Theorem evls1fval 22642

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 3471	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
3	2	adantr 486	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑆 ∈ V)
4		simpr 490	. . 3 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑅 ∈ 𝒫 𝐵)
5		ovex 7453	. . . . . 6 ⊢ (𝐵 ↑m (𝐵 ↑m 1o)) ∈ V
6	5	mptex 7229	. . . . 5 ⊢ (𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∈ V
7		fvex 6898	. . . . 5 ⊢ (𝐸‘𝑅) ∈ V
8	6, 7	coex 7942	. . . 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 6885	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
11	10	adantr 486	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = (Base‘𝑆))
12		evls1fval.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
13	11, 12	eqtr4di 2813	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (Base‘𝑠) = 𝐵)
14	13	csbeq1d 3850	. . . . 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 6899	. . . . . . 7 ⊢ 𝐵 ∈ V
16	15	a1i 11	. . . . . 6 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝐵 ∈ V)
17		id 23	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → 𝑏 = 𝐵)
18		oveq1 7427	. . . . . . . . . 10 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m 1o) = (𝐵 ↑m 1o))
19	17, 18	oveq12d 7438	. . . . . . . . 9 ⊢ (𝑏 = 𝐵 → (𝑏 ↑m (𝑏 ↑m 1o)) = (𝐵 ↑m (𝐵 ↑m 1o)))
20		mpteq1 5193	. . . . . . . . . 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 5191	. . . . . . . 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 487	. . . . . 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 3882	. . . . 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 7428	. . . . . . . . 9 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = (1o evalSub 𝑆))
27		evls1fval.e	. . . . . . . . 9 ⊢ 𝐸 = (1o evalSub 𝑆)
28	26, 27	eqtr4di 2813	. . . . . . . 8 ⊢ (𝑠 = 𝑆 → (1o evalSub 𝑠) = 𝐸)
29	28	adantr 486	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (1o evalSub 𝑠) = 𝐸)
30		simpr 490	. . . . . . 7 ⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅)
31	29, 30	fveq12d 6892	. . . . . 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 2799	. . . 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 2813	. . . . 5 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = 𝐵)
35	34	pweqd 4573	. . . 4 ⊢ (𝑠 = 𝑆 → 𝒫 (Base‘𝑠) = 𝒫 𝐵)
36		df-evls1 22638	. . . 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 7573	. . 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 1398	. 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 2807	1 ⊢ ((𝑆 ∈ 𝑉 ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑m (𝐵 ↑m 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ (𝐸‘𝑅)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  Vcvv 3450  ⦋csb 3846  𝒫 cpw 4556  {csn 4584   ↦ cmpt 5185   × cxp 5648   ∘ ccom 5654  ‘cfv 6538  (class class class)co 7420  1_oc1o 8472   ↑_m cmap 8850  Basecbs 17392   evalSub ces 22386   evalSub₁ ces1 22636

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-evls1 22638

This theorem is used by:  evls1val  22643  evls1rhm  22645  evls1sca  22646  evl1fval1lem  22653