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Theorem evl1fval 22332
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 6921 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 id 22 . . . . . . . . 9 (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4 fveq2 6906 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 evl1fval.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2795 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
73, 6sylan9eqr 2799 . . . . . . . 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 5237 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
1110coeq2d 5873 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
129, 11mpteq12dv 5233 . . . . . 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 2795 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
1712, 16coeq12d 5875 . . . . 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 3935 . . . 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 22320 . . . 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 7952 . . . 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 2789 . 2 (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26 fvprc 6898 . . . . 5 𝑅 ∈ V → (eval1𝑅) = ∅)
271, 26eqtrid 2789 . . . 4 𝑅 ∈ V → 𝑂 = ∅)
28 co02 6280 . . . 4 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
2927, 28eqtr4di 2795 . . 3 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30 df-evl 22099 . . . . . . 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 2789 . . . 4 𝑅 ∈ V → 𝑄 = ∅)
3433coeq2d 5873 . . 3 𝑅 ∈ V → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
3529, 34eqtr4d 2780 . 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 1540  wcel 2108  Vcvv 3480  csb 3899  c0 4333  {csn 4626  cmpt 5225   × cxp 5683  ccom 5689  cfv 6561  (class class class)co 7431  1oc1o 8499  m cmap 8866  Basecbs 17247   evalSub ces 22096   eval cevl 22097  eval1ce1 22318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-evl 22099  df-evl1 22320
This theorem is referenced by:  evl1val  22333  evl1fval1lem  22334  evl1rhm  22336  pf1rcl  22353
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