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Theorem cytpval 27507
 Description: Substitutions for the Nth cyclotomic polynomial. (Contributed by Stefan O'Rear, 5-Sep-2015.)
Hypotheses
Ref Expression
cytpval.t mulGrpflds
cytpval.o
cytpval.p Poly1fld
cytpval.x var1fld
cytpval.q mulGrp
cytpval.m
cytpval.a algSc
Assertion
Ref Expression
cytpval CytP g
Distinct variable group:   ,
Allowed substitution hints:   ()   ()   ()   ()   ()   ()   ()

Proof of Theorem cytpval
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cytpval.p . . . . . . 7 Poly1fld
21eqcomi 2442 . . . . . 6 Poly1fld
32fveq2i 5733 . . . . 5 mulGrpPoly1fld mulGrp
4 cytpval.q . . . . 5 mulGrp
53, 4eqtr4i 2461 . . . 4 mulGrpPoly1fld
65a1i 11 . . 3 mulGrpPoly1fld
7 cytpval.o . . . . . . . 8
8 cytpval.t . . . . . . . . 9 mulGrpflds
98fveq2i 5733 . . . . . . . 8 mulGrpflds
107, 9eqtri 2458 . . . . . . 7 mulGrpflds
1110cnveqi 5049 . . . . . 6 mulGrpflds
1211imaeq1i 5202 . . . . 5 mulGrpflds
13 sneq 3827 . . . . . 6
1413imaeq2d 5205 . . . . 5
1512, 14syl5eqr 2484 . . . 4 mulGrpflds
16 cytpval.x . . . . . . 7 var1fld
17 cytpval.a . . . . . . . . 9 algSc
181fveq2i 5733 . . . . . . . . 9 algSc algScPoly1fld
1917, 18eqtri 2458 . . . . . . . 8 algScPoly1fld
2019fveq1i 5731 . . . . . . 7 algScPoly1fld
21 cytpval.m . . . . . . . 8
221fveq2i 5733 . . . . . . . 8 Poly1fld
2321, 22eqtri 2458 . . . . . . 7 Poly1fld
2416, 20, 23oveq123i 6097 . . . . . 6 var1fldPoly1fldalgScPoly1fld
2524eqcomi 2442 . . . . 5 var1fldPoly1fldalgScPoly1fld
2625a1i 11 . . . 4 var1fldPoly1fldalgScPoly1fld
2715, 26mpteq12dv 4289 . . 3 mulGrpflds var1fldPoly1fldalgScPoly1fld
286, 27oveq12d 6101 . 2 mulGrpPoly1fld g mulGrpflds var1fldPoly1fldalgScPoly1fld g
29 df-cytp 27501 . 2 CytP mulGrpPoly1fld g mulGrpflds var1fldPoly1fldalgScPoly1fld
30 ovex 6108 . 2 g
3128, 29, 30fvmpt 5808 1 CytP g
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726   cdif 3319  csn 3816   cmpt 4268  ccnv 4879  cima 4883  cfv 5456  (class class class)co 6083  cc 8990  cc0 8992  cn 10002   ↾s cress 13472   g cgsu 13726  csg 14690  cod 15165  mulGrpcmgp 15650  algSccascl 16373  var1cv1 16572  Poly1cpl1 16573  ℂfldccnfld 16705  CytPccytp 27500 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-cytp 27501
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