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Theorem idomrootle 26677
Description: No element of an integral domain can have more than  N  N-th roots. (Contributed by Stefan O'Rear, 11-Sep-2015.)
Hypotheses
Ref Expression
idomrootle.b  |-  B  =  ( Base `  R
)
idomrootle.e  |-  .^  =  (.g
`  (mulGrp `  R )
)
Assertion
Ref Expression
idomrootle  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 { y  e.  B  |  ( N 
.^  y )  =  X } )  <_  N )
Distinct variable groups:    y, B    y, N    y, R    y, X
Allowed substitution hint:    .^ ( y)

Proof of Theorem idomrootle
StepHypRef Expression
1 eqid 2253 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2253 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2253 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
4 eqid 2253 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
5 eqid 2253 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
6 eqid 2253 . . 3  |-  ( 0g
`  (Poly1 `  R ) )  =  ( 0g `  (Poly1 `  R ) )
7 simp1 960 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. IDomn )
8 isidom 15877 . . . . . . . . 9  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
98simplbi 448 . . . . . . . 8  |-  ( R  e. IDomn  ->  R  e.  CRing )
107, 9syl 17 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  CRing )
11 crngrng 15186 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
1210, 11syl 17 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Ring )
131ply1rng 16158 . . . . . 6  |-  ( R  e.  Ring  ->  (Poly1 `  R
)  e.  Ring )
1412, 13syl 17 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Ring )
15 rnggrp 15181 . . . . 5  |-  ( (Poly1 `  R )  e.  Ring  -> 
(Poly1 `
 R )  e. 
Grp )
1614, 15syl 17 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Grp )
17 eqid 2253 . . . . . . 7  |-  (mulGrp `  (Poly1 `  R ) )  =  (mulGrp `  (Poly1 `  R
) )
1817rngmgp 15182 . . . . . 6  |-  ( (Poly1 `  R )  e.  Ring  -> 
(mulGrp `  (Poly1 `  R
) )  e.  Mnd )
1914, 18syl 17 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  (Poly1 `  R ) )  e.  Mnd )
20 simp3 962 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN )
21 eqid 2253 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
2221, 1, 2vr1cl 16126 . . . . . 6  |-  ( R  e.  Ring  ->  (var1 `  R
)  e.  ( Base `  (Poly1 `  R ) ) )
2312, 22syl 17 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (var1 `  R )  e.  (
Base `  (Poly1 `  R
) ) )
2417, 2mgpbas 15166 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (mulGrp `  (Poly1 `  R ) ) )
25 eqid 2253 . . . . . 6  |-  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
2624, 25mulgnncl 14417 . . . . 5  |-  ( ( (mulGrp `  (Poly1 `  R
) )  e.  Mnd  /\  N  e.  NN  /\  (var1 `  R )  e.  (
Base `  (Poly1 `  R
) ) )  -> 
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) ) )
2719, 20, 23, 26syl3anc 1187 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) ) )
28 eqid 2253 . . . . . . 7  |-  (algSc `  (Poly1 `  R ) )  =  (algSc `  (Poly1 `  R
) )
29 idomrootle.b . . . . . . 7  |-  B  =  ( Base `  R
)
301, 28, 29, 2ply1sclf 16193 . . . . . 6  |-  ( R  e.  Ring  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
3112, 30syl 17 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
32 simp2 961 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  X  e.  B )
33 ffvelrn 5515 . . . . 5  |-  ( ( (algSc `  (Poly1 `  R
) ) : B --> ( Base `  (Poly1 `  R
) )  /\  X  e.  B )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
3431, 32, 33syl2anc 645 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
35 eqid 2253 . . . . 5  |-  ( -g `  (Poly1 `  R ) )  =  ( -g `  (Poly1 `  R ) )
362, 35grpsubcl 14381 . . . 4  |-  ( ( (Poly1 `  R )  e. 
Grp  /\  ( N
(.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( (algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )
3716, 27, 34, 36syl3anc 1187 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )
383, 1, 2deg1xrcl 19300 . . . . . . . . . 10  |-  ( ( (algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) )  ->  ( ( deg1  `  R
) `  ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  RR* )
3934, 38syl 17 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  RR* )
40 0xr 8758 . . . . . . . . . 10  |-  0  e.  RR*
4140a1i 12 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  e.  RR* )
42 nnre 9633 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
4342rexrd 8761 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR* )
44433ad2ant3 983 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  RR* )
453, 1, 29, 28deg1sclle 19330 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
4612, 32, 45syl2anc 645 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
47 nngt0 9655 . . . . . . . . . 10  |-  ( N  e.  NN  ->  0  <  N )
48473ad2ant3 983 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  <  N )
4939, 41, 44, 46, 48xrlelttrd 10370 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <  N
)
508simprbi 452 . . . . . . . . . . 11  |-  ( R  e. IDomn  ->  R  e. Domn )
51 domnnzr 15868 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
5250, 51syl 17 . . . . . . . . . 10  |-  ( R  e. IDomn  ->  R  e. NzRing )
537, 52syl 17 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. NzRing )
54 nnnn0 9851 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  NN0 )
55543ad2ant3 983 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN0 )
563, 1, 21, 17, 25deg1pw 19338 . . . . . . . . 9  |-  ( ( R  e. NzRing  /\  N  e. 
