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Theorem idomrootle 27522
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 2285 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2285 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2285 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
4 eqid 2285 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
5 eqid 2285 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
6 eqid 2285 . . 3  |-  ( 0g
`  (Poly1 `  R ) )  =  ( 0g `  (Poly1 `  R ) )
7 simp1 955 . . 3  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. IDomn )
8 isidom 16047 . . . . . . . . 9  |-  ( R  e. IDomn 
<->  ( R  e.  CRing  /\  R  e. Domn ) )
98simplbi 446 . . . . . . . 8  |-  ( R  e. IDomn  ->  R  e.  CRing )
107, 9syl 15 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  CRing )
11 crngrng 15353 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
1210, 11syl 15 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Ring )
131ply1rng 16328 . . . . . 6  |-  ( R  e.  Ring  ->  (Poly1 `  R
)  e.  Ring )
1412, 13syl 15 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Ring )
15 rnggrp 15348 . . . . 5  |-  ( (Poly1 `  R )  e.  Ring  -> 
(Poly1 `
 R )  e. 
Grp )
1614, 15syl 15 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (Poly1 `  R )  e.  Grp )
17 eqid 2285 . . . . . . 7  |-  (mulGrp `  (Poly1 `  R ) )  =  (mulGrp `  (Poly1 `  R
) )
1817rngmgp 15349 . . . . . 6  |-  ( (Poly1 `  R )  e.  Ring  -> 
(mulGrp `  (Poly1 `  R
) )  e.  Mnd )
1914, 18syl 15 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  (Poly1 `  R ) )  e.  Mnd )
20 simp3 957 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN )
21 eqid 2285 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
2221, 1, 2vr1cl 16296 . . . . . 6  |-  ( R  e.  Ring  ->  (var1 `  R
)  e.  ( Base `  (Poly1 `  R ) ) )
2312, 22syl 15 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (var1 `  R )  e.  (
Base `  (Poly1 `  R
) ) )
2417, 2mgpbas 15333 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (mulGrp `  (Poly1 `  R ) ) )
25 eqid 2285 . . . . . 6  |-  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
2624, 25mulgnncl 14584 . . . . 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 1182 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) )  e.  ( Base `  (Poly1 `  R ) ) )
28 eqid 2285 . . . . . . 7  |-  (algSc `  (Poly1 `  R ) )  =  (algSc `  (Poly1 `  R
) )
29 idomrootle.b . . . . . . 7  |-  B  =  ( Base `  R
)
301, 28, 29, 2ply1sclf 16363 . . . . . 6  |-  ( R  e.  Ring  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
3112, 30syl 15 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (algSc `  (Poly1 `  R ) ) : B --> ( Base `  (Poly1 `  R ) ) )
32 simp2 956 . . . . 5  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  X  e.  B )
33 ffvelrn 5665 . . . . 5  |-  ( ( (algSc `  (Poly1 `  R
) ) : B --> ( Base `  (Poly1 `  R
) )  /\  X  e.  B )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
3431, 32, 33syl2anc 642 . . . 4  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
(algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) ) )
35 eqid 2285 . . . . 5  |-  ( -g `  (Poly1 `  R ) )  =  ( -g `  (Poly1 `  R ) )
362, 35grpsubcl 14548 . . . 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 1182 . . 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 19470 . . . . . . . . . 10  |-  ( ( (algSc `  (Poly1 `  R
) ) `  X
)  e.  ( Base `  (Poly1 `  R ) )  ->  ( ( deg1  `  R
) `  ( (algSc `  (Poly1 `  R ) ) `
 X ) )  e.  RR* )
3934, 38syl 15 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  e.  RR* )
40 0xr 8880 . . . . . . . . . 10  |-  0  e.  RR*
4140a1i 10 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  e.  RR* )
42 nnre 9755 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
4342rexrd 8883 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  RR* )
44433ad2ant3 978 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  RR* )
453, 1, 29, 28deg1sclle 19500 . . . . . . . . . 10  |-  ( ( R  e.  Ring  /\  X  e.  B )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
4612, 32, 45syl2anc 642 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <_  0
)
47 nngt0 9777 . . . . . . . . . 10  |-  ( N  e.  NN  ->  0  <  N )
48473ad2ant3 978 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  <  N )
4939, 41, 44, 46, 48xrlelttrd 10493 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( (algSc `  (Poly1 `  R
) ) `  X
) )  <  N
)
508simprbi 450 . . . . . . . . . . 11  |-  ( R  e. IDomn  ->  R  e. Domn )
51 domnnzr 16038 . . . . . . . . . . 11  |-  ( R  e. Domn  ->  R  e. NzRing )
5250, 51syl 15 . . . . . . . . . 10  |-  ( R  e. IDomn  ->  R  e. NzRing )
537, 52syl 15 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e. NzRing )
54 nnnn0 9974 . . . . . . . . . 10  |-  ( N  e.  NN  ->  N  e.  NN0 )
55543ad2ant3 978 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  N  e.  NN0 )
563, 1, 21, 17, 25deg1pw 19508 . . . . . . . . 9  |-  ( ( R  e. NzRing  /\  N  e. 
