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Theorem evlval 19891
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 6919 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3 fveq2 6437 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 evlval.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2879 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65adantl 475 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
72, 6fveq12d 6444 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8 df-evl 19874 . . . 4 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9 fvex 6450 . . . 4 ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
107, 8, 9ovmpt2a 7056 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
118mpt2ndm0 7140 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12 0fv 6477 . . . . 5 (∅‘𝐵) = ∅
1311, 12syl6eqr 2879 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14 reldmevls 19884 . . . . . 6 Rel dom evalSub
1514ovprc 6947 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
1615fveq1d 6439 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
1713, 16eqtr4d 2864 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
1810, 17pm2.61i 177 . 2 (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
191, 18eqtri 2849 1 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 386   = wceq 1656  wcel 2164  Vcvv 3414  c0 4146  cfv 6127  (class class class)co 6910  Basecbs 16229   evalSub ces 19871   eval cevl 19872
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-id 5252  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-iota 6090  df-fun 6129  df-fv 6135  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-evls 19873  df-evl 19874
This theorem is referenced by:  evlrhm  19892  evlsscasrng  19893  evlsvarsrng  19895  evl1fval1lem  20061  evl1sca  20065  evl1var  20067  pf1rcl  20080  mpfpf1  20082  pf1ind  20086  mzpmfp  38149
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