	. . . . . . . 8 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> f e. ( J Cn J ) )
40	27, 39	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( x e. K |-> ( f ` x ) ) e. ( J Cn J ) )
41		simp3 1063	. . . . . . . 8 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> g e. ( J Cn J ) )
42	36, 41	eqeltrrd 2702	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( x e. K |-> ( g ` x ) ) e. ( J Cn J ) )
43		eqid 2622	. . . . . . . . . 10 |- ( +f ` R ) = ( +f ` R )
44	1, 2, 43	plusffval 17247	. . . . . . . . 9 |- ( +f ` R ) = ( y e. K , z e. K |-> ( y ( +g ` R ) z ) )
45	19, 43	tgpcn 21888	. . . . . . . . . 10 |- ( R e. TopGrp -> ( +f ` R ) e. ( ( J tX J ) Cn J ) )
46	17, 18, 45	3syl 18	. . . . . . . . 9 |- ( ph -> ( +f ` R ) e. ( ( J tX J ) Cn J ) )
47	44, 46	syl5eqelr 2706	. . . . . . . 8 |- ( ph -> ( y e. K , z e. K |-> ( y ( +g ` R ) z ) ) e. ( ( J tX J ) Cn J ) )
48	47	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( y e. K , z e. K |-> ( y ( +g ` R ) z ) ) e. ( ( J tX J ) Cn J ) )
49		oveq12 6659	. . . . . . 7 |- ( ( y = ( f ` x ) /\ z = ( g ` x ) ) -> ( y ( +g ` R ) z ) = ( ( f ` x ) ( +g ` R ) ( g ` x ) ) )
50	38, 40, 42, 38, 38, 48, 49	cnmpt12 21470	. . . . . 6 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( x e. K |-> ( ( f ` x ) ( +g ` R ) ( g ` x ) ) ) e. ( J Cn J ) )
51	37, 50	eqeltrd 2701	. . . . 5 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( f oF ( +g ` R ) g ) e. ( J Cn J ) )
52	51	3adant2l 1320	. . . 4 |- ( ( ph /\ ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ g e. ( J Cn J ) ) -> ( f oF ( +g ` R ) g ) e. ( J Cn J ) )
53	52	3adant3l 1322	. . 3 |- ( ( ph /\ ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ ( g e. ran (eval1 ` R ) /\ g e. ( J Cn J ) ) ) -> ( f oF ( +g ` R ) g ) e. ( J Cn J ) )
54	53	3expb 1266	. 2 |- ( ( ph /\ ( ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ ( g e. ran (eval1 ` R ) /\ g e. ( J Cn J ) ) ) ) -> ( f oF ( +g ` R ) g ) e. ( J Cn J ) )
55	7, 8, 9, 27, 36	offval2 6914	. . . . . 6 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( f oF ( .r ` R ) g ) = ( x e. K |-> ( ( f ` x ) ( .r ` R ) ( g ` x ) ) ) )
56		eqid 2622	. . . . . . . . . . 11 |- (mulGrp ` R ) = (mulGrp ` R )
57	56, 1	mgpbas 18495	. . . . . . . . . 10 |- K = ( Base ` (mulGrp ` R ) )
58	56, 3	mgpplusg 18493	. . . . . . . . . 10 |- ( .r ` R ) = ( +g ` (mulGrp ` R ) )
59		eqid 2622	. . . . . . . . . 10 |- ( +f ` (mulGrp ` R ) ) = ( +f ` (mulGrp ` R ) )
60	57, 58, 59	plusffval 17247	. . . . . . . . 9 |- ( +f ` (mulGrp ` R ) ) = ( y e. K , z e. K |-> ( y ( .r ` R ) z ) )
61	19, 59	mulrcn 21982	. . . . . . . . . 10 |- ( R e. TopRing -> ( +f ` (mulGrp ` R ) ) e. ( ( J tX J ) Cn J ) )
62	17, 61	syl 17	. . . . . . . . 9 |- ( ph -> ( +f ` (mulGrp ` R ) ) e. ( ( J tX J ) Cn J ) )
63	60, 62	syl5eqelr 2706	. . . . . . . 8 |- ( ph -> ( y e. K , z e. K |-> ( y ( .r ` R ) z ) ) e. ( ( J tX J ) Cn J ) )
64	63	3ad2ant1 1082	. . . . . . 7 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( y e. K , z e. K |-> ( y ( .r ` R ) z ) ) e. ( ( J tX J ) Cn J ) )
65		oveq12 6659	. . . . . . 7 |- ( ( y = ( f ` x ) /\ z = ( g ` x ) ) -> ( y ( .r ` R ) z ) = ( ( f ` x ) ( .r ` R ) ( g ` x ) ) )
66	38, 40, 42, 38, 38, 64, 65	cnmpt12 21470	. . . . . 6 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( x e. K |-> ( ( f ` x ) ( .r ` R ) ( g ` x ) ) ) e. ( J Cn J ) )
67	55, 66	eqeltrd 2701	. . . . 5 |- ( ( ph /\ f e. ( J Cn J ) /\ g e. ( J Cn J ) ) -> ( f oF ( .r ` R ) g ) e. ( J Cn J ) )
68	67	3adant2l 1320	. . . 4 |- ( ( ph /\ ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ g e. ( J Cn J ) ) -> ( f oF ( .r ` R ) g ) e. ( J Cn J ) )
69	68	3adant3l 1322	. . 3 |- ( ( ph /\ ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ ( g e. ran (eval1 ` R ) /\ g e. ( J Cn J ) ) ) -> ( f oF ( .r ` R ) g ) e. ( J Cn J ) )
