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Theorem evlval 19697
 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.
StepHypRef Expression
1 evlval.q . 2 𝑄 = (𝐼 eval 𝑅)
2 oveq12 6810 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3 fveq2 6340 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 evlval.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2800 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65adantl 473 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
72, 6fveq12d 6346 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8 df-evl 19680 . . . 4 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9 fvex 6350 . . . 4 ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
107, 8, 9ovmpt2a 6944 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
118mpt2ndm0 7028 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12 0fv 6376 . . . . 5 (∅‘𝐵) = ∅
1311, 12syl6eqr 2800 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14 reldmevls 19690 . . . . . 6 Rel dom evalSub
1514ovprc 6834 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
1615fveq1d 6342 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
1713, 16eqtr4d 2785 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
1810, 17pm2.61i 176 . 2 (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
191, 18eqtri 2770 1 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1620   ∈ wcel 2127  Vcvv 3328  ∅c0 4046  ‘cfv 6037  (class class class)co 6801  Basecbs 16030   evalSub ces 19677   eval cevl 19678 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-sbc 3565  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-uni 4577  df-br 4793  df-opab 4853  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-iota 6000  df-fun 6039  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-evls 19679  df-evl 19680 This theorem is referenced by:  evlrhm  19698  evlsscasrng  19699  evlsvarsrng  19701  evl1fval1lem  19867  evl1sca  19871  evl1var  19873  pf1rcl  19886  mpfpf1  19888  pf1ind  19892  mzpmfp  37781
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