NN0 )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5753, 55, 56syl2anc 645 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5849, 57breqtrrd 3946 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) )
591, 3, 12, 2, 35, 27, 34, 58deg1sub 19326 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  =  ( ( deg1  `  R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) )
6059, 57eqtrd 2285 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  =  N )
6160, 55eqeltrd 2327 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  NN0 )
623, 1, 6, 2deg1nn0clb 19308 . . . . 5  |-  ( ( R  e.  Ring  /\  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  (
Base `  (Poly1 `  R
) ) )  -> 
( ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  =/=  ( 0g `  (Poly1 `  R ) )  <-> 
( ( deg1  `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  NN0 )
)
6312, 37, 62syl2anc 645 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) )  =/=  ( 0g `  (Poly1 `  R ) )  <->  ( ( deg1  `  R ) `  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) )  e. 
NN0 ) )
6461, 63mpbird 225 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) )  =/=  ( 0g `  (Poly1 `  R ) ) )
651, 2, 3, 4, 5, 6, 7, 37, 64fta1g 19385 . 2  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 ( `' ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) " { ( 0g `  R ) } ) )  <_ 
( ( deg1  `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) )
66 eqid 2253 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
67 eqid 2253 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
68 fvex 5391 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
6929, 68eqeltri 2323 . . . . . . . 8  |-  B  e. 
_V
7069a1i 12 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  B  e.  _V )
714, 1, 66, 29evl1rhm 19244 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
7210, 71syl 17 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
732, 67rhmf 15339 . . . . . . . . 9  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
7472, 73syl 17 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) ) )
75 ffvelrn 5515 . . . . . . . 8  |-  ( ( (eval1 `  R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) )  /\  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  ( Base `  (Poly1 `  R ) ) )  ->  ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) )  e.  ( Base `  ( R  ^s  B ) ) )
7674, 37, 75syl2anc 645 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  e.  ( Base `  ( R  ^s  B ) ) )
7766, 29, 67, 7, 70, 76pwselbas 13262 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) : B --> B )
78 ffn 5246 . . . . . 6  |-  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) : B --> B  -> 
( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B )
7977, 78syl 17 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(eval1 `
 R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B )
80 fniniseg2 5500 . . . . 5  |-  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) )  Fn  B  -> 
( `' ( (eval1 `  R ) `  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) } )
8179, 80syl 17 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( `' ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) } )
8210adantr 453 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  CRing )
83 simpr 449 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  y  e.  B )
844, 21, 29, 1, 2, 82, 83evl1vard 19248 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
(var1 `  R )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  (var1 `  R
) ) `  y
)  =  y ) )
85 idomrootle.e . . . . . . . . . 10  |-  .^  =  (.g
`  (mulGrp `  R )
)
86 simpl3 965 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  N  e.  NN )
8786, 54syl 17 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  N  e.  NN0 )
884, 1, 29, 2, 82, 83, 84, 25, 85, 87evl1expd 19253 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  ( N
(.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ) `  y )  =  ( N  .^  y ) ) )
89 simpl2 964 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  X  e.  B )
904, 1, 29, 28, 2, 82, 89, 83evl1scad 19246 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( (algSc `  (Poly1 `  R ) ) `  X )  e.  (
Base `  (Poly1 `  R
) )  /\  (
( (eval1 `  R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) ) `  y
)  =  X ) )
91 eqid 2253 . . . . . . . . 9  |-  ( -g `  R )  =  (
-g `  R )
924, 1, 29, 2, 82, 83, 88, 90, 35, 91evl1subd 19250 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  ( Base `  (Poly1 `  R ) )  /\  ( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( ( N  .^  y
) ( -g `  R
) X ) ) )
9392simprd 451 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( ( N 
.^  y ) (
-g `  R ) X ) )
9493eqeq1d 2261 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( 0g `  R )  <-> 
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R ) ) )