NN0 )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5753, 55, 56syl2anc 642 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (
( deg1  `
 R ) `  ( N (.g `  (mulGrp `  (Poly1 `  R ) ) ) (var1 `  R ) ) )  =  N )
5849, 57breqtrrd 4051 . . . . . . 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 19496 . . . . . 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 2317 . . . . 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 2359 . . . 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 19478 . . . . 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 642 . . . 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 223 . . 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 19555 . 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 2285 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
67 eqid 2285 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
68 fvex 5541 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
6929, 68eqeltri 2355 . . . . . . . 8  |-  B  e. 
_V
7069a1i 10 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  B  e.  _V )
714, 1, 66, 29evl1rhm 19414 . . . . . . . . . 10  |-  ( R  e.  CRing  ->  (eval1 `  R
)  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
7210, 71syl 15 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) ) )
732, 67rhmf 15506 . . . . . . . . 9  |-  ( (eval1 `  R )  e.  ( (Poly1 `  R ) RingHom  ( R  ^s  B ) )  -> 
(eval1 `
 R ) : ( Base `  (Poly1 `  R ) ) --> (
Base `  ( R  ^s  B ) ) )
7472, 73syl 15 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (eval1 `  R ) : (
Base `  (Poly1 `  R
) ) --> ( Base `  ( R  ^s  B ) ) )
75 ffvelrn 5665 . . . . . . . 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 642 . . . . . . 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 13390 . . . . . 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 5391 . . . . . 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 15 . . . . 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 5650 . . . . 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 15 . . . 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 451 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  CRing )
83 simpr 447 . . . . . . . . 9  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  y  e.  B )
844, 21, 29, 1, 2, 82, 83evl1vard 19418 . . . . . . . . . 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 960 . . . . . . . . . . 11  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  N  e.  NN )
8786, 54syl 15 . . . . . . . . . 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 19423 . . . . . . . . 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 959 . . . . . . . . . 10  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  X  e.  B )
904, 1, 29, 28, 2, 82, 89, 83evl1scad 19416 . . . . . . . . 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 2285 . . . . . . . . 9  |-  ( -g `  R )  =  (
-g `  R )
924, 1, 29, 2, 82, 83, 88, 90, 35, 91evl1subd 19420 . . . . . . . 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 449 . . . . . . 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 2293 . . . . . 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 15348 . . . . . . . . 9  |-  ( R  e.  Ring  ->  R  e. 
Grp )
9612, 95syl 15 . . . . . . . 8  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Grp )
9796adantr 451 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  R  e.  Grp )
98 eqid 2285 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
9998rngmgp 15349 . . . . . . . . . 10  |-  ( R  e.  Ring  ->  (mulGrp `  R )  e.  Mnd )
10012, 99syl 15 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  (mulGrp `  R )  e.  Mnd )
101100adantr 451 . . . . . . . 8  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  (mulGrp `  R )  e.  Mnd )
10298, 29mgpbas 15333 . . . . . . . . 9  |-  B  =  ( Base `  (mulGrp `  R ) )
103102, 85mulgnncl 14584 . . . . . . . 8  |-  ( ( (mulGrp `  R )  e.  Mnd  /\  N  e.  NN  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
