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Theorem coe1subfv 18181
Description: A particular coefficient of a subtraction. (Contributed by Stefan O'Rear, 23-Mar-2015.)
Hypotheses
Ref Expression
coe1sub.y  |-  Y  =  (Poly1 `  R )
coe1sub.b  |-  B  =  ( Base `  Y
)
coe1sub.p  |-  .-  =  ( -g `  Y )
coe1sub.q  |-  N  =  ( -g `  R
)
Assertion
Ref Expression
coe1subfv  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
 X )  =  ( ( (coe1 `  F
) `  X ) N ( (coe1 `  G
) `  X )
) )

Proof of Theorem coe1subfv
StepHypRef Expression
1 simpl1 1000 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Ring )
2 coe1sub.y . . . . . . . . 9  |-  Y  =  (Poly1 `  R )
32ply1ring 18163 . . . . . . . 8  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
4 ringgrp 17077 . . . . . . . 8  |-  ( Y  e.  Ring  ->  Y  e. 
Grp )
53, 4syl 16 . . . . . . 7  |-  ( R  e.  Ring  ->  Y  e. 
Grp )
6 coe1sub.b . . . . . . . 8  |-  B  =  ( Base `  Y
)
7 coe1sub.p . . . . . . . 8  |-  .-  =  ( -g `  Y )
86, 7grpsubcl 15992 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G
)  e.  B )
95, 8syl3an1 1262 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G )  e.  B )
109adantr 465 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( F  .-  G )  e.  B
)
11 simpl3 1002 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  G  e.  B
)
12 simpr 461 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  X  e.  NN0 )
13 eqid 2443 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
14 eqid 2443 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
152, 6, 13, 14coe1addfv 18180 . . . . 5  |-  ( ( ( R  e.  Ring  /\  ( F  .-  G
)  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  (
( F  .-  G
) ( +g  `  Y
) G ) ) `
 X )  =  ( ( (coe1 `  ( F  .-  G ) ) `
 X ) ( +g  `  R ) ( (coe1 `  G ) `  X ) ) )
161, 10, 11, 12, 15syl31anc 1232 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  (
( F  .-  G
) ( +g  `  Y
) G ) ) `
 X )  =  ( ( (coe1 `  ( F  .-  G ) ) `
 X ) ( +g  `  R ) ( (coe1 `  G ) `  X ) ) )
1753ad2ant1 1018 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  Y  e.  Grp )
1817adantr 465 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  Y  e.  Grp )
19 simpl2 1001 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  F  e.  B
)
206, 13, 7grpnpcan 16004 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( ( F  .-  G ) ( +g  `  Y ) G )  =  F )
2118, 19, 11, 20syl3anc 1229 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( F 
.-  G ) ( +g  `  Y ) G )  =  F )
2221fveq2d 5860 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  (coe1 `  ( ( F 
.-  G ) ( +g  `  Y ) G ) )  =  (coe1 `  F ) )
2322fveq1d 5858 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  (
( F  .-  G
) ( +g  `  Y
) G ) ) `
 X )  =  ( (coe1 `  F ) `  X ) )
2416, 23eqtr3d 2486 . . 3  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( (coe1 `  ( F  .-  G
) ) `  X
) ( +g  `  R
) ( (coe1 `  G
) `  X )
)  =  ( (coe1 `  F ) `  X
) )
25 ringgrp 17077 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
26253ad2ant1 1018 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  R  e.  Grp )
2726adantr 465 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Grp )
28 eqid 2443 . . . . . . 7  |-  (coe1 `  F
)  =  (coe1 `  F
)
29 eqid 2443 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
3028, 6, 2, 29coe1f 18124 . . . . . 6  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
31303ad2ant2 1019 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
