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Theorem evl1fval 19914
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 6365 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
3 id 22 . . . . . . . . 9 (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
4 fveq2 6353 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
5 evl1fval.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
64, 5syl6eqr 2812 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
73, 6sylan9eqr 2816 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
87oveq1d 6829 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏𝑚 1𝑜) = (𝐵𝑚 1𝑜))
97, 8oveq12d 6832 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏𝑚 (𝑏𝑚 1𝑜)) = (𝐵𝑚 (𝐵𝑚 1𝑜)))
107mpteq1d 4890 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑦𝑏 ↦ (1𝑜 × {𝑦})) = (𝑦𝐵 ↦ (1𝑜 × {𝑦})))
1110coeq2d 5440 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
129, 11mpteq12dv 4885 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))))
13 simpl 474 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
1413oveq2d 6830 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = (1𝑜 eval 𝑅))
15 evl1fval.q . . . . . . 7 𝑄 = (1𝑜 eval 𝑅)
1614, 15syl6eqr 2812 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = 𝑄)
1712, 16coeq12d 5442 . . . . 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 3701 . . . 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 19903 . . . 4 eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
20 ovex 6842 . . . . . 6 (𝐵𝑚 (𝐵𝑚 1𝑜)) ∈ V
2120mptex 6651 . . . . 5 (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∈ V
22 ovex 6842 . . . . . 6 (1𝑜 eval 𝑅) ∈ V
2315, 22eqeltri 2835 . . . . 5 𝑄 ∈ V
2421, 23coex 7284 . . . 4 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄) ∈ V
2518, 19, 24fvmpt 6445 . . 3 (𝑅 ∈ V → (eval1𝑅) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
261, 25syl5eq 2806 . 2 (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
27 fvprc 6347 . . . . 5 𝑅 ∈ V → (eval1𝑅) = ∅)
281, 27syl5eq 2806 . . . 4 𝑅 ∈ V → 𝑂 = ∅)
29 co02 5810 . . . 4 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅) = ∅
3028, 29syl6eqr 2812 . . 3 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅))
31 df-evl 19729 . . . . . . 7 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
3231reldmmpt2 6937 . . . . . 6 Rel dom eval
3332ovprc2 6849 . . . . 5 𝑅 ∈ V → (1𝑜 eval 𝑅) = ∅)
3415, 33syl5eq 2806 . . . 4 𝑅 ∈ V → 𝑄 = ∅)
3534coeq2d 5440 . . 3 𝑅 ∈ V → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅))
3630, 35eqtr4d 2797 . 2 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
3726, 36pm2.61i 176 1 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 383   = wceq 1632  wcel 2139  Vcvv 3340  csb 3674  c0 4058  {csn 4321  cmpt 4881   × cxp 5264  ccom 5270  cfv 6049  (class class class)co 6814  1𝑜c1o 7723  𝑚 cmap 8025  Basecbs 16079   evalSub ces 19726   eval cevl 19727  eval1ce1 19901
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-evl 19729  df-evl1 19903
This theorem is referenced by:  evl1val  19915  evl1fval1lem  19916  evl1rhm  19918  pf1rcl  19935
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