95 rnggrp 15181 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9612, 95syl 17 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Grp )
9796adantr 453 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  Grp )
98 eqid 2253 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
9998rngmgp 15182 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
10012, 99syl 17 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  R )  e.  Mnd )
101100adantr 453 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
10298, 29mgpbas 15166 . . . . . . . . 9  |-  B  =  ( Base `  (mulGrp `  R ) )
103102, 85mulgnncl 14417 . . . . . . . 8  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  N  e.  NN  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
104101, 86, 83, 103syl3anc 1187 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
10529, 5, 91grpsubeq0 14387 . . . . . . 7  |-  ( ( R  e.  Grp  /\  ( N  .^  y )  e.  B  /\  X  e.  B )  ->  (
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R )  <->  ( N  .^  y )  =  X ) )
10697, 104, 89, 105syl3anc 1187 . . . . . 6  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( N  .^  y ) ( -g `  R ) X )  =  ( 0g `  R )  <->  ( N  .^  y )  =  X ) )
10794, 106bitrd 246 . . . . 5  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (
( ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) `  y )  =  ( 0g `  R )  <-> 
( N  .^  y
)  =  X ) )
108107rabbidva 2718 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  { y  e.  B  |  ( ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) `  y )  =  ( 0g `  R ) }  =  { y  e.  B  |  ( N  .^  y )  =  X } )
10981, 108eqtrd 2285 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( `' ( (eval1 `  R
) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) ( -g `  (Poly1 `  R ) ) ( (algSc `  (Poly1 `  R
) ) `  X
) ) ) " { ( 0g `  R ) } )  =  { y  e.  B  |  ( N 
.^  y )  =  X } )
110109fveq2d 5381 . 2  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 ( `' ( (eval1 `  R ) `  ( ( N (.g `  (mulGrp `  (Poly1 `  R
) ) ) (var1 `  R ) ) (
-g `  (Poly1 `  R
) ) ( (algSc `  (Poly1 `  R ) ) `
 X ) ) ) " { ( 0g `  R ) } ) )  =  ( # `  {
y  e.  B  | 
( N  .^  y
)  =  X }
) )
11165, 110, 603brtr3d 3949 1  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( # `
 { y  e.  B  |  ( N 
.^  y )  =  X } )  <_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621    =/= wne 2412   {crab 2512   _Vcvv 2727   {csn 3544   class class class wbr 3920   `'ccnv 4579   "cima 4583    Fn wfn 4587   -->wf 4588   ` cfv 4592  (class class class)co 5710   0cc0 8617   RR*cxr 8746    < clt 8747    <_ cle 8748   NNcn 9626   NN0cn0 9844   #chash 11215   Basecbs 13022    ^s cpws 13221   0gc0g 13274   Mndcmnd 14196   Grpcgrp 14197   -gcsg 14200  .gcmg 14201  mulGrpcmgp 15160   Ringcrg 15172   CRingccrg 15173   RingHom crh 15329  NzRingcnzr 15841  Domncdomn 15853  IDomncidom 15854  algSccascl 15884  var1cv1 16083  Poly1cpl1 16084  eval1ce1 16086   deg1 cdg1 19272
This theorem is referenced by:  idomodle  26678
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-15 2102  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-iin 3806  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-of 5930  df-ofr 5931  df-1st 5974  df-2nd 5975  df-tpos 6086  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-2o 6366  df-oadd 6369  df-er 6546  df-map 6660  df-pm 6661  df-ixp 6704  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-cda 7678  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-n 9627  df-2 9684  df-3 9685  df-4 9686  df-5 9687  df-6 9688  df-7 9689  df-8 9690  df-9 9691  df-10 9692  df-n0 9845  df-z 9904  df-dec 10004  df-uz 10110  df-fz 10661  df-fzo 10749  df-seq 10925  df-hash 11216  df-struct 13024  df-ndx 13025  df-slot 13026  df-base 13027  df-sets 13028  df-ress 13029  df-plusg 13095  df-mulr 13096  df-starv 13097  df-sca 13098  df-vsca 13099  df-tset 13101  df-ple 13102  df-ds 13104  df-hom 13106  df-cco 13107  df-prds 13222  df-pws 13224  df-0g 13278  df-gsum 13279  df-mre 13361  df-mrc 13362  df-acs 13363  df-mnd 14202  df-mhm 14250  df-submnd 14251  df-grp 14324  df-minusg 14325  df-sbg 14326  df-mulg 14327  df-subg 14453  df-ghm 14516  df-cntz 14628  df-cmn 14926  df-abl 14927  df-mgp 15161  df-ring 15175  df-cring 15176  df-ur 15177  df-oppr 15240  df-dvdsr 15258  df-unit 15259  df-invr 15289  df-rnghom 15331  df-subrg 15378  df-lmod 15464  df-lss 15525  df-lsp 15564  df-nzr 15842  df-rlreg 15856  df-domn 15857  df-idom 15858  df-assa 15885  df-asp 15886  df-ascl 15887  df-psr 15930  df-mvr 15931  df-mpl 15932  df-evls 15933  df-evl 15934  df-opsr 15938  df-psr1 16089  df-vr1 16090  df-ply1 16091  df-evl1 16093  df-coe1 16094  df-cnfld 16210  df-mdeg 19273  df-deg1 19274  df-mon1 19348  df-uc1p 19349  df-q1p 19350  df-r1p 19351
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