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Theorem coe1subfv 16651
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 960 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Ring )
2 coe1sub.y . . . . . . . . 9  |-  Y  =  (Poly1 `  R )
32ply1rng 16634 . . . . . . . 8  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
4 rnggrp 15661 . . . . . . . 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 14861 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G
)  e.  B )
95, 8syl3an1 1217 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G )  e.  B )
109adantr 452 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( F  .-  G )  e.  B
)
11 simpl3 962 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  G  e.  B
)
12 simpr 448 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  X  e.  NN0 )
13 eqid 2435 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
14 eqid 2435 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
152, 6, 13, 14coe1addfv 16650 . . . . 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 1187 . . . 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 978 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  Y  e.  Grp )
1817adantr 452 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  Y  e.  Grp )
19 simpl2 961 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  F  e.  B
)
206, 13, 7grpnpcan 14872 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( ( F  .-  G ) ( +g  `  Y ) G )  =  F )
2118, 19, 11, 20syl3anc 1184 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( F 
.-  G ) ( +g  `  Y ) G )  =  F )
2221fveq2d 5724 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  (coe1 `  ( ( F 
.-  G ) ( +g  `  Y ) G ) )  =  (coe1 `  F ) )
2322fveq1d 5722 . . . 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 2469 . . 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 rnggrp 15661 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
26253ad2ant1 978 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  R  e.  Grp )
2726adantr 452 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Grp )
28 eqid 2435 . . . . . . 7  |-  (coe1 `  F
)  =  (coe1 `  F
)
29 eqid 2435 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
3028, 6, 2, 29coe1f 16601 . . . . . 6  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
31303ad2ant2 979 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
3231ffvelrnda 5862 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  F
) `  X )  e.  ( Base `  R
) )
33 eqid 2435 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
3433, 6, 2, 29coe1f 16601 . . . . . 6  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
35343ad2ant3 980 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
3635ffvelrnda 5862 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  G
) `  X )  e.  ( Base `  R
) )
37 eqid 2435 . . . . . . 7  |-  (coe1 `  ( F  .-  G ) )  =  (coe1 `  ( F  .-  G ) )
3837, 6, 2, 29coe1f 16601 . . . . . 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 5862 . . . 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 14868 . . . 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 1186 . . 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 224 . 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 2440 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 set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725   -->wf 5442   ` cfv 5446  (class class class)co 6073   NN0cn0 10213   Basecbs 13461   +g cplusg 13521   Grpcgrp 14677   -gcsg 14680   Ringcrg 15652  Poly1cpl1 16563  coe1cco1 16566
This theorem is referenced by:  deg1sublt  20025  ply1remlem  20077
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-of 6297  df-ofr 6298  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-map 7012  df-pm 7013  df-ixp 7056  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-nn 9993  df-2 10050  df-3 10051  df-4 10052  df-5 10053  df-6 10054  df-7 10055  df-8 10056  df-9 10057  df-10 10058  df-n0 10214  df-z 10275  df-uz 10481  df-fz 11036  df-fzo 11128  df-seq 11316  df-hash 11611  df-struct 13463  df-ndx 13464  df-slot 13465  df-base 13466  df-sets 13467  df-ress 13468  df-plusg 13534  df-mulr 13535  df-sca 13537  df-vsca 13538  df-tset 13540  df-ple 13541  df-0g 13719  df-gsum 13720  df-mre 13803  df-mrc 13804  df-acs 13806  df-mnd 14682  df-mhm 14730  df-submnd 14731  df-grp 14804  df-minusg 14805  df-sbg 14806  df-mulg 14807  df-subg 14933  df-ghm 14996  df-cntz 15108  df-cmn 15406  df-abl 15407  df-mgp 15641  df-rng 15655  df-ur 15657  df-subrg 15858  df-psr 16409  df-mpl 16411  df-opsr 16417  df-psr1 16568  df-ply1 16570  df-coe1 16573
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