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	⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))

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.

Step	Hyp	Ref	Expression
1		evls1fval.q	. 2 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
2		elex 3426	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
3	2	adantr 484	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4		simpr 488	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5		ovex 7246	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
6	5	mptex 7039	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
7		fvex 6730	. . . . 5 ⊢ (𝐸‘𝑅) ∈ V
8	6, 7	coex 7708	. . . 4 ⊢ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)) ∈ V
9	8	a1i 11	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)) ∈ V)
10		fveq2 6717	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
11	10	adantr 484	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12		evls1fval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2796	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
14	13	csbeq1d 3815	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ⦋𝐵 / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
15	12	fvexi 6731	. . . . . . 7 ⊢ 𝐵 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝐵 ∈ V)
17		id 22	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
18		oveq1 7220	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
19	17, 18	oveq12d 7231	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
20		mpteq1 5143	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
21	20	coeq2d 5731	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
22	19, 21	mpteq12dv 5140	. . . . . . . 8 ⊢ (𝑏 = 𝐵 → (𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) = (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))))
23	22	coeq1d 5730	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
24	23	adantl 485	. . . . . 6 ⊢ (((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) ∧ 𝑏 = 𝐵) → ((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
25	16, 24	csbied 3849	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ⦋𝐵 / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (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)
28	26, 27	eqtr4di 2796	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
29	28	adantr 484	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30		simpr 488	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
31	29, 30	fveq12d 6724	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸‘𝑅))
32	31	coeq2d 5731	. . . . 5 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))
33	14, 25, 32	3eqtrd 2781	. . . 4 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ⦋(Base‘𝑠) / 𝑏⦌((𝑥 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))
34	10, 12	eqtr4di 2796	. . . . 5 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
35	34	pweqd 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 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
37	33, 35, 36	ovmpox 7362	. . 3 ⊢ ((𝑆 ∈ V ∧ 𝑅 ∈ 𝒫 𝐵 ∧ ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)) ∈ V) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))
38	3, 4, 9, 37	syl3anc 1373	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))
39	1, 38	syl5eq 2790	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))

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  1_oc1o 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