Theorem evlval 20767

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 7144	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3		fveq2 6645	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		evlval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2851	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	adantl 485	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
7	2, 6	fveq12d 6652	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8		df-evl 20746	. . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9		fvex 6658	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7284	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
11	8	mpondm0 7366	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12		0fv 6684	. . . . 5 ⊢ (∅‘𝐵) = ∅
13	11, 12	eqtr4di 2851	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14		reldmevls 20756	. . . . . 6 ⊢ Rel dom evalSub
15	14	ovprc 7173	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
16	15	fveq1d 6647	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
17	13, 16	eqtr4d 2836	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
18	10, 17	pm2.61i 185	. 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
19	1, 18	eqtri 2821	1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ∅c0 4243  ‘cfv 6324  (class class class)co 7135  Basecbs 16475   evalSub ces 20743   eval cevl 20744

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-evls 20745  df-evl 20746

This theorem is referenced by:  evlrhm  20768  evlsscasrng  20769  evlsvarsrng  20771  evl1fval1lem  20954  evl1sca  20958  evl1var  20960  pf1rcl  20973  mpfpf1  20975  pf1ind  20979  mzpmfp  39688