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Theorem evls1fval 21838
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 3493 . . . 4 (𝑆𝑉𝑆 ∈ V)
32adantr 482 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4 simpr 486 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5 ovex 7442 . . . . . 6 (𝐵m (𝐵m 1o)) ∈ V
65mptex 7225 . . . . 5 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ V
7 fvex 6905 . . . . 5 (𝐸𝑅) ∈ V
86, 7coex 7921 . . . 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 6892 . . . . . . . 8 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
1110adantr 482 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12 evls1fval.b . . . . . . 7 𝐵 = (Base‘𝑆)
1311, 12eqtr4di 2791 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
1413csbeq1d 3898 . . . . 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 6906 . . . . . . 7 𝐵 ∈ V
1615a1i 11 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝐵 ∈ V)
17 id 22 . . . . . . . . . 10 (𝑏 = 𝐵𝑏 = 𝐵)
18 oveq1 7416 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏m 1o) = (𝐵m 1o))
1917, 18oveq12d 7427 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏m (𝑏m 1o)) = (𝐵m (𝐵m 1o)))
20 mpteq1 5242 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
2120coeq2d 5863 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
2219, 21mpteq12dv 5240 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
2322coeq1d 5862 . . . . . . 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 483 . . . . . 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 3932 . . . . 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 7417 . . . . . . . . 9 (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27 evls1fval.e . . . . . . . . 9 𝐸 = (1o evalSub 𝑆)
2826, 27eqtr4di 2791 . . . . . . . 8 (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
2928adantr 482 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30 simpr 486 . . . . . . 7 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝑟 = 𝑅)
3129, 30fveq12d 6899 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸𝑅))
3231coeq2d 5863 . . . . 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 2777 . . . 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 2791 . . . . 5 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
3534pweqd 4620 . . . 4 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
36 df-evls1 21834 . . . 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 7561 . . 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 1372 . 2 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))
391, 38eqtrid 2785 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 397   = wceq 1542  wcel 2107  Vcvv 3475  csb 3894  𝒫 cpw 4603  {csn 4629  cmpt 5232   × cxp 5675  ccom 5681  cfv 6544  (class class class)co 7409  1oc1o 8459  m cmap 8820  Basecbs 17144   evalSub ces 21633   evalSub1 ces1 21832
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7412  df-oprab 7413  df-mpo 7414  df-evls1 21834
This theorem is referenced by:  evls1val  21839  evls1rhm  21841  evls1sca  21842  evl1fval1lem  21849
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