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Theorem evls1fval 21591
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 3460 . . . 4 (𝑆𝑉𝑆 ∈ V)
32adantr 482 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4 simpr 486 . . 3 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5 ovex 7375 . . . . . 6 (𝐵m (𝐵m 1o)) ∈ V
65mptex 7160 . . . . 5 (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∈ V
7 fvex 6843 . . . . 5 (𝐸𝑅) ∈ V
86, 7coex 7850 . . . 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 6830 . . . . . . . 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 2795 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
1413csbeq1d 3851 . . . . 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 6844 . . . . . . 7 𝐵 ∈ V
1615a1i 11 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → 𝐵 ∈ V)
17 id 22 . . . . . . . . . 10 (𝑏 = 𝐵𝑏 = 𝐵)
18 oveq1 7349 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑏m 1o) = (𝐵m 1o))
1917, 18oveq12d 7360 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑏m (𝑏m 1o)) = (𝐵m (𝐵m 1o)))
20 mpteq1 5190 . . . . . . . . . 10 (𝑏 = 𝐵 → (𝑦𝑏 ↦ (1o × {𝑦})) = (𝑦𝐵 ↦ (1o × {𝑦})))
2120coeq2d 5809 . . . . . . . . 9 (𝑏 = 𝐵 → (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦}))))
2219, 21mpteq12dv 5188 . . . . . . . 8 (𝑏 = 𝐵 → (𝑥 ∈ (𝑏m (𝑏m 1o)) ↦ (𝑥 ∘ (𝑦𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))))
2322coeq1d 5808 . . . . . . 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 3885 . . . . 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 7350 . . . . . . . . 9 (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27 evls1fval.e . . . . . . . . 9 𝐸 = (1o evalSub 𝑆)
2826, 27eqtr4di 2795 . . . . . . . 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 6837 . . . . . 6 ((𝑠 = 𝑆𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸𝑅))
3231coeq2d 5809 . . . . 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 2795 . . . . 5 (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
3534pweqd 4569 . . . 4 (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
36 df-evls1 21587 . . . 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 7493 . . 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 1371 . 2 ((𝑆𝑉𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵m (𝐵m 1o)) ↦ (𝑥 ∘ (𝑦𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸𝑅)))
391, 38eqtrid 2789 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 1541  wcel 2106  Vcvv 3442  csb 3847  𝒫 cpw 4552  {csn 4578  cmpt 5180   × cxp 5623  ccom 5629  cfv 6484  (class class class)co 7342  1oc1o 8365  m cmap 8691  Basecbs 17010   evalSub ces 21386   evalSub1 ces1 21585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5234  ax-sep 5248  ax-nul 5255  ax-pow 5313  ax-pr 5377  ax-un 7655
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3351  df-rab 3405  df-v 3444  df-sbc 3732  df-csb 3848  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4858  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5181  df-id 5523  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6436  df-fun 6486  df-fn 6487  df-f 6488  df-f1 6489  df-fo 6490  df-f1o 6491  df-fv 6492  df-ov 7345  df-oprab 7346  df-mpo 7347  df-evls1 21587
This theorem is referenced by:  evls1val  21592  evls1rhm  21594  evls1sca  21595  evl1fval1lem  21602
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