Theorem evlval 22142

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 7457	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3		fveq2 6920	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		evlval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2798	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	adantl 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
7	2, 6	fveq12d 6927	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8		df-evl 22122	. . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9		fvex 6933	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7605	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
11	8	mpondm0 7690	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12		0fv 6964	. . . . 5 ⊢ (∅‘𝐵) = ∅
13	11, 12	eqtr4di 2798	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14		reldmevls 22131	. . . . . 6 ⊢ Rel dom evalSub
15	14	ovprc 7486	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
16	15	fveq1d 6922	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
17	13, 16	eqtr4d 2783	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
18	10, 17	pm2.61i 182	. 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
19	1, 18	eqtri 2768	1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ‘cfv 6573  (class class class)co 7448  Basecbs 17258   evalSub ces 22119   eval cevl 22120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-iota 6525  df-fun 6575  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-evls 22121  df-evl 22122

This theorem is referenced by:  evlrhm  22143  evlsscasrng  22144  evlsvarsrng  22146  evl1fval1lem  22355  evl1sca  22359  evl1var  22361  pf1rcl  22374  mpfpf1  22376  pf1ind  22380  evlsevl  42526  mhphf4  42555  mzpmfp  42703