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Theorem coe1subfv 16588
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 16571 . . . . . . . 8  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
4 rnggrp 15598 . . . . . . . 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 14798 . . . . . . 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 2389 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
14 eqid 2389 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
152, 6, 13, 14coe1addfv 16587 . . . . 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 14809 . . . . . . 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 5674 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  (coe1 `  ( ( F 
.-  G ) ( +g  `  Y ) G ) )  =  (coe1 `  F ) )
2322fveq1d 5672 . . . 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 2423 . . 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 15598 . . . . . 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 2389 . . . . . . 7  |-  (coe1 `  F
)  =  (coe1 `  F
)
29 eqid 2389 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
3028, 6, 2, 29coe1f 16538 . . . . . 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 5811 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  F
) `  X )  e.  ( Base `  R
) )
33 eqid 2389 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
3433, 6, 2, 29coe1f 16538 . . . . . 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 5811 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  G
) `  X )  e.  ( Base `  R
) )
37 eqid 2389 . . . . . . 7  |-  (coe1 `  ( F  .-  G ) )  =  (coe1 `  ( F  .-  G ) )
3837, 6, 2, 29coe1f 16538 . . . . . 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 5811 . . . 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 14805 . . . 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 2394 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 1649    e. wcel 1717   -->wf 5392   ` cfv 5396  (class class class)co 6022   NN0cn0 10155   Basecbs 13398   +g cplusg 13458   Grpcgrp 14614   -gcsg 14617   Ringcrg 15589  Poly1cpl1 16500  coe1cco1 16503
This theorem is referenced by:  deg1sublt  19902  ply1remlem  19954
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643  ax-inf2 7531  ax-cnex 8981  ax-resscn 8982  ax-1cn 8983  ax-icn 8984  ax-addcl 8985  ax-addrcl 8986  ax-mulcl 8987  ax-mulrcl 8988  ax-mulcom 8989  ax-addass 8990  ax-mulass 8991  ax-distr 8992  ax-i2m1 8993  ax-1ne0 8994  ax-1rid 8995  ax-rnegex 8996  ax-rrecex 8997  ax-cnre 8998  ax-pre-lttri 8999  ax-pre-lttrn 9000  ax-pre-ltadd 9001  ax-pre-mulgt0 9002
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rmo 2659  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-pss 3281  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-tp 3767  df-op 3768  df-uni 3960  df-int 3995  df-iun 4039  df-iin 4040  df-br 4156  df-opab 4210  df-mpt 4211  df-tr 4246  df-eprel 4437  df-id 4441  df-po 4446  df-so 4447  df-fr 4484  df-se 4485  df-we 4486  df-ord 4527  df-on 4528  df-lim 4529  df-suc 4530  df-om 4788  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-isom 5405  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-of 6246  df-ofr 6247  df-1st 6290  df-2nd 6291  df-riota 6487  df-recs 6571  df-rdg 6606  df-1o 6662  df-2o 6663  df-oadd 6666  df-er 6843  df-map 6958  df-pm 6959  df-ixp 7002  df-en 7048  df-dom 7049  df-sdom 7050  df-fin 7051  df-oi 7414  df-card 7761  df-pnf 9057  df-mnf 9058  df-xr 9059  df-ltxr 9060  df-le 9061  df-sub 9227  df-neg 9228  df-nn 9935  df-2 9992  df-3 9993  df-4 9994  df-5 9995  df-6 9996  df-7 9997  df-8 9998  df-9 9999  df-10 10000  df-n0 10156  df-z 10217  df-uz 10423  df-fz 10978  df-fzo 11068  df-seq 11253  df-hash 11548  df-struct 13400  df-ndx 13401  df-slot 13402  df-base 13403  df-sets 13404  df-ress 13405  df-plusg 13471  df-mulr 13472  df-sca 13474  df-vsca 13475  df-tset 13477  df-ple 13478  df-0g 13656  df-gsum 13657  df-mre 13740  df-mrc 13741  df-acs 13743  df-mnd 14619  df-mhm 14667  df-submnd 14668  df-grp 14741  df-minusg 14742  df-sbg 14743  df-mulg 14744  df-subg 14870  df-ghm 14933  df-cntz 15045  df-cmn 15343  df-abl 15344  df-mgp 15578  df-rng 15592  df-ur 15594  df-subrg 15795  df-psr 16346  df-mpl 16348  df-opsr 16354  df-psr1 16505  df-ply1 16507  df-coe1 16510
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