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Theorem evls1fval 19505
 Description: Value of the univariate polynomial evaluation map function. (Contributed by AV, 7-Sep-2019.)
Hypotheses
Ref Expression
evls1fval.q 𝑄 = (𝑆 evalSub1 𝑅)
evls1fval.e 𝐸 = (1𝑜 evalSub 𝑆)
evls1fval.b 𝐵 = (Base‘𝑆)
Assertion
Ref Expression
evls1fval ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
Distinct variable group:   𝑥,𝐵,𝑦
Allowed substitution hints:   𝑄(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐸(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem evls1fval
Dummy variables 𝑏 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evls1fval.q . 2 𝑄 = (𝑆 evalSub1 𝑅)
2 elex 3185 . . . 4 (𝑆𝑉𝑆 ∈ V)
32adantr 480 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4 simpr 476 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5 ovex 6577 . . . . . 6 (𝐵𝑚 (𝐵𝑚 1𝑜)) ∈ V
65mptex 6390 . . . . 5 (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∈ V
7 fvex 6113 . . . . 5 (𝐸𝑅) ∈ V
86, 7coex 7011 . . . 4 ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)) ∈ V
98a1i 11 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)) ∈ V)
10 fveq2 6103 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1110adantr 480 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12 evls1fval.b . . . . . . 7 𝐵 = (Base‘𝑆)
1311, 12syl6eqr 2662 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
1413csbeq1d 3506 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = 𝐵 / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
15 fvex 6113 . . . . . . . 8 (Base‘𝑆) ∈ V
1612, 15eqeltri 2684 . . . . . . 7 𝐵 ∈ V
1716a1i 11 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝐵 ∈ V)
18 id 22 . . . . . . . . . 10 (𝑏 = 𝐵𝑏 = 𝐵)
19 oveq1 6556 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏𝑚 1𝑜) = (𝐵𝑚 1𝑜))
2018, 19oveq12d 6567 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏𝑚 (𝑏𝑚 1𝑜)) = (𝐵𝑚 (𝐵𝑚 1𝑜)))
21 mpteq1 4665 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (1𝑜 × {𝑦})) = (𝑦𝐵 ↦ (1𝑜 × {𝑦})))
2221coeq2d 5206 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦}))))
2320, 22mpteq12dv 4663 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))))
2423coeq1d 5205 . . . . . . 7 (𝑏 = 𝐵 → ((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
2524adantl 481 . . . . . 6 (((𝑠 = 𝑆𝑟 = 𝑅) ∧ 𝑏 = 𝐵) → ((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
2617, 25csbied 3526 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝐵 / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
27 oveq2 6557 . . . . . . . . 9 (𝑠 = 𝑆 → (1𝑜 evalSub 𝑠) = (1𝑜 evalSub 𝑆))
28 evls1fval.e . . . . . . . . 9 𝐸 = (1𝑜 evalSub 𝑆)
2927, 28syl6eqr 2662 . . . . . . . 8 (𝑠 = 𝑆 → (1𝑜 evalSub 𝑠) = 𝐸)
3029adantr 480 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (1𝑜 evalSub 𝑠) = 𝐸)
31 simpr 476 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3230, 31fveq12d 6109 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → ((1𝑜 evalSub 𝑠)‘𝑟) = (𝐸𝑅))
3332coeq2d 5206 . . . . 5 ((𝑠 = 𝑆𝑟 = 𝑅) → ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
3414, 26, 333eqtrd 2648 . . . 4 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
3510, 12syl6eqr 2662 . . . . 5 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
3635pweqd 4113 . . . 4 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
37 df-evls1 19501 . . . 4 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏𝑚 (𝑏𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
3834, 36, 37ovmpt2x 6687 . . 3 ((𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 𝐵 ∧ ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)) ∈ V) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
393, 4, 9, 38syl3anc 1318 . 2 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
401, 39syl5eq 2656 1 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵𝑚 (𝐵𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸𝑅)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⦋csb 3499  𝒫 cpw 4108  {csn 4125   ↦ cmpt 4643   × cxp 5036   ∘ ccom 5042  ‘cfv 5804  (class class class)co 6549  1𝑜c1o 7440   ↑𝑚 cmap 7744  Basecbs 15695   evalSub ces 19325   evalSub1 ces1 19499 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evls1 19501 This theorem is referenced by:  evls1val  19506  evls1rhm  19508  evls1sca  19509  evl1fval1lem  19515
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