Theorem evlsval 19519

Description: Value of the polynomial evaluation map function. (Contributed by Stefan O'Rear, 11-Mar-2015.) (Revised by AV, 18-Sep-2021.)

Hypotheses

Ref	Expression
evlsval.q	⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
evlsval.w	⊢ 𝑊 = (𝐼 mPoly 𝑈)
evlsval.v	⊢ 𝑉 = (𝐼 mVar 𝑈)
evlsval.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
evlsval.t	⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑𝑚 𝐼))
evlsval.b	⊢ 𝐵 = (Base‘𝑆)
evlsval.a	⊢ 𝐴 = (algSc‘𝑊)
evlsval.x	⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))
evlsval.y	⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))

Assertion

Ref	Expression
evlsval	⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)))

Distinct variable groups:   𝑓,𝐼,𝑔,𝑥   𝑅,𝑓,𝑥   𝑆,𝑓,𝑔,𝑥   𝑇,𝑓   𝑓,𝑊

Allowed substitution hints:   𝐴(𝑥,𝑓,𝑔)   𝐵(𝑥,𝑓,𝑔)   𝑄(𝑥,𝑓,𝑔)   𝑅(𝑔)   𝑇(𝑥,𝑔)   𝑈(𝑥,𝑓,𝑔)   𝑉(𝑥,𝑓,𝑔)   𝑊(𝑥,𝑔)   𝑋(𝑥,𝑓,𝑔)   𝑌(𝑥,𝑓,𝑔)   𝑍(𝑥,𝑓,𝑔)