70	69	3expb 1266	. 2 |- ( ( ph /\ ( ( f e. ran (eval1 ` R ) /\ f e. ( J Cn J ) ) /\ ( g e. ran (eval1 ` R ) /\ g e. ( J Cn J ) ) ) ) -> ( f oF ( .r ` R ) g ) e. ( J Cn J ) )
71		eleq1 2689	. 2 |- ( h = ( K X. { f } ) -> ( h e. ( J Cn J ) <-> ( K X. { f } ) e. ( J Cn J ) ) )
72		eleq1 2689	. 2 |- ( h = ( _I |` K ) -> ( h e. ( J Cn J ) <-> ( _I |` K ) e. ( J Cn J ) ) )
73		eleq1 2689	. 2 |- ( h = f -> ( h e. ( J Cn J ) <-> f e. ( J Cn J ) ) )
74		eleq1 2689	. 2 |- ( h = g -> ( h e. ( J Cn J ) <-> g e. ( J Cn J ) ) )
75		eleq1 2689	. 2 |- ( h = ( f oF ( +g ` R ) g ) -> ( h e. ( J Cn J ) <-> ( f oF ( +g ` R ) g ) e. ( J Cn J ) ) )
76		eleq1 2689	. 2 |- ( h = ( f oF ( .r ` R ) g ) -> ( h e. ( J Cn J ) <-> ( f oF ( .r ` R ) g ) e. ( J Cn J ) ) )
77		eleq1 2689	. 2 |- ( h = ( E ` F ) -> ( h e. ( J Cn J ) <-> ( E ` F ) e. ( J Cn J ) ) )
78	21	adantr 481	. . 3 |- ( ( ph /\ f e. K ) -> J e. (TopOn ` K ) )
79		simpr 477	. . 3 |- ( ( ph /\ f e. K ) -> f e. K )
80		cnconst2 21087	. . 3 |- ( ( J e. (TopOn ` K ) /\ J e. (TopOn ` K ) /\ f e. K ) -> ( K X. { f } ) e. ( J Cn J ) )
81	78, 78, 79, 80	syl3anc 1326	. 2 |- ( ( ph /\ f e. K ) -> ( K X. { f } ) e. ( J Cn J ) )
82		idcn 21061	. . 3 |- ( J e. (TopOn ` K ) -> ( _I |` K ) e. ( J Cn J ) )
83	21, 82	syl 17	. 2 |- ( ph -> ( _I |` K ) e. ( J Cn J ) )
84		pl1cn.1	. . . . 5 |- ( ph -> R e. CRing )
85		pl1cn.e	. . . . . . 7 |- E = (eval1 ` R )
86		pl1cn.p	. . . . . . 7 |- P = (Poly1 ` R )
87		eqid 2622	. . . . . . 7 |- ( R ^s K ) = ( R ^s K )
88	85, 86, 87, 1	evl1rhm 19696	. . . . . 6 |- ( R e. CRing -> E e. ( P RingHom ( R ^s K ) ) )
89		pl1cn.b	. . . . . . 7 |- B = ( Base ` P )
90		eqid 2622	. . . . . . 7 |- ( Base ` ( R ^s K ) ) = ( Base ` ( R ^s K ) )
91	89, 90	rhmf 18726	. . . . . 6 |- ( E e. ( P RingHom ( R ^s K ) ) -> E : B --> ( Base ` ( R ^s K ) ) )
92		ffn 6045	. . . . . 6 |- ( E : B --> ( Base ` ( R ^s K ) ) -> E Fn B )
93		dffn3 6054	. . . . . . 7 |- ( E Fn B <-> E : B --> ran E )
94	93	biimpi 206	. . . . . 6 |- ( E Fn B -> E : B --> ran E )
95	88, 91, 92, 94	4syl 19	. . . . 5 |- ( R e. CRing -> E : B --> ran E )
96	84, 95	syl 17	. . . 4 |- ( ph -> E : B --> ran E )
97		pl1cn.3	. . . 4 |- ( ph -> F e. B )
98	96, 97	ffvelrnd 6360	. . 3 |- ( ph -> ( E ` F ) e. ran E )
99	85	rneqi 5352	. . 3 |- ran E = ran (eval1 ` R )
100	98, 99	syl6eleq 2711	. 2 |- ( ph -> ( E ` F ) e. ran (eval1 ` R ) )
101	1, 2, 3, 4, 54, 70, 71, 72, 73, 74, 75, 76, 77, 81, 83, 100	pf1ind 19719	1 |- ( ph -> ( E ` F ) e. ( J Cn J ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   {csn 4177   U.cuni 4436   |-> cmpt 4729   _I cid 5023   X. cxp 5112   ran crn 5115   |` cres 5116   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   oFcof 6895   Basecbs 15857   +g cplusg 15941   .rcmulr 15942   TopOpenctopn 16082   ^_s cpws 16107   +fcplusf 17239  mulGrpcmgp 18489   CRingccrg 18548   RingHom crh 18712  Poly₁cpl1 19547  eval₁ce1 19679  TopOnctopon 20715   Cn ccn 21028   tX ctx 21363   TopGrpctgp 21875   TopRingctrg 21959

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-inf2 8538  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-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-int 4476  df-iun 4522  df-iin 4523  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  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-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  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-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-plusf 17241  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-evl 19507  df-psr1 19550  df-ply1 19552  df-evl1 19681  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cn 21031  df-cnp 21032  df-tx 21365  df-tmd 21876  df-tgp 21877  df-trg 21963

This theorem is referenced by: (None)