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Theorem evls1fval 21235
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 3426 . . . 4 (𝑆𝑉𝑆 ∈ V)
32adantr 484 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4 simpr 488 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5 ovex 7246 . . . . . 6 (𝐵m (𝐵m 1o)) ∈ V
65mptex 7039 . . . . 5 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ V
7 fvex 6730 . . . . 5 (𝐸𝑅) ∈ V
86, 7coex 7708 . . . 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 6717 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1110adantr 484 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12 evls1fval.b . . . . . . 7 𝐵 = (Base‘𝑆)
1311, 12eqtr4di 2796 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
1413csbeq1d 3815 . . . . 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)))
1512fvexi 6731 . . . . . . 7 𝐵 ∈ V
1615a1i 11 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝐵 ∈ V)
17 id 22 . . . . . . . . . 10 (𝑏 = 𝐵𝑏 = 𝐵)
18 oveq1 7220 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏m 1o) = (𝐵m 1o))
1917, 18oveq12d 7231 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏m (𝑏m 1o)) = (𝐵m (𝐵m 1o)))
20 mpteq1 5143 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
2120coeq2d 5731 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
2219, 21mpteq12dv 5140 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
2322coeq1d 5730 . . . . . . 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)))
2423adantl 485 . . . . . 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)))
2516, 24csbied 3849 . . . . 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)))
26 oveq2 7221 . . . . . . . . 9 (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27 evls1fval.e . . . . . . . . 9 𝐸 = (1o evalSub 𝑆)
2826, 27eqtr4di 2796 . . . . . . . 8 (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
2928adantr 484 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30 simpr 488 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3129, 30fveq12d 6724 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸𝑅))
3231coeq2d 5731 . . . . 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)))
3314, 25, 323eqtrd 2781 . . . 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)))
3410, 12eqtr4di 2796 . . . . 5 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
3534pweqd 4532 . . . 4 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
36 df-evls1 21231 . . . 4 evalSub1 = (𝑠 ∈ V, 𝑟 ∈ 𝒫 (Base‘𝑠) ↦ (Base‘𝑠) / 𝑏((𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
3733, 35, 36ovmpox 7362 . . 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)))
383, 4, 9, 37syl3anc 1373 . 2 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))
391, 38syl5eq 2790 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 399   = wceq 1543  wcel 2110  Vcvv 3408  csb 3811  𝒫 cpw 4513  {csn 4541  cmpt 5135   × cxp 5549  ccom 5555  cfv 6380  (class class class)co 7213  1oc1o 8195  m cmap 8508  Basecbs 16760   evalSub ces 21030   evalSub1 ces1 21229
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-iun 4906  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-ov 7216  df-oprab 7217  df-mpo 7218  df-evls1 21231
This theorem is referenced by:  evls1val  21236  evls1rhm  21238  evls1sca  21239  evl1fval1lem  21246
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