MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psrrng Unicode version

Theorem psrrng 16155
Description: The ring of power series is a ring. (Contributed by Mario Carneiro, 29-Dec-2014.)
Hypotheses
Ref Expression
psrrng.s  |-  S  =  ( I mPwSer  R )
psrrng.i  |-  ( ph  ->  I  e.  V )
psrrng.r  |-  ( ph  ->  R  e.  Ring )
Assertion
Ref Expression
psrrng  |-  ( ph  ->  S  e.  Ring )

Proof of Theorem psrrng
Dummy variables  x  f  y  z  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqidd 2284 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2284 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 eqidd 2284 . 2  |-  ( ph  ->  ( .r `  S
)  =  ( .r
`  S ) )
4 psrrng.s . . 3  |-  S  =  ( I mPwSer  R )
5 psrrng.i . . 3  |-  ( ph  ->  I  e.  V )
6 psrrng.r . . . 4  |-  ( ph  ->  R  e.  Ring )
7 rnggrp 15346 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
86, 7syl 15 . . 3  |-  ( ph  ->  R  e.  Grp )
94, 5, 8psrgrp 16143 . 2  |-  ( ph  ->  S  e.  Grp )
10 eqid 2283 . . 3  |-  ( Base `  S )  =  (
Base `  S )
11 eqid 2283 . . 3  |-  ( .r
`  S )  =  ( .r `  S
)
1263ad2ant1 976 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Ring )
13 simp2 956 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
14 simp3 957 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
154, 10, 11, 12, 13, 14psrmulcl 16133 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( .r `  S ) y )  e.  (
Base `  S )
)
165adantr 451 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  I  e.  V )
176adantr 451 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Ring )
18 eqid 2283 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
19 simpr1 961 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
20 simpr2 962 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
21 simpr3 963 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  z  e.  ( Base `  S )
)
224, 16, 17, 18, 11, 10, 19, 20, 21psrass1 16150 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( .r `  S ) y ) ( .r `  S
) z )  =  ( x ( .r
`  S ) ( y ( .r `  S ) z ) ) )
23 eqid 2283 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
244, 16, 17, 18, 11, 10, 19, 20, 21, 23psrdi 16151 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( x
( .r `  S
) ( y ( +g  `  S ) z ) )  =  ( ( x ( .r `  S ) y ) ( +g  `  S ) ( x ( .r `  S
) z ) ) )
254, 16, 17, 18, 11, 10, 19, 20, 21, 23psrdir 16152 . 2  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  ( (
x ( +g  `  S
) y ) ( .r `  S ) z )  =  ( ( x ( .r
`  S ) z ) ( +g  `  S
) ( y ( .r `  S ) z ) ) )
26 eqid 2283 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
27 eqid 2283 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
28 eqid 2283 . . 3  |-  ( r  e.  { f  e.  ( NN0  ^m  I
)  |  ( `' f " NN )  e.  Fin }  |->  if ( r  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) )  =  ( r  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( r  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) )
294, 5, 6, 18, 26, 27, 28, 10psr1cl 16147 . 2  |-  ( ph  ->  ( r  e.  {
f  e.  ( NN0 
^m  I )  |  ( `' f " NN )  e.  Fin } 
|->  if ( r  =  ( I  X.  {
0 } ) ,  ( 1r `  R
) ,  ( 0g
`  R ) ) )  e.  ( Base `  S ) )
305adantr 451 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
316adantr 451 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Ring )
32 simpr 447 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
334, 30, 31, 18, 26, 27, 28, 10, 11, 32psrlidm 16148 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( (
r  e.  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  |->  if ( r  =  ( I  X.  { 0 } ) ,  ( 1r `  R ) ,  ( 0g `  R ) ) ) ( .r `  S
) x )  =  x )
344, 30, 31, 18, 26, 27, 28, 10, 11, 32psrridm 16149 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  ( x
( .r `  S
) ( r  e. 
{ f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }  |->  if ( r  =  ( I  X.  { 0 } ) ,  ( 1r
`  R ) ,  ( 0g `  R
) ) ) )  =  x )
351, 2, 3, 9, 15, 22, 24, 25, 29, 33, 34isrngd 15375 1  |-  ( ph  ->  S  e.  Ring )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   {crab 2547   ifcif 3565   {csn 3640    e. cmpt 4077    X. cxp 4687   `'ccnv 4688   "cima 4692   ` cfv 5255  (class class class)co 5858    ^m cmap 6772   Fincfn 6863   0cc0 8737   NNcn 9746   NN0cn0 9965   Basecbs 13148   +g cplusg 13208   .rcmulr 13209   0gc0g 13400   Grpcgrp 14362   Ringcrg 15337   1rcur 15339   mPwSer cmps 16087
This theorem is referenced by:  psr1  16156  psrcrng  16157  psrassa  16158  subrgpsr  16163  mplsubrg  16184  opsrrng  16323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-ofr 6079  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-pm 6775  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-fzo 10871  df-seq 11047  df-hash 11338  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-sca 13224  df-vsca 13225  df-tset 13227  df-0g 13404  df-gsum 13405  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-mhm 14415  df-submnd 14416  df-grp 14489  df-minusg 14490  df-mulg 14492  df-ghm 14681  df-cntz 14793  df-cmn 15091  df-abl 15092  df-mgp 15326  df-rng 15340  df-ur 15342  df-psr 16098
  Copyright terms: Public domain W3C validator