Theorem evls1fval 22440

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 3478	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
3	2	adantr 485	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4		simpr 489	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5		ovex 7433	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
6	5	mptex 7211	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
7		fvex 6884	. . . . 5 ⊢ (𝐸‘𝑅) ∈ V
8	6, 7	coex 7915	. . . 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 6871	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
11	10	adantr 485	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12		evls1fval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2818	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
14	13	csbeq1d 3859	. . . . 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 6885	. . . . . . 7 ⊢ 𝐵 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝐵 ∈ V)
17		id 23	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
18		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
19	17, 18	oveq12d 7418	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
20		mpteq1 5194	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑦 ∈ 𝑏 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
21	20	coeq2d 5839	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑥 ∘ (𝑦 ∈ 𝑏 ↦ (1o × {𝑦}))) = (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
22	19, 21	mpteq12dv 5192	. . . . . . . 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 5838	. . . . . . 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 486	. . . . . 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 3891	. . . . 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 7408	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27		evls1fval.e	. . . . . . . . 9 ⊢ 𝐸 = (1o evalSub 𝑆)
28	26, 27	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
29	28	adantr 485	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30		simpr 489	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
31	29, 30	fveq12d 6878	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((1o evalSub 𝑠)‘𝑟) = (𝐸‘𝑅))
32	31	coeq2d 5839	. . . . 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 2804	. . . 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 2818	. . . . 5 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
35	34	pweqd 4575	. . . 4 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
36		df-evls1 22436	. . . 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 7553	. . 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 1394	. 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 2812	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 400   = wceq 1563   ∈ wcel 2145  Vcvv 3457  ⦋csb 3855  𝒫 cpw 4558  {csn 4585   ↦ cmpt 5186   × cxp 5650   ∘ ccom 5656  ‘cfv 6525  (class class class)co 7400  1_oc1o 8434   ↑_m cmap 8812  Basecbs 17259   evalSub ces 22183   evalSub₁ ces1 22434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-evls1 22436

This theorem is referenced by:  evls1val  22441  evls1rhm  22443  evls1sca  22444  evl1fval1lem  22451