Theorem evlval 19524

Description: Value of the simple/same ring evaluation map. (Contributed by Stefan O'Rear, 19-Mar-2015.) (Revised by Mario Carneiro, 12-Jun-2015.)

Hypotheses

Ref	Expression
evlval.q	|- Q = ( I eval R )
evlval.b	|- B = ( Base ` R )

Assertion

Ref	Expression
evlval	|- Q = ( ( I evalSub R ) ` B )

Proof of Theorem evlval

Dummy variables i r are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlval.q	. 2 |- Q = ( I eval R )
2		oveq12 6659	. . . . 5 |- ( ( i = I /\ r = R ) -> ( i evalSub r ) = ( I evalSub R ) )
3		fveq2 6191	. . . . . . 7 |- ( r = R -> ( Base ` r ) = ( Base ` R ) )
4		evlval.b	. . . . . . 7 |- B = ( Base ` R )
5	3, 4	syl6eqr 2674	. . . . . 6 |- ( r = R -> ( Base ` r ) = B )
6	5	adantl 482	. . . . 5 |- ( ( i = I /\ r = R ) -> ( Base ` r ) = B )
7	2, 6	fveq12d 6197	. . . 4 |- ( ( i = I /\ r = R ) -> ( ( i evalSub r ) ` ( Base ` r ) ) = ( ( I evalSub R ) ` B ) )
8		df-evl 19507	. . . 4 |- eval = ( i e. _V , r e. _V |-> ( ( i evalSub r ) ` ( Base ` r ) ) )
9		fvex 6201	. . . 4 |- ( ( I evalSub R ) ` B ) e. _V
10	7, 8, 9	ovmpt2a 6791	. . 3 |- ( ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( ( I evalSub R ) ` B ) )
11	8	mpt2ndm0 6875	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = (/) )
12		0fv 6227	. . . . 5 |- ( (/) ` B ) = (/)
13	11, 12	syl6eqr 2674	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( (/) ` B ) )
14		reldmevls 19517	. . . . . 6 |- Rel dom evalSub
15	14	ovprc 6683	. . . . 5 |- ( -. ( I e. _V /\ R e. _V ) -> ( I evalSub R ) = (/) )
16	15	fveq1d 6193	. . . 4 |- ( -. ( I e. _V /\ R e. _V ) -> ( ( I evalSub R ) ` B ) = ( (/) ` B ) )
17	13, 16	eqtr4d 2659	. . 3 |- ( -. ( I e. _V /\ R e. _V ) -> ( I eval R ) = ( ( I evalSub R ) ` B ) )
18	10, 17	pm2.61i 176	. 2 |- ( I eval R ) = ( ( I evalSub R ) ` B )
19	1, 18	eqtri 2644	1 |- Q = ( ( I evalSub R ) ` B )

Syntax hints:   -. wn 3   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   ` cfv 5888  (class class class)co 6650   Basecbs 15857   evalSub ces 19504   eval cevl 19505

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-8 1992  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-pow 4843  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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  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-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-evls 19506  df-evl 19507

This theorem is referenced by:  evlrhm  19525  evlsscasrng  19526  evlsvarsrng  19528  evl1fval1lem  19694  evl1sca  19698  evl1var  19700  pf1rcl  19713  mpfpf1  19715  pf1ind  19719  mzpmfp  37310