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	⊢ ((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 3185	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
3	2	adantr 480	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4		simpr 476	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5		ovex 6577	. . . . . 6 ⊢ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ∈ V
6	5	mptex 6390	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∈ V
7		fvex 6113	. . . . 5 ⊢ (𝐸‘𝑅) ∈ V
8	6, 7	coex 7011	. . . 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 6103	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
11	10	adantr 480	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12		evls1fval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	syl6eqr 2662	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
14	13	csbeq1d 3506	. . . . 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		fvex 6113	. . . . . . . 8 ⊢ (Base‘𝑆) ∈ V
16	12, 15	eqeltri 2684	. . . . . . 7 ⊢ 𝐵 ∈ V
17	16	a1i 11	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝐵 ∈ V)
18		id 22	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
19		oveq1 6556	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ↑𝑚 1𝑜) = (𝐵 ↑𝑚 1𝑜))
20	18, 19	oveq12d 6567	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) = (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)))
21		mpteq1 4665	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))
22	21	coeq2d 5206	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
23	20, 22	mpteq12dv 4663	. . . . . . . 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})))))
24	23	coeq1d 5205	. . . . . . 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)))
25	24	adantl 481	. . . . . 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)))
26	17, 25	csbied 3526	. . . . 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)))
27		oveq2 6557	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (1𝑜 evalSub 𝑠) = (1𝑜 evalSub 𝑆))
28		evls1fval.e	. . . . . . . . 9 ⊢ 𝐸 = (1𝑜 evalSub 𝑆)
29	27, 28	syl6eqr 2662	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1𝑜 evalSub 𝑠) = 𝐸)
30	29	adantr 480	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (1𝑜 evalSub 𝑠) = 𝐸)
31		simpr 476	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
32	30, 31	fveq12d 6109	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((1𝑜 evalSub 𝑠)‘𝑟) = (𝐸‘𝑅))
33	32	coeq2d 5206	. . . . 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)))
34	14, 26, 33	3eqtrd 2648	. . . 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)))
35	10, 12	syl6eqr 2662	. . . . 5 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
36	35	pweqd 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 ∈ (𝑏 ↑𝑚 (𝑏 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑠)‘𝑟)))
38	34, 36, 37	ovmpt2x 6687	. . 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)))
39	3, 4, 9, 38	syl3anc 1318	. 2 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → (𝑆 evalSub1 𝑅) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ (𝐸‘𝑅)))
40	1, 39	syl5eq 2656	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 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  1_𝑜c1o 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