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Theorem evl1fval 20966
 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 6665 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 id 22 . . . . . . . . 9 (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4 fveq2 6650 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 evl1fval.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
64, 5eqtr4di 2851 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
73, 6sylan9eqr 2855 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
87oveq1d 7155 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏m 1o) = (𝐵m 1o))
97, 8oveq12d 7158 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏m (𝑏m 1o)) = (𝐵m (𝐵m 1o)))
107mpteq1d 5120 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
1110coeq2d 5698 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
129, 11mpteq12dv 5116 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
13 simpl 486 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
1413oveq2d 7156 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = (1o eval 𝑅))
15 evl1fval.q . . . . . . 7 𝑄 = (1o eval 𝑅)
1614, 15eqtr4di 2851 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1o eval 𝑟) = 𝑄)
1712, 16coeq12d 5700 . . . . 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 3864 . . . 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 20954 . . . 4 eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ (1o eval 𝑟)))
20 ovex 7173 . . . . . 6 (𝐵m (𝐵m 1o)) ∈ V
2120mptex 6968 . . . . 5 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ V
2215ovexi 7174 . . . . 5 𝑄 ∈ V
2321, 22coex 7624 . . . 4 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) ∈ V
2418, 19, 23fvmpt 6750 . . 3 (𝑅 ∈ V → (eval1𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
251, 24syl5eq 2845 . 2 (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
26 fvprc 6642 . . . . 5 𝑅 ∈ V → (eval1𝑅) = ∅)
271, 26syl5eq 2845 . . . 4 𝑅 ∈ V → 𝑂 = ∅)
28 co02 6083 . . . 4 ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅) = ∅
2927, 28eqtr4di 2851 . . 3 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
30 df-evl 20756 . . . . . . 7 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
3130reldmmpo 7270 . . . . . 6 Rel dom eval
3231ovprc2 7180 . . . . 5 𝑅 ∈ V → (1o eval 𝑅) = ∅)
3315, 32syl5eq 2845 . . . 4 𝑅 ∈ V → 𝑄 = ∅)
3433coeq2d 5698 . . 3 𝑅 ∈ V → ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ ∅))
3529, 34eqtr4d 2836 . 2 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄))
3625, 35pm2.61i 185 1 𝑂 = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ 𝑄)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  ⦋csb 3828  ∅c0 4243  {csn 4525   ↦ cmpt 5111   × cxp 5518   ∘ ccom 5524  ‘cfv 6327  (class class class)co 7140  1oc1o 8085   ↑m cmap 8396  Basecbs 16482   evalSub ces 20753   eval cevl 20754  eval1ce1 20952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7448 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-ov 7143  df-oprab 7144  df-mpo 7145  df-evl 20756  df-evl1 20954 This theorem is referenced by:  evl1val  20967  evl1fval1lem  20968  evl1rhm  20970  pf1rcl  20987
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