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

Theorem psrrng 17830
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 2461 . 2  |-  ( ph  ->  ( Base `  S
)  =  ( Base `  S ) )
2 eqidd 2461 . 2  |-  ( ph  ->  ( +g  `  S
)  =  ( +g  `  S ) )
3 eqidd 2461 . 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 16984 . . . 4  |-  ( R  e.  Ring  ->  R  e. 
Grp )
86, 7syl 16 . . 3  |-  ( ph  ->  R  e.  Grp )
94, 5, 8psrgrp 17815 . 2  |-  ( ph  ->  S  e.  Grp )
10 eqid 2460 . . 3  |-  ( Base `  S )  =  (
Base `  S )
11 eqid 2460 . . 3  |-  ( .r
`  S )  =  ( .r `  S
)
1263ad2ant1 1012 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  R  e.  Ring )
13 simp2 992 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  x  e.  (
Base `  S )
)
14 simp3 993 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  y  e.  (
Base `  S )
)
154, 10, 11, 12, 13, 14psrmulcl 17805 . 2  |-  ( (
ph  /\  x  e.  ( Base `  S )  /\  y  e.  ( Base `  S ) )  ->  ( x ( .r `  S ) y )  e.  (
Base `  S )
)
165adantr 465 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  I  e.  V )
176adantr 465 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  R  e.  Ring )
18 eqid 2460 . . 3  |-  { f  e.  ( NN0  ^m  I )  |  ( `' f " NN )  e.  Fin }  =  { f  e.  ( NN0  ^m  I )  |  ( `' f
" NN )  e. 
Fin }
19 simpr1 997 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  x  e.  ( Base `  S )
)
20 simpr2 998 . . 3  |-  ( (
ph  /\  ( x  e.  ( Base `  S
)  /\  y  e.  ( Base `  S )  /\  z  e.  ( Base `  S ) ) )  ->  y  e.  ( Base `  S )
)
21 simpr3 999 . . 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 17824 . 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 2460 . . 3  |-  ( +g  `  S )  =  ( +g  `  S )
244, 16, 17, 18, 11, 10, 19, 20, 21, 23psrdi 17825 . 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 17826 . 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 2460 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
27 eqid 2460 . . 3  |-  ( 1r
`  R )  =  ( 1r `  R
)
28 eqid 2460 . . 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 17819 . 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 465 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  I  e.  V )
316adantr 465 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  R  e.  Ring )
32 simpr 461 . . 3  |-  ( (
ph  /\  x  e.  ( Base `  S )
)  ->  x  e.  ( Base `  S )
)
334, 30, 31, 18, 26, 27, 28, 10, 11, 32psrlidm 17820 . 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 17822 . 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 17013 1  |-  ( ph  ->  S  e.  Ring )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   {crab 2811   ifcif 3932   {csn 4020    |-> cmpt 4498    X. cxp 4990   `'ccnv 4991   "cima 4995   ` cfv 5579  (class class class)co 6275    ^m cmap 7410   Fincfn 7506   0cc0 9481   NNcn 10525   NN0cn0 10784   Basecbs 14479   +g cplusg 14544   .rcmulr 14545   0gc0g 14684   Grpcgrp 15716   1rcur 16936   Ringcrg 16979   mPwSer cmps 17764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  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 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-ofr 6516  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-map 7412  df-pm 7413  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-oi 7924  df-card 8309  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-n0 10785  df-z 10854  df-uz 11072  df-fz 11662  df-fzo 11782  df-seq 12064  df-hash 12361  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-sca 14560  df-vsca 14561  df-tset 14563  df-0g 14686  df-gsum 14687  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-mhm 15770  df-submnd 15771  df-grp 15851  df-minusg 15852  df-mulg 15854  df-ghm 16053  df-cntz 16143  df-cmn 16589  df-abl 16590  df-mgp 16925  df-ur 16937  df-rng 16981  df-psr 17769
This theorem is referenced by:  psr1  17831  psrcrng  17832  psrassa  17833  subrgpsr  17838  mplsubrg  17866  opsrrng  18050
  Copyright terms: Public domain W3C validator