Theorem evlval 19920

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 6931	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3		fveq2 6446	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		evlval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	syl6eqr 2832	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	adantl 475	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
7	2, 6	fveq12d 6453	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8		df-evl 19903	. . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9		fvex 6459	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
10	7, 8, 9	ovmpt2a 7068	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
11	8	mpt2ndm0 7152	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12		0fv 6486	. . . . 5 ⊢ (∅‘𝐵) = ∅
13	11, 12	syl6eqr 2832	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14		reldmevls 19913	. . . . . 6 ⊢ Rel dom evalSub
15	14	ovprc 6959	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
16	15	fveq1d 6448	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
17	13, 16	eqtr4d 2817	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
18	10, 17	pm2.61i 177	. 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
19	1, 18	eqtri 2802	1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 386   = wceq 1601   ∈ wcel 2107  Vcvv 3398  ∅c0 4141  ‘cfv 6135  (class class class)co 6922  Basecbs 16255   evalSub ces 19900   eval cevl 19901

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-iota 6099  df-fun 6137  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-evls 19902  df-evl 19903

This theorem is referenced by:  evlrhm  19921  evlsscasrng  19922  evlsvarsrng  19924  evl1fval1lem  20090  evl1sca  20094  evl1var  20096  pf1rcl  20109  mpfpf1  20111  pf1ind  20115  mzpmfp  38270