Proof of Theorem evlsval

Dummy variables 𝑏 𝑖 𝑟 𝑠 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		evlsval.q	. . . 4 ⊢ 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
2		elex 3212	. . . . 5 ⊢ (𝐼 ∈ 𝑍 → 𝐼 ∈ V)
3		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
4	3	adantl 482	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
5	4	csbeq1d 3540	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))) = ⦋(Base‘𝑆) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))))
6		fvex 6201	. . . . . . . . . 10 ⊢ (Base‘𝑆) ∈ V
7	6	a1i 11	. . . . . . . . 9 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (Base‘𝑆) ∈ V)
8		simplr 792	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑠 = 𝑆)
9	8	fveq2d 6195	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (SubRing‘𝑠) = (SubRing‘𝑆))
10		simpll 790	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑖 = 𝐼)
11		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (𝑠 ↾s 𝑟) = (𝑆 ↾s 𝑟))
12	11	ad2antlr 763	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑠 ↾s 𝑟) = (𝑆 ↾s 𝑟))
13	10, 12	oveq12d 6668	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠 ↾s 𝑟)) = (𝐼 mPoly (𝑆 ↾s 𝑟)))
14	13	csbeq1d 3540	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))))
15		ovexd 6680	. . . . . . . . . . . 12 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝐼 mPoly (𝑆 ↾s 𝑟)) ∈ V)
16		simprr 796	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))
17		simplr 792	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → 𝑠 = 𝑆)
18		simprl 794	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → 𝑏 = (Base‘𝑆))
19		simpll 790	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → 𝑖 = 𝐼)
20	18, 19	oveq12d 6668	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑏 ↑𝑚 𝑖) = ((Base‘𝑆) ↑𝑚 𝐼))
21	17, 20	oveq12d 6668	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑠 ↑s (𝑏 ↑𝑚 𝑖)) = (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))
22	16, 21	oveq12d 6668	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖))) = ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))))
23	16	fveq2d 6195	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (algSc‘𝑤) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟))))
24	23	coeq2d 5284	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑓 ∘ (algSc‘𝑤)) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))))
25	20	xpeq1d 5138	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → ((𝑏 ↑𝑚 𝑖) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
26	25	mpteq2dv 4745	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
27	24, 26	eqeq12d 2637	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))))
28	17	oveq1d 6665	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑠 ↾s 𝑟) = (𝑆 ↾s 𝑟))
29	19, 28	oveq12d 6668	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑖 mVar (𝑠 ↾s 𝑟)) = (𝐼 mVar (𝑆 ↾s 𝑟)))
30	29	coeq2d 5284	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))))
31	20	mpteq1d 4738	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))
32	19, 31	mpteq12dv 4733	. . . . . . . . . . . . . . . 16 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))
33	30, 32	eqeq12d 2637	. . . . . . . . . . . . . . 15 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → ((𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))
34	27, 33	anbi12d 747	. . . . . . . . . . . . . 14 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
35	22, 34	riotaeqbidv 6614	. . . . . . . . . . . . 13 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟)))) → (℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
36	35	anassrs 680	. . . . . . . . . . . 12 ⊢ ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) ∧ 𝑤 = (𝐼 mPoly (𝑆 ↾s 𝑟))) → (℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
37	15, 36	csbied 3560	. . . . . . . . . . 11 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
38	14, 37	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
39	9, 38	mpteq12dv 4733	. . . . . . . . 9 ⊢ (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))))
40	7, 39	csbied 3560	. . . . . . . 8 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑆) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))))
41	5, 40	eqtrd 2656	. . . . . . 7 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))))
42		df-evls 19506	. . . . . . 7 ⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑𝑚 𝑖) ↦ (𝑔‘𝑥)))))))
43		fvex 6201	. . . . . . . 8 ⊢ (SubRing‘𝑆) ∈ V
44	43	mptex 6486	. . . . . . 7 ⊢ (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))) ∈ V
45	41, 42, 44	ovmpt2a 6791	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (𝐼 evalSub 𝑆) = (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))))
46	45	fveq1d 6193	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → ((𝐼 evalSub 𝑆)‘𝑅) = ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅))
47	2, 46	sylan 488	. . . 4 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) → ((𝐼 evalSub 𝑆)‘𝑅) = ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅))
48	1, 47	syl5eq 2668	. . 3 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) → 𝑄 = ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅))
49	48	3adant3 1081	. 2 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅))
50		oveq2 6658	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑆 ↾s 𝑟) = (𝑆 ↾s 𝑅))
51	50	oveq2d 6666	. . . . . . 7 ⊢ (𝑟 = 𝑅 → (𝐼 mPoly (𝑆 ↾s 𝑟)) = (𝐼 mPoly (𝑆 ↾s 𝑅)))
52	51	oveq1d 6665	. . . . . 6 ⊢ (𝑟 = 𝑅 → ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))) = ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))))
