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Theorem evlval 19443
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 6613 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (𝑖 evalSub 𝑟) = (𝐼 evalSub 𝑅))
3 fveq2 6148 . . . . . . 7 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
4 evlval.b . . . . . . 7 𝐵 = (Base‘𝑅)
53, 4syl6eqr 2673 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
65adantl 482 . . . . 5 ((𝑖 = 𝐼𝑟 = 𝑅) → (Base‘𝑟) = 𝐵)
72, 6fveq12d 6154 . . . 4 ((𝑖 = 𝐼𝑟 = 𝑅) → ((𝑖 evalSub 𝑟)‘(Base‘𝑟)) = ((𝐼 evalSub 𝑅)‘𝐵))
8 df-evl 19426 . . . 4 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
9 fvex 6158 . . . 4 ((𝐼 evalSub 𝑅)‘𝐵) ∈ V
107, 8, 9ovmpt2a 6744 . . 3 ((𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ((𝐼 evalSub 𝑅)‘𝐵))
118mpt2ndm0 6828 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = ∅)
12 0fv 6184 . . . . 5 (∅‘𝐵) = ∅
1311, 12syl6eqr 2673 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 eval 𝑅) = (∅‘𝐵))
14 reldmevls 19436 . . . . . 6 Rel dom evalSub
1514ovprc 6636 . . . . 5 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → (𝐼 evalSub 𝑅) = ∅)
1615fveq1d 6150 . . . 4 (¬ (𝐼 ∈ V ∧ 𝑅 ∈ V) → ((𝐼 evalSub 𝑅)‘𝐵) = (∅‘𝐵))
1713, 16eqtr4d 2658 . . 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 2643 1 𝑄 = ((𝐼 evalSub 𝑅)‘𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  c0 3891  cfv 5847  (class class class)co 6604  Basecbs 15781   evalSub ces 19423   eval cevl 19424
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-evls 19425  df-evl 19426
This theorem is referenced by:  evlrhm  19444  evlsscasrng  19445  evlsvarsrng  19447  evl1fval1lem  19613  evl1sca  19617  evl1var  19619  pf1rcl  19632  mpfpf1  19634  pf1ind  19638  mzpmfp  36787
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