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Theorem coe1subfv 16614
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 16597 . . . . . . . 8  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
4 rnggrp 15624 . . . . . . . 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 14824 . . . . . . 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 2404 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
14 eqid 2404 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
152, 6, 13, 14coe1addfv 16613 . . . . 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 14835 . . . . . . 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 5691 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  (coe1 `  ( ( F 
.-  G ) ( +g  `  Y ) G ) )  =  (coe1 `  F ) )
2322fveq1d 5689 . . . 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 2438 . . 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 15624 . . . . . 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 2404 . . . . . . 7  |-  (coe1 `  F
)  =  (coe1 `  F
)
29 eqid 2404 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
3028, 6, 2, 29coe1f 16564 . . . . . 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 5829 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  F
) `  X )  e.  ( Base `  R
) )
33 eqid 2404 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
3433, 6, 2, 29coe1f 16564 . . . . . 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 5829 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  G
) `  X )  e.  ( Base `  R
) )
37 eqid 2404 . . . . . . 7  |-  (coe1 `  ( F  .-  G ) )  =  (coe1 `  ( F  .-  G ) )
3837, 6, 2, 29coe1f 16564 . . . . . 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 5829 . . . 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 14831 . . . 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 2409 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 1721   -->wf 5409   ` cfv 5413  (class class class)co 6040   NN0cn0 10177   Basecbs 13424   +g cplusg 13484   Grpcgrp 14640   -gcsg 14643   Ringcrg 15615  Poly1cpl1 16526  coe1cco1 16529
This theorem is referenced by:  deg1sublt  19986  ply1remlem  20038
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-ofr 6265  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-2o 6684  df-oadd 6687  df-er 6864  df-map 6979  df-pm 6980  df-ixp 7023  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-5 10017  df-6 10018  df-7 10019  df-8 10020  df-9 10021  df-10 10022  df-n0 10178  df-z 10239  df-uz 10445  df-fz 11000  df-fzo 11091  df-seq 11279  df-hash 11574  df-struct 13426  df-ndx 13427  df-slot 13428  df-base 13429  df-sets 13430  df-ress 13431  df-plusg 13497  df-mulr 13498  df-sca 13500  df-vsca 13501  df-tset 13503  df-ple 13504  df-0g 13682  df-gsum 13683  df-mre 13766  df-mrc 13767  df-acs 13769  df-mnd 14645  df-mhm 14693  df-submnd 14694  df-grp 14767  df-minusg 14768  df-sbg 14769  df-mulg 14770  df-subg 14896  df-ghm 14959  df-cntz 15071  df-cmn 15369  df-abl 15370  df-mgp 15604  df-rng 15618  df-ur 15620  df-subrg 15821  df-psr 16372  df-mpl 16374  df-opsr 16380  df-psr1 16531  df-ply1 16533  df-coe1 16536
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