Theorem reldmevls 19517

Description: Well-behaved binary operation property of evalSub. (Contributed by Stefan O'Rear, 19-Mar-2015.)

Assertion

Ref	Expression
reldmevls	|- Rel dom evalSub

Proof of Theorem reldmevls

Dummy variables b f g i r s w x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-evls 19506	. 2 |- evalSub = ( i e. _V , s e. CRing |-> [_ ( Base ` s ) / b ]_ ( r e. (SubRing ` s ) |-> [_ ( i mPoly ( s ↾s r ) ) / w ]_ ( iota_ f e. ( w RingHom ( s ^s ( b ^m i ) ) ) ( ( f o. (algSc ` w ) ) = ( x e. r |-> ( ( b ^m i ) X. { x } ) ) /\ ( f o. ( i mVar ( s ↾s r ) ) ) = ( x e. i |-> ( g e. ( b ^m i ) |-> ( g ` x ) ) ) ) ) ) )
2	1	reldmmpt2 6771	1 |- Rel dom evalSub

Syntax hints:   /\ wa 384   = wceq 1483   _Vcvv 3200   [_csb 3533   {csn 4177   |-> cmpt 4729   X. cxp 5112   dom cdm 5114   o. ccom 5118   Rel wrel 5119   ` cfv 5888   iota_crio 6610  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   ↾_s cress 15858   ^_s cpws 16107   CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776  algSccascl 19311   mVar cmvr 19352   mPoly cmpl 19353   evalSub ces 19504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-oprab 6654  df-mpt2 6655  df-evls 19506

This theorem is referenced by:  mpfrcl  19518  evlval  19524