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Theorem idomrootle 26879
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 2258 . . 3  |-  (Poly1 `  R
)  =  (Poly1 `  R
)
2 eqid 2258 . . 3  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (Poly1 `  R ) )
3 eqid 2258 . . 3  |-  ( deg1  `  R
)  =  ( deg1  `  R
)
4 eqid 2258 . . 3  |-  (eval1 `  R
)  =  (eval1 `  R
)
5 eqid 2258 . . 3  |-  ( 0g
`  R )  =  ( 0g `  R
)
6 eqid 2258 . . 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 16008 . . . . . . . . 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 15314 . . . . . . 7  |-  ( R  e.  CRing  ->  R  e.  Ring )
1210, 11syl 17 . . . . . 6  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  R  e.  Ring )
131ply1rng 16289 . . . . . 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 15309 . . . . 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 2258 . . . . . . 7  |-  (mulGrp `  (Poly1 `  R ) )  =  (mulGrp `  (Poly1 `  R
) )
1817rngmgp 15310 . . . . . 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 2258 . . . . . . 7  |-  (var1 `  R
)  =  (var1 `  R
)
2221, 1, 2vr1cl 16257 . . . . . 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 15294 . . . . . 6  |-  ( Base `  (Poly1 `  R ) )  =  ( Base `  (mulGrp `  (Poly1 `  R ) ) )
25 eqid 2258 . . . . . 6  |-  (.g `  (mulGrp `  (Poly1 `  R ) ) )  =  (.g `  (mulGrp `  (Poly1 `  R ) ) )
2624, 25mulgnncl 14545 . . . . 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 2258 . . . . . . 7  |-  (algSc `  (Poly1 `  R ) )  =  (algSc `  (Poly1 `  R
) )
29 idomrootle.b . . . . . . 7  |-  B  =  ( Base `  R
)
301, 28, 29, 2ply1sclf 16324 . . . . . 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 5597 . . . . 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 2258 . . . . 5  |-  ( -g `  (Poly1 `  R ) )  =  ( -g `  (Poly1 `  R ) )
362, 35grpsubcl 14509 . . . 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 19431 . . . . . . . . . 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 8846 . . . . . . . . . 10  |-  0  e.  RR*
4140a1i 12 . . . . . . . . 9  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  0  e.  RR* )
42 nnre 9721 . . . . . . . . . . 11  |-  ( N  e.  NN  ->  N  e.  RR )
4342rexrd 8849 . . . . . . . . . 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 19461 . . . . . . . . . 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 9743 . . . . . . . . . 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 10459 . . . . . . . 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 15999 . . . . . . . . . . 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 9940 . . . . . . . . . 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 19469 . . . . . . . . 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 4023 . . . . . . 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 19457 . . . . . 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 2290 . . . . 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 2332 . . . 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 19439 . . . . 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 19516 . 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 2258 . . . . . . 7  |-  ( R  ^s  B )  =  ( R  ^s  B )
67 eqid 2258 . . . . . . 7  |-  ( Base `  ( R  ^s  B ) )  =  ( Base `  ( R  ^s  B ) )
68 fvex 5472 . . . . . . . . 9  |-  ( Base `  R )  e.  _V
6929, 68eqeltri 2328 . . . . . . . 8  |-  B  e. 
_V
7069a1i 12 . . . . . . 7  |-  ( ( R  e. IDomn  /\  X  e.  B  /\  N  e.  NN )  ->  B  e.  _V )
714, 1, 66, 29evl1rhm 19375 . . . . . . . . . 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 15467 . . . . . . . . 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 5597 . . . . . . . 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 13351 . . . . . 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 5327 . . . . . 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 5582 . . . . 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 19379 . . . . . . . . . 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 19384 . . . . . . . . 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 19377 . . . . . . . . 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 2258 . . . . . . . . 9  |-  ( -g `  R )  =  (
-g `  R )
924, 1, 29, 2, 82, 83, 88, 90, 35, 91evl1subd 19381 . . . . . . . 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 2266 . . . . . 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 15309 . . . . . . . . 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 2258 . . . . . . . . . . 11  |-  (mulGrp `  R )  =  (mulGrp `  R )