104101, 86, 83, 103syl3anc 1182 . . . . . . 7  |-  ( ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  /\  y  e.  B )  ->  ( N  .^  y )  e.  B )
10529, 5, 91grpsubeq0 14554 . . . . . . 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 1182 . . . . . 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 244 . . . . 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 2781 . . . 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 2317 . . 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 5531 . 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 4054 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 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1625    e. wcel 1686    =/= wne 2448   {crab 2549   _Vcvv 2790   {csn 3642   class class class wbr 4025   `'ccnv 4690   "cima 4694    Fn wfn 5252   -->wf 5253   ` cfv 5257  (class class class)co 5860   0cc0 8739   RR*cxr 8868    < clt 8869    <_ cle 8870   NNcn 9748   NN0cn0 9967   #chash 11339   Basecbs 13150    ^s cpws 13349   0gc0g 13402   Mndcmnd 14363   Grpcgrp 14364   -gcsg 14367  .gcmg 14368  mulGrpcmgp 15327   Ringcrg 15339   CRingccrg 15340   RingHom crh 15496  NzRingcnzr 16011  Domncdomn 16023  IDomncidom 16024  algSccascl 16054  var1cv1 16253  Poly1cpl1 16254  eval1ce1 16256   deg1 cdg1 19442
This theorem is referenced by:  idomodle  27523
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-inf2 7344  ax-cnex 8795  ax-resscn 8796  ax-1cn 8797  ax-icn 8798  ax-addcl 8799  ax-addrcl 8800  ax-mulcl 8801  ax-mulrcl 8802  ax-mulcom 8803  ax-addass 8804  ax-mulass 8805  ax-distr 8806  ax-i2m1 8807  ax-1ne0 8808  ax-1rid 8809  ax-rnegex 8810  ax-rrecex 8811  ax-cnre 8812  ax-pre-lttri 8813  ax-pre-lttrn 8814  ax-pre-ltadd 8815  ax-pre-mulgt0 8816  ax-pre-sup 8817  ax-addf 8818  ax-mulf 8819
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4307  df-id 4311  df-po 4316  df-so 4317  df-fr 4354  df-se 4355  df-we 4356  df-ord 4397  df-on 4398  df-lim 4399  df-suc 4400  df-om 4659  df-xp 4697  df-rel 4698  df-cnv 4699  df-co 4700  df-dm 4701  df-rn 4702  df-res 4703  df-ima 4704  df-iota 5221  df-fun 5259  df-fn 5260  df-f 5261  df-f1 5262  df-fo 5263  df-f1o 5264  df-fv 5265  df-isom 5266  df-ov 5863  df-oprab 5864  df-mpt2 5865  df-of 6080  df-ofr 6081  df-1st 6124  df-2nd 6125  df-tpos 6236  df-riota 6306  df-recs 6390  df-rdg 6425  df-1o 6481  df-2o 6482  df-oadd 6485  df-er 6662  df-map 6776  df-pm 6777  df-ixp 6820  df-en 6866  df-dom 6867  df-sdom 6868  df-fin 6869  df-sup 7196  df-oi 7227  df-card 7574  df-cda 7796  df-pnf 8871  df-mnf 8872  df-xr 8873  df-ltxr 8874  df-le 8875  df-sub 9041  df-neg 9042  df-nn 9749  df-2 9806  df-3 9807  df-4 9808  df-5 9809  df-6 9810  df-7 9811  df-8 9812  df-9 9813  df-10 9814  df-n0 9968  df-z 10027  df-dec 10127  df-uz 10233  df-fz 10785  df-fzo 10873  df-seq 11049  df-hash 11340  df-struct 13152  df-ndx 13153  df-slot 13154  df-base 13155  df-sets 13156  df-ress 13157  df-plusg 13223  df-mulr 13224  df-starv 13225  df-sca 13226  df-vsca 13227  df-tset 13229  df-ple 13230  df-ds 13232  df-hom 13234  df-cco 13235  df-prds 13350  df-pws 13352  df-0g 13406  df-gsum 13407  df-mre 13490  df-mrc 13491  df-acs 13493  df-mnd 14369  df-mhm 14417  df-submnd 14418  df-grp 14491  df-minusg 14492  df-sbg 14493  df-mulg 14494  df-subg 14620  df-ghm 14683  df-cntz 14795  df-cmn 15093  df-abl 15094  df-mgp 15328  df-rng 15342  df-cring 15343  df-ur 15344  df-oppr 15407  df-dvdsr 15425  df-unit 15426  df-invr 15456  df-rnghom 15498  df-subrg 15545  df-lmod 15631  df-lss 15692  df-lsp 15731  df-nzr 16012  df-rlreg 16026  df-domn 16027  df-idom 16028  df-assa 16055  df-asp 16056  df-ascl 16057  df-psr 16100  df-mvr 16101  df-mpl 16102  df-evls 16103  df-evl 16104  df-opsr 16108  df-psr1 16259  df-vr1 16260  df-ply1 16261  df-evl1 16263  df-coe1 16264  df-cnfld 16380  df-mdeg 19443  df-deg1 19444  df-mon1 19518  df-uc1p 19519  df-q1p 19520  df-r1p 19521
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