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Theorem evlval 21967
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 7410 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3 fveq2 6881 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 evlval.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4eqtr4di 2782 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65adantl 481 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
72, 6fveq12d 6888 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8 df-evl 21945 . . . 4 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9 fvex 6894 . . . 4 ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
107, 8, 9ovmpoa 7555 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
118mpondm0 7640 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12 0fv 6925 . . . . 5 (∅‘𝐵) = ∅
1311, 12eqtr4di 2782 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14 reldmevls 21956 . . . . . 6 Rel dom evalSub
1514ovprc 7439 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
1615fveq1d 6883 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
1713, 16eqtr4d 2767 . . 3 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
1810, 17pm2.61i 182 . 2 (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵)
191, 18eqtri 2752 1 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1533  wcel 2098  Vcvv 3466  c0 4314  cfv 6533  (class class class)co 7401  Basecbs 17142   evalSub ces 21942   eval cevl 21943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-iota 6485  df-fun 6535  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-evls 21944  df-evl 21945
This theorem is referenced by:  evlrhm  21968  evlsscasrng  21969  evlsvarsrng  21971  evl1fval1lem  22170  evl1sca  22174  evl1var  22176  pf1rcl  22189  mpfpf1  22191  pf1ind  22195  evlsevl  41598  mhphf4  41627  mzpmfp  41940
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