Theorem evlval 20302

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 7159	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3		fveq2 6664	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		evlval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	syl6eqr 2874	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	adantl 484	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
7	2, 6	fveq12d 6671	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8		df-evl 20281	. . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9		fvex 6677	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7299	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
11	8	mpondm0 7380	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12		0fv 6703	. . . . 5 ⊢ (∅‘𝐵) = ∅
13	11, 12	syl6eqr 2874	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14		reldmevls 20291	. . . . . 6 ⊢ Rel dom evalSub
15	14	ovprc 7188	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
16	15	fveq1d 6666	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
17	13, 16	eqtr4d 2859	. . 3 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
18	10, 17	pm2.61i 184	. 2 ⊢ (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
19	1, 18	eqtri 2844	1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 398   = wceq 1533   ∈ wcel 2110  Vcvv 3494  ∅c0 4290  ‘cfv 6349  (class class class)co 7150  Basecbs 16477   evalSub ces 20278   eval cevl 20279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-iota 6308  df-fun 6351  df-fv 6357  df-ov 7153  df-oprab 7154  df-mpo 7155  df-evls 20280  df-evl 20281

This theorem is referenced by:  evlrhm  20303  evlsscasrng  20304  evlsvarsrng  20306  evl1fval1lem  20487  evl1sca  20491  evl1var  20493  pf1rcl  20506  mpfpf1  20508  pf1ind  20512  mzpmfp  39337