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Theorem evl1fval 22348
Description: Value of the simple/same ring evaluation map. (Contributed by Mario Carneiro, 12-Jun-2015.)
Hypotheses
Ref Expression
evl1fval.o 𝑂 = (eval1𝑅)
evl1fval.q 𝑄 = (1o eval 𝑅)
evl1fval.b 𝐵 = (Base‘𝑅)
Assertion
Ref Expression
evl1fval 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝑄   𝑥,𝑅
Allowed substitution hints:   𝑄(𝑦)   𝑅(𝑦)   𝑂(𝑥,𝑦)

Proof of Theorem evl1fval
Dummy variables 𝑖 𝑟 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evl1fval.o . . 3 𝑂 = (eval1𝑅)
2 fvexd 6922 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 id 22 . . . . . . . . 9 (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 evl1fval.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2793 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
73, 6sylan9eqr 2797 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
87oveq1d 7446 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏m 1o) = (𝐵m 1o))
97, 8oveq12d 7449 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏m (𝑏m 1o)) = (𝐵m (𝐵m 1o)))
107mpteq1d 5243 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
1110coeq2d 5876 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
129, 11mpteq12dv 5239 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
13 simpl 482 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
1413oveq2d 7447 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅))
15 evl1fval.q . . . . . . 7 𝑄 = (1o eval 𝑅)
1614, 15eqtr4di 2793 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
1712, 16coeq12d 5878 . . . . 5 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → ((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
182, 17csbied 3946 . . . 4 (𝑟 = 𝑅(Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
19 df-evl1 22336 . . . 4 eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
20 ovex 7464 . . . . . 6 (𝐵m (𝐵m 1o)) ∈ V
2120mptex 7243 . . . . 5 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ V
2215ovexi 7465 . . . . 5 𝑄 ∈ V
2321, 22coex 7953 . . . 4 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V
2418, 19, 23fvmpt 7016 . . 3 (𝑅 ∈ V → (eval1𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
251, 24eqtrid 2787 . 2 (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26 fvprc 6899 . . . . 5 𝑅 ∈ V → (eval1𝑅) = ∅)
271, 26eqtrid 2787 . . . 4 𝑅 ∈ V → 𝑂 = ∅)
28 co02 6282 . . . 4 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
2927, 28eqtr4di 2793 . . 3 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30 df-evl 22117 . . . . . . 7 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
3130reldmmpo 7567 . . . . . 6 Rel dom eval
3231ovprc2 7471 . . . . 5 𝑅 ∈ V → (1o eval 𝑅) = ∅)
3315, 32eqtrid 2787 . . . 4 𝑅 ∈ V → 𝑄 = ∅)
3433coeq2d 5876 . . 3 𝑅 ∈ V → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
3529, 34eqtr4d 2778 . 2 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
3625, 35pm2.61i 182 1 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  csb 3908  c0 4339  {csn 4631  cmpt 5231   × cxp 5687  ccom 5693  cfv 6563  (class class class)co 7431  1oc1o 8498  m cmap 8865  Basecbs 17245   evalSub ces 22114   eval cevl 22115  eval1ce1 22334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-evl 22117  df-evl1 22336
This theorem is referenced by:  evl1val  22349  evl1fval1lem  22350  evl1rhm  22352  pf1rcl  22369
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