3231ffvelrnda 6016 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  F
) `  X )  e.  ( Base `  R
) )
33 eqid 2443 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
3433, 6, 2, 29coe1f 18124 . . . . . 6  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
35343ad2ant3 1020 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
3635ffvelrnda 6016 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  G
) `  X )  e.  ( Base `  R
) )
37 eqid 2443 . . . . . . 7  |-  (coe1 `  ( F  .-  G ) )  =  (coe1 `  ( F  .-  G ) )
3837, 6, 2, 29coe1f 18124 . . . . . 6  |-  ( ( F  .-  G )  e.  B  ->  (coe1 `  ( F  .-  G ) ) : NN0 --> ( Base `  R ) )
399, 38syl 16 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  ( F  .-  G ) ) : NN0 --> ( Base `  R ) )
4039ffvelrnda 6016 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
 X )  e.  ( Base `  R
) )
41 coe1sub.q . . . . 5  |-  N  =  ( -g `  R
)
4229, 14, 41grpsubadd 16000 . . . 4  |-  ( ( R  e.  Grp  /\  ( ( (coe1 `  F
) `  X )  e.  ( Base `  R
)  /\  ( (coe1 `  G ) `  X
)  e.  ( Base `  R )  /\  (
(coe1 `  ( F  .-  G ) ) `  X )  e.  (
Base `  R )
) )  ->  (
( ( (coe1 `  F
) `  X ) N ( (coe1 `  G
) `  X )
)  =  ( (coe1 `  ( F  .-  G
) ) `  X
)  <->  ( ( (coe1 `  ( F  .-  G
) ) `  X
) ( +g  `  R
) ( (coe1 `  G
) `  X )
)  =  ( (coe1 `  F ) `  X
) ) )
4327, 32, 36, 40, 42syl13anc 1231 . . 3  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( ( (coe1 `  F ) `  X ) N ( (coe1 `  G ) `  X ) )  =  ( (coe1 `  ( F  .-  G ) ) `  X )  <->  ( (
(coe1 `  ( F  .-  G ) ) `  X ) ( +g  `  R ) ( (coe1 `  G ) `  X
) )  =  ( (coe1 `  F ) `  X ) ) )
4424, 43mpbird 232 . 2  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( (coe1 `  F ) `  X
) N ( (coe1 `  G ) `  X
) )  =  ( (coe1 `  ( F  .-  G ) ) `  X ) )
4544eqcomd 2451 1  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  ( F  .-  G ) ) `
 X )  =  ( ( (coe1 `  F
) `  X ) N ( (coe1 `  G
) `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804   -->wf 5574   ` cfv 5578  (class class class)co 6281   NN0cn0 10801   Basecbs 14509   +g cplusg 14574   Grpcgrp 15927   -gcsg 15929   Ringcrg 17072  Poly1cpl1 18090  coe1cco1 18091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-iin 4318  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-of 6525  df-ofr 6526  df-om 6686  df-1st 6785  df-2nd 6786  df-supp 6904  df-recs 7044  df-rdg 7078  df-1o 7132  df-2o 7133  df-oadd 7136  df-er 7313  df-map 7424  df-pm 7425  df-ixp 7472  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-fsupp 7832  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-nn 10543  df-2 10600  df-3 10601  df-4 10602  df-5 10603  df-6 10604  df-7 10605  df-8 10606  df-9 10607  df-10 10608  df-n0 10802  df-z 10871  df-uz 11091  df-fz 11682  df-fzo 11804  df-seq 12087  df-hash 12385  df-struct 14511  df-ndx 14512  df-slot 14513  df-base 14514  df-sets 14515  df-ress 14516  df-plusg 14587  df-mulr 14588  df-sca 14590  df-vsca 14591  df-tset 14593  df-ple 14594  df-0g 14716  df-gsum 14717  df-mre 14860  df-mrc 14861  df-acs 14863  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-mhm 15840  df-submnd 15841  df-grp 15931  df-minusg 15932  df-sbg 15933  df-mulg 15934  df-subg 16072  df-ghm 16139  df-cntz 16229  df-cmn 16674  df-abl 16675  df-mgp 17016  df-ur 17028  df-ring 17074  df-subrg 17301  df-psr 17879  df-mpl 17881  df-opsr 17883  df-psr1 18093  df-ply1 18095  df-coe1 18096
This theorem is referenced by:  deg1sublt  22384  ply1remlem  22436
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