Theorem evlval 22055

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 7367	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3		fveq2 6834	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4		evlval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑅)
5	3, 4	eqtr4di 2789	. . . . . 6 ⊢ (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
6	5	adantl 481	. . . . 5 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
7	2, 6	fveq12d 6841	. . . 4 ⊢ ((𝑖 = 𝐼 ∧ 𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8		df-evl 22030	. . . 4 ⊢ eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9		fvex 6847	. . . 4 ⊢ ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
10	7, 8, 9	ovmpoa 7513	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
11	8	mpondm0 7598	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12		0fv 6875	. . . . 5 ⊢ (∅‘𝐵) = ∅
13	11, 12	eqtr4di 2789	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14		reldmevls 22039	. . . . . 6 ⊢ Rel dom evalSub
15	14	ovprc 7396	. . . . 5 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
16	15	fveq1d 6836	. . . 4 ⊢ (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
17	13, 16	eqtr4d 2774	. . 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 2759	1 ⊢ 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1541   ∈ wcel 2113  Vcvv 3440  ∅c0 4285  ‘cfv 6492  (class class class)co 7358  Basecbs 17136   evalSub ces 22027   eval cevl 22028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-evls 22029  df-evl 22030

This theorem is referenced by:  evlrhm  22056  evlsscasrng  22060  evlsvarsrng  22062  evl1fval1lem  22274  evl1sca  22278  evl1var  22280  pf1rcl  22293  mpfpf1  22295  pf1ind  22299  evlextv  33707  evlsevl  42813  mhphf4  42839  mzpmfp  42985