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	⊢ ((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 3460	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
3	2	adantr 482	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4		simpr 486	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5		ovex 7375	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
6	5	mptex 7160	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
7		fvex 6843	. . . . 5 ⊢ (𝐸‘𝑅) ∈ V
8	6, 7	coex 7850	. . . 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 6830	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
11	10	adantr 482	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12		evls1fval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2795	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
14	13	csbeq1d 3851	. . . . 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 6844	. . . . . . 7 ⊢ 𝐵 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝐵 ∈ V)
17		id 22	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
18		oveq1 7349	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
19	17, 18	oveq12d 7360	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
20		mpteq1 5190	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
21	20	coeq2d 5809	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
22	19, 21	mpteq12dv 5188	. . . . . . . 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 5808	. . . . . . 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 483	. . . . . 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 3885	. . . . 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 7350	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27		evls1fval.e	. . . . . . . . 9 ⊢ 𝐸 = (1o evalSub 𝑆)
28	26, 27	eqtr4di 2795	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
29	28	adantr 482	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30		simpr 486	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
31	29, 30	fveq12d 6837	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸‘𝑅))
32	31	coeq2d 5809	. . . . 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 2795	. . . . 5 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
35	34	pweqd 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 ∈ (𝑏 ↑m (𝑏 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑠)‘𝑟)))
37	33, 35, 36	ovmpox 7493	. . 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 1371	. 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	eqtrid 2789	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 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  1_oc1o 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