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Theorem evlsval 20863
 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 = (𝑆s 𝑅)
evlsval.t 𝑇 = (𝑆s (𝐵m 𝐼))
evlsval.b 𝐵 = (Base‘𝑆)
evlsval.a 𝐴 = (algSc‘𝑊)
evlsval.x 𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))
evlsval.y 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))
Assertion
Ref Expression
evlsval ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 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.
StepHypRef Expression
1 evlsval.q . . . 4 𝑄 = ((𝐼 evalSub 𝑆)‘𝑅)
2 elex 3428 . . . . 5 (𝐼𝑍𝐼 ∈ V)
3 fveq2 6663 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
43adantl 485 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
54csbeq1d 3811 . . . . . . . 8 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))) = (Base‘𝑆) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))))
6 fvex 6676 . . . . . . . . . 10 (Base‘𝑆) ∈ V
76a1i 11 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑆) ∈ V)
8 simplr 768 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑠 = 𝑆)
98fveq2d 6667 . . . . . . . . . 10 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (SubRing‘𝑠) = (SubRing‘𝑆))
10 simpll 766 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑖 = 𝐼)
11 oveq1 7163 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
1211ad2antlr 726 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑠s 𝑟) = (𝑆s 𝑟))
1310, 12oveq12d 7174 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
1413csbeq1d 3811 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))) = (𝐼 mPoly (𝑆s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))))
15 ovexd 7191 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝐼 mPoly (𝑆s 𝑟)) ∈ V)
16 simprr 772 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))
17 simplr 768 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑠 = 𝑆)
18 simprl 770 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑏 = (Base‘𝑆))
19 simpll 766 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑖 = 𝐼)
2018, 19oveq12d 7174 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑏m 𝑖) = ((Base‘𝑆) ↑m 𝐼))
2117, 20oveq12d 7174 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s (𝑏m 𝑖)) = (𝑆s ((Base‘𝑆) ↑m 𝐼)))
2216, 21oveq12d 7174 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑤 RingHom (𝑠s (𝑏m 𝑖))) = ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼))))
2316fveq2d 6667 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (algSc‘𝑤) = (algSc‘(𝐼 mPoly (𝑆s 𝑟))))
2423coeq2d 5708 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (algSc‘𝑤)) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))))
2520xpeq1d 5557 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → ((𝑏m 𝑖) × {𝑥}) = (((Base‘𝑆) ↑m 𝐼) × {𝑥}))
2625mpteq2dv 5132 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})))
2724, 26eqeq12d 2774 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥}))))
2817oveq1d 7171 . . . . . . . . . . . . . . . . . 18 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s 𝑟) = (𝑆s 𝑟))
2919, 28oveq12d 7174 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
3029coeq2d 5708 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
3120mpteq1d 5125 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))
3219, 31mpteq12dv 5121 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))
3330, 32eqeq12d 2774 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → ((𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))
3427, 33anbi12d 633 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
3522, 34riotaeqbidv 7117 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
3635anassrs 471 . . . . . . . . . . . 12 ((((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟))) → (𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
3715, 36csbied 3843 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝐼 mPoly (𝑆s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
3814, 37eqtrd 2793 . . . . . . . . . 10 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))))) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
