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Theorem coe1subfv 17618
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 984 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Ring )
2 coe1sub.y . . . . . . . . 9  |-  Y  =  (Poly1 `  R )
32ply1rng 17601 . . . . . . . 8  |-  ( R  e.  Ring  ->  Y  e. 
Ring )
4 rnggrp 16586 . . . . . . . 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 15586 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G
)  e.  B )
95, 8syl3an1 1244 . . . . . 6  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  ( F  .-  G )  e.  B )
109adantr 462 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( F  .-  G )  e.  B
)
11 simpl3 986 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  G  e.  B
)
12 simpr 458 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  X  e.  NN0 )
13 eqid 2433 . . . . . 6  |-  ( +g  `  Y )  =  ( +g  `  Y )
14 eqid 2433 . . . . . 6  |-  ( +g  `  R )  =  ( +g  `  R )
152, 6, 13, 14coe1addfv 17617 . . . . 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 1214 . . . 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 1002 . . . . . . . 8  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  Y  e.  Grp )
1817adantr 462 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  Y  e.  Grp )
19 simpl2 985 . . . . . . 7  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  F  e.  B
)
206, 13, 7grpnpcan 15597 . . . . . . 7  |-  ( ( Y  e.  Grp  /\  F  e.  B  /\  G  e.  B )  ->  ( ( F  .-  G ) ( +g  `  Y ) G )  =  F )
2118, 19, 11, 20syl3anc 1211 . . . . . 6  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( ( F 
.-  G ) ( +g  `  Y ) G )  =  F )
2221fveq2d 5683 . . . . 5  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  (coe1 `  ( ( F 
.-  G ) ( +g  `  Y ) G ) )  =  (coe1 `  F ) )
2322fveq1d 5681 . . . 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 2467 . . 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 16586 . . . . . 6  |-  ( R  e.  Ring  ->  R  e. 
Grp )
26253ad2ant1 1002 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  R  e.  Grp )
2726adantr 462 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  R  e.  Grp )
28 eqid 2433 . . . . . . 7  |-  (coe1 `  F
)  =  (coe1 `  F
)
29 eqid 2433 . . . . . . 7  |-  ( Base `  R )  =  (
Base `  R )
3028, 6, 2, 29coe1f 17566 . . . . . 6  |-  ( F  e.  B  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
31303ad2ant2 1003 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  F ) : NN0 --> (
Base `  R )
)
3231ffvelrnda 5831 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  F
) `  X )  e.  ( Base `  R
) )
33 eqid 2433 . . . . . . 7  |-  (coe1 `  G
)  =  (coe1 `  G
)
3433, 6, 2, 29coe1f 17566 . . . . . 6  |-  ( G  e.  B  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
35343ad2ant3 1004 . . . . 5  |-  ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  ->  (coe1 `  G ) : NN0 --> (
Base `  R )
)
3635ffvelrnda 5831 . . . 4  |-  ( ( ( R  e.  Ring  /\  F  e.  B  /\  G  e.  B )  /\  X  e.  NN0 )  ->  ( (coe1 `  G
) `  X )  e.  ( Base `  R
) )
37 eqid 2433 . . . . . . 7  |-  (coe1 `  ( F  .-  G ) )  =  (coe1 `  ( F  .-  G ) )
3837, 6, 2, 29coe1f 17566 . . . . . 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 5831 . . . 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 15593 . . . 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 1213 . . 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 2438 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 958    = wceq 1362    e. wcel 1755   -->wf 5402   ` cfv 5406  (class class class)co 6080   NN0cn0 10567   Basecbs 14157   +g cplusg 14221   Grpcgrp 15393   -gcsg 15396   Ringcrg 16577  Poly1cpl1 17528  coe1cco1 17531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-inf2 7835  ax-cnex 9326  ax-resscn 9327  ax-1cn 9328  ax-icn 9329  ax-addcl 9330  ax-addrcl 9331  ax-mulcl 9332  ax-mulrcl 9333  ax-mulcom 9334  ax-addass 9335  ax-mulass 9336  ax-distr 9337  ax-i2m1 9338  ax-1ne0 9339  ax-1rid 9340  ax-rnegex 9341  ax-rrecex 9342  ax-cnre 9343  ax-pre-lttri 9344  ax-pre-lttrn 9345  ax-pre-ltadd 9346  ax-pre-mulgt0 9347
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-iin 4162  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-se 4667  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-isom 5415  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-of 6309  df-ofr 6310  df-om 6466  df-1st 6566  df-2nd 6567  df-supp 6680  df-recs 6818  df-rdg 6852  df-1o 6908  df-2o 6909  df-oadd 6912  df-er 7089  df-map 7204  df-pm 7205  df-ixp 7252  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-fsupp 7609  df-oi 7712  df-card 8097  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-sub 9585  df-neg 9586  df-nn 10311  df-2 10368  df-3 10369  df-4 10370  df-5 10371  df-6 10372  df-7 10373  df-8 10374  df-9 10375  df-10 10376  df-n0 10568  df-z 10635  df-uz 10850  df-fz 11425  df-fzo 11533  df-seq 11791  df-hash 12088  df-struct 14159  df-ndx 14160  df-slot 14161  df-base 14162  df-sets 14163  df-ress 14164  df-plusg 14234  df-mulr 14235  df-sca 14237  df-vsca 14238  df-tset 14240  df-ple 14241  df-0g 14363  df-gsum 14364  df-mre 14507  df-mrc 14508  df-acs 14510  df-mnd 15398  df-mhm 15447  df-submnd 15448  df-grp 15525  df-minusg 15526  df-sbg 15527  df-mulg 15528  df-subg 15658  df-ghm 15725  df-cntz 15815  df-cmn 16259  df-abl 16260  df-mgp 16566  df-rng 16580  df-ur 16582  df-subrg 16787  df-psr 17351  df-mpl 17353  df-opsr 17359  df-psr1 17533  df-ply1 17535  df-coe1 17538
This theorem is referenced by:  deg1sublt  21467  ply1remlem  21519
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