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Theorem evl1fval 19611
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 fvex 6158 . . . . . 6 (Base‘𝑟) ∈ V
32a1i 11 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) ∈ V)
4 id 22 . . . . . . . . 9 (𝑏 = (Base‘𝑟) → 𝑏 = (Base‘𝑟))
5 fveq2 6148 . . . . . . . . . 10 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
6 evl1fval.b . . . . . . . . . 10 𝐵 = (Base‘𝑅)
75, 6syl6eqr 2673 . . . . . . . . 9 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
84, 7sylan9eqr 2677 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑏 = 𝐵)
98oveq1d 6619 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏𝑚 1𝑜) = (𝐵𝑚 1𝑜))
108, 9oveq12d 6622 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑏𝑚 (𝑏𝑚 1𝑜)) = (𝐵𝑚 (𝐵𝑚 1𝑜)))
118mpteq1d 4698 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑦𝑏 ↦ (1𝑜 × {𝑦})) = (𝑦𝐵 ↦ (1𝑜 × {𝑦})))
1211coeq2d 5244 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
1310, 12mpteq12dv 4693 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))))
14 simpl 473 . . . . . . . 8 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → 𝑟 = 𝑅)
1514oveq2d 6620 . . . . . . 7 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = (1𝑜 eval 𝑅))
16 evl1fval.q . . . . . . 7 𝑄 = (1𝑜 eval 𝑅)
1715, 16syl6eqr 2673 . . . . . 6 ((𝑟 = 𝑅𝑏 = (Base‘𝑟)) → (1𝑜 eval 𝑟) = 𝑄)
1813, 17coeq12d 5246 . . . . 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))
193, 18csbied 3541 . . . 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))
20 df-evl1 19600 . . . 4 eval1 = (𝑟 ∈ V ↦ (Base‘𝑟) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ (1𝑜 eval 𝑟)))
21 ovex 6632 . . . . . 6 (𝐵𝑚 (𝐵𝑚 1𝑜)) ∈ V
2221mptex 6440 . . . . 5 (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∈ V
23 ovex 6632 . . . . . 6 (1𝑜 eval 𝑅) ∈ V
2416, 23eqeltri 2694 . . . . 5 𝑄 ∈ V
2522, 24coex 7065 . . . 4 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄) ∈ V
2619, 20, 25fvmpt 6239 . . 3 (𝑅 ∈ V → (eval1𝑅) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
271, 26syl5eq 2667 . 2 (𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
28 fvprc 6142 . . . . 5 𝑅 ∈ V → (eval1𝑅) = ∅)
291, 28syl5eq 2667 . . . 4 𝑅 ∈ V → 𝑂 = ∅)
30 co02 5608 . . . 4 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅) = ∅
3129, 30syl6eqr 2673 . . 3 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅))
32 df-evl 19426 . . . . . . 7 eval = (𝑖 ∈ V, 𝑟 ∈ V ↦ ((𝑖 evalSub 𝑟)‘(Base‘𝑟)))
3332reldmmpt2 6724 . . . . . 6 Rel dom eval
3433ovprc2 6638 . . . . 5 𝑅 ∈ V → (1𝑜 eval 𝑅) = ∅)
3516, 34syl5eq 2667 . . . 4 𝑅 ∈ V → 𝑄 = ∅)
3635coeq2d 5244 . . 3 𝑅 ∈ V → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ∅))
3731, 36eqtr4d 2658 . 2 𝑅 ∈ V → 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄))
3827, 37pm2.61i 176 1 𝑂 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  csb 3514  c0 3891  {csn 4148  cmpt 4673   × cxp 5072  ccom 5078  cfv 5847  (class class class)co 6604  1𝑜c1o 7498  𝑚 cmap 7802  Basecbs 15781   evalSub ces 19423   eval cevl 19424  eval1ce1 19598
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-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-evl 19426  df-evl1 19600
This theorem is referenced by:  evl1val  19612  evl1fval1lem  19613  evl1rhm  19615  pf1rcl  19632
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