53	51	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟))) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
54	53	coeq2d 5284	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))))
55		mpteq1 4737	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
56	54, 55	eqeq12d 2637	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))))
57	50	oveq2d 6666	. . . . . . . . 9 ⊢ (𝑟 = 𝑅 → (𝐼 mVar (𝑆 ↾s 𝑟)) = (𝐼 mVar (𝑆 ↾s 𝑅)))
58	57	coeq2d 5284	. . . . . . . 8 ⊢ (𝑟 = 𝑅 → (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))))
59	58	eqeq1d 2624	. . . . . . 7 ⊢ (𝑟 = 𝑅 → ((𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))
60	56, 59	anbi12d 747	. . . . . 6 ⊢ (𝑟 = 𝑅 → (((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
61	52, 60	riotaeqbidv 6614	. . . . 5 ⊢ (𝑟 = 𝑅 → (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
62		eqid 2622	. . . . 5 ⊢ (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
63		riotaex 6615	. . . . 5 ⊢ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))) ∈ V
64	61, 62, 63	fvmpt 6282	. . . 4 ⊢ (𝑅 ∈ (SubRing‘𝑆) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
65		evlsval.w	. . . . . . . . 9 ⊢ 𝑊 = (𝐼 mPoly 𝑈)
66		evlsval.u	. . . . . . . . . 10 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
67	66	oveq2i 6661	. . . . . . . . 9 ⊢ (𝐼 mPoly 𝑈) = (𝐼 mPoly (𝑆 ↾s 𝑅))
68	65, 67	eqtri 2644	. . . . . . . 8 ⊢ 𝑊 = (𝐼 mPoly (𝑆 ↾s 𝑅))
69		evlsval.t	. . . . . . . . 9 ⊢ 𝑇 = (𝑆 ↑s (𝐵 ↑𝑚 𝐼))
70		evlsval.b	. . . . . . . . . . 11 ⊢ 𝐵 = (Base‘𝑆)
71	70	oveq1i 6660	. . . . . . . . . 10 ⊢ (𝐵 ↑𝑚 𝐼) = ((Base‘𝑆) ↑𝑚 𝐼)
72	71	oveq2i 6661	. . . . . . . . 9 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 𝐼)) = (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))
73	69, 72	eqtri 2644	. . . . . . . 8 ⊢ 𝑇 = (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))
74	68, 73	oveq12i 6662	. . . . . . 7 ⊢ (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))
75	74	a1i 11	. . . . . 6 ⊢ (⊤ → (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼))))
76		evlsval.a	. . . . . . . . . . 11 ⊢ 𝐴 = (algSc‘𝑊)
77	68	fveq2i 6194	. . . . . . . . . . 11 ⊢ (algSc‘𝑊) = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
78	76, 77	eqtri 2644	. . . . . . . . . 10 ⊢ 𝐴 = (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))
79	78	coeq2i 5282	. . . . . . . . 9 ⊢ (𝑓 ∘ 𝐴) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅))))
80		evlsval.x	. . . . . . . . . 10 ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥}))
81	71	xpeq1i 5135	. . . . . . . . . . 11 ⊢ ((𝐵 ↑𝑚 𝐼) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})
82	81	mpteq2i 4741	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑅 ↦ ((𝐵 ↑𝑚 𝐼) × {𝑥})) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
83	80, 82	eqtri 2644	. . . . . . . . 9 ⊢ 𝑋 = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
84	79, 83	eqeq12i 2636	. . . . . . . 8 ⊢ ((𝑓 ∘ 𝐴) = 𝑋 ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
85		evlsval.v	. . . . . . . . . . 11 ⊢ 𝑉 = (𝐼 mVar 𝑈)
86	66	oveq2i 6661	. . . . . . . . . . 11 ⊢ (𝐼 mVar 𝑈) = (𝐼 mVar (𝑆 ↾s 𝑅))
87	85, 86	eqtri 2644	. . . . . . . . . 10 ⊢ 𝑉 = (𝐼 mVar (𝑆 ↾s 𝑅))
88	87	coeq2i 5282	. . . . . . . . 9 ⊢ (𝑓 ∘ 𝑉) = (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅)))
89		evlsval.y	. . . . . . . . . 10 ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))
90		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑥) = (𝑔‘𝑥)
91	71, 90	mpteq12i 4742	. . . . . . . . . . 11 ⊢ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))
92	91	mpteq2i 4741	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝐵 ↑𝑚 𝐼) ↦ (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))
93	89, 92	eqtri 2644	. . . . . . . . 9 ⊢ 𝑌 = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))
94	88, 93	eqeq12i 2636	. . . . . . . 8 ⊢ ((𝑓 ∘ 𝑉) = 𝑌 ↔ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))
95	84, 94	anbi12i 733	. . . . . . 7 ⊢ (((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))
96	95	a1i 11	. . . . . 6 ⊢ (⊤ → (((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
97	75, 96	riotaeqbidv 6614	. . . . 5 ⊢ (⊤ → (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))
98	97	trud 1493	. . . 4 ⊢ (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑅)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑅)))) = (𝑥 ∈ 𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑅))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥)))))
99	64, 98	syl6eqr 2674	. . 3 ⊢ (𝑅 ∈ (SubRing‘𝑆) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅) = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)))
100	99	3ad2ant3 1084	. 2 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 ↾s 𝑟)) RingHom (𝑆 ↑s ((Base‘𝑆) ↑𝑚 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆 ↾s 𝑟)))) = (𝑥 ∈ 𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔‘𝑥))))))‘𝑅) = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)))
101	49, 100	eqtrd 2656	1 ⊢ ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (℩𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = 𝑌)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ⊤wtru 1484   ∈ wcel 1990  Vcvv 3200  ⦋csb 3533  {csn 4177   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  ‘cfv 5888  ℩crio 6610  (class class class)co 6650   ↑_𝑚 cmap 7857  Basecbs 15857   ↾_s cress 15858   ↑_s cpws 16107  CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776  algSccascl 19311   mVar cmvr 19352   mPoly cmpl 19353   evalSub ces 19504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-evls 19506

This theorem is referenced by:  evlsval2  19520