9998rngmgp 15310 . . . . . . . . . 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 15294 . . . . . . . . 9  |-  B  =  ( Base `  (mulGrp `  R ) )
103102, 85mulgnncl 14545 . . . . . . . 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 14515 . . . . . . 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 2754 . . . 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 2290 . . 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 5462 . 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 4026 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 2421   {crab 2522   _Vcvv 2763   {csn 3614   class class class wbr 3997   `'ccnv 4660   "cima 4664    Fn wfn 4668   -->wf 4669   ` cfv 4673  (class class class)co 5792   0cc0 8705   RR*cxr 8834    < clt 8835    <_ cle 8836   NNcn 9714   NN0cn0 9933   #chash 11304   Basecbs 13111    ^s cpws 13310   0gc0g 13363   Mndcmnd 14324   Grpcgrp 14325   -gcsg 14328  .gcmg 14329  mulGrpcmgp 15288   Ringcrg 15300   CRingccrg 15301   RingHom crh 15457  NzRingcnzr 15972  Domncdomn 15984  IDomncidom 15985  algSccascl 16015  var1cv1 16214  Poly1cpl1 16215  eval1ce1 16217   deg1 cdg1 19403
This theorem is referenced by:  idomodle  26880
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 1927  ax-15 2106  ax-ext 2239  ax-rep 4105  ax-sep 4115  ax-nul 4123  ax-pow 4160  ax-pr 4186  ax-un 4484  ax-inf2 7310  ax-cnex 8761  ax-resscn 8762  ax-1cn 8763  ax-icn 8764  ax-addcl 8765  ax-addrcl 8766  ax-mulcl 8767  ax-mulrcl 8768  ax-mulcom 8769  ax-addass 8770  ax-mulass 8771  ax-distr 8772  ax-i2m1 8773  ax-1ne0 8774  ax-1rid 8775  ax-rnegex 8776  ax-rrecex 8777  ax-cnre 8778  ax-pre-lttri 8779  ax-pre-lttrn 8780  ax-pre-ltadd 8781  ax-pre-mulgt0 8782  ax-pre-sup 8783  ax-addf 8784  ax-mulf 8785
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 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-nel 2424  df-ral 2523  df-rex 2524  df-reu 2525  df-rmo 2526  df-rab 2527  df-v 2765  df-sbc 2967  df-csb 3057  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-pss 3143  df-nul 3431  df-if 3540  df-pw 3601  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-uni 3802  df-int 3837  df-iun 3881  df-iin 3882  df-br 3998  df-opab 4052  df-mpt 4053  df-tr 4088  df-eprel 4277  df-id 4281  df-po 4286  df-so 4287  df-fr 4324  df-se 4325  df-we 4326  df-ord 4367  df-on 4368  df-lim 4369  df-suc 4370  df-om 4629  df-xp 4675  df-rel 4676  df-cnv 4677  df-co 4678  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fun 4683  df-fn 4684  df-f 4685  df-f1 4686  df-fo 4687  df-f1o 4688  df-fv 4689  df-isom 4690  df-ov 5795  df-oprab 5796  df-mpt2 5797  df-of 6012  df-ofr 6013  df-1st 6056  df-2nd 6057  df-tpos 6168  df-iota 6225  df-riota 6272  df-recs 6356  df-rdg 6391  df-1o 6447  df-2o 6448  df-oadd 6451  df-er 6628  df-map 6742  df-pm 6743  df-ixp 6786  df-en 6832  df-dom 6833  df-sdom 6834  df-fin 6835  df-sup 7162  df-oi 7193  df-card 7540  df-cda 7762  df-pnf 8837  df-mnf 8838  df-xr 8839  df-ltxr 8840  df-le 8841  df-sub 9007  df-neg 9008  df-n 9715  df-2 9772  df-3 9773  df-4 9774  df-5 9775  df-6 9776  df-7 9777  df-8 9778  df-9 9779  df-10 9780  df-n0 9934  df-z 9993  df-dec 10093  df-uz 10199  df-fz 10750  df-fzo 10838  df-seq 11014  df-hash 11305  df-struct 13113  df-ndx 13114  df-slot 13115  df-base 13116  df-sets 13117  df-ress 13118  df-plusg 13184  df-mulr 13185  df-starv 13186  df-sca 13187  df-vsca 13188  df-tset 13190  df-ple 13191  df-ds 13193  df-hom 13195  df-cco 13196  df-prds 13311  df-pws 13313  df-0g 13367  df-gsum 13368  df-mre 13451  df-mrc 13452  df-acs 13454  df-mnd 14330  df-mhm 14378  df-submnd 14379  df-grp 14452  df-minusg 14453  df-sbg 14454  df-mulg 14455  df-subg 14581  df-ghm 14644  df-cntz 14756  df-cmn 15054  df-abl 15055  df-mgp 15289  df-ring 15303  df-cring 15304  df-ur 15305  df-oppr 15368  df-dvdsr 15386  df-unit 15387  df-invr 15417  df-rnghom 15459  df-subrg 15506  df-lmod 15592  df-lss 15653  df-lsp 15692  df-nzr 15973  df-rlreg 15987  df-domn 15988  df-idom 15989  df-assa 16016  df-asp 16017  df-ascl 16018  df-psr 16061  df-mvr 16062  df-mpl 16063  df-evls 16064  df-evl 16065  df-opsr 16069  df-psr1 16220  df-vr1 16221  df-ply1 16222  df-evl1 16224  df-coe1 16225  df-cnfld 16341  df-mdeg 19404  df-deg1 19405  df-mon1 19479  df-uc1p 19480  df-q1p 19481  df-r1p 19482
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