399, 38mpteq12dv 5121 . . . . . . . . 9 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))))
407, 39csbied 3843 . . . . . . . 8 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑆) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))))
415, 40eqtrd 2793 . . . . . . 7 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))))
42 df-evls 20849 . . . . . . 7 evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ (Base‘𝑠) / 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (𝑖 mPoly (𝑠s 𝑟)) / 𝑤(𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)))))))
43 fvex 6676 . . . . . . . 8 (SubRing‘𝑆) ∈ V
4443mptex 6983 . . . . . . 7 (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))) ∈ V
4541, 42, 44ovmpoa 7306 . . . . . 6 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (𝐼 evalSub 𝑆) = (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))))
4645fveq1d 6665 . . . . 5 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → ((𝐼 evalSub 𝑆)‘𝑅) = ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅))
472, 46sylan 583 . . . 4 ((𝐼𝑍𝑆 ∈ CRing) → ((𝐼 evalSub 𝑆)‘𝑅) = ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅))
481, 47syl5eq 2805 . . 3 ((𝐼𝑍𝑆 ∈ CRing) → 𝑄 = ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅))
49483adant3 1129 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅))
50 oveq2 7164 . . . . . . . 8 (𝑟 = 𝑅 → (𝑆s 𝑟) = (𝑆s 𝑅))
5150oveq2d 7172 . . . . . . 7 (𝑟 = 𝑅 → (𝐼 mPoly (𝑆s 𝑟)) = (𝐼 mPoly (𝑆s 𝑅)))
5251oveq1d 7171 . . . . . 6 (𝑟 = 𝑅 → ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼))) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼))))
5351fveq2d 6667 . . . . . . . . 9 (𝑟 = 𝑅 → (algSc‘(𝐼 mPoly (𝑆s 𝑟))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
5453coeq2d 5708 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))))
55 mpteq1 5124 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})))
5654, 55eqeq12d 2774 . . . . . . 7 (𝑟 = 𝑅 → ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥}))))
5750oveq2d 7172 . . . . . . . . 9 (𝑟 = 𝑅 → (𝐼 mVar (𝑆s 𝑟)) = (𝐼 mVar (𝑆s 𝑅)))
5857coeq2d 5708 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))))
5958eqeq1d 2760 . . . . . . 7 (𝑟 = 𝑅 → ((𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))
6056, 59anbi12d 633 . . . . . 6 (𝑟 = 𝑅 → (((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
6152, 60riotaeqbidv 7117 . . . . 5 (𝑟 = 𝑅 → (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
62 eqid 2758 . . . . 5 (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
63 riotaex 7118 . . . . 5 (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))) ∈ V
6461, 62, 63fvmpt 6764 . . . 4 (𝑅 ∈ (SubRing‘𝑆) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
65 evlsval.w . . . . . . . . 9 𝑊 = (𝐼 mPoly 𝑈)
66 evlsval.u . . . . . . . . . 10 𝑈 = (𝑆s 𝑅)
6766oveq2i 7167 . . . . . . . . 9 (𝐼 mPoly 𝑈) = (𝐼 mPoly (𝑆s 𝑅))
6865, 67eqtri 2781 . . . . . . . 8 𝑊 = (𝐼 mPoly (𝑆s 𝑅))
69 evlsval.t . . . . . . . . 9 𝑇 = (𝑆s (𝐵m 𝐼))
70 evlsval.b . . . . . . . . . . 11 𝐵 = (Base‘𝑆)
7170oveq1i 7166 . . . . . . . . . 10 (𝐵m 𝐼) = ((Base‘𝑆) ↑m 𝐼)
7271oveq2i 7167 . . . . . . . . 9 (𝑆s (𝐵m 𝐼)) = (𝑆s ((Base‘𝑆) ↑m 𝐼))
7369, 72eqtri 2781 . . . . . . . 8 𝑇 = (𝑆s ((Base‘𝑆) ↑m 𝐼))
7468, 73oveq12i 7168 . . . . . . 7 (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))
7574a1i 11 . . . . . 6 (⊤ → (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼))))
76 evlsval.a . . . . . . . . . . 11 𝐴 = (algSc‘𝑊)
7768fveq2i 6666 . . . . . . . . . . 11 (algSc‘𝑊) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
7876, 77eqtri 2781 . . . . . . . . . 10 𝐴 = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
7978coeq2i 5706 . . . . . . . . 9 (𝑓𝐴) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
80 evlsval.x . . . . . . . . . 10 𝑋 = (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥}))
8171xpeq1i 5554 . . . . . . . . . . 11 ((𝐵m 𝐼) × {𝑥}) = (((Base‘𝑆) ↑m 𝐼) × {𝑥})
8281mpteq2i 5128 . . . . . . . . . 10 (𝑥𝑅 ↦ ((𝐵m 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥}))
8380, 82eqtri 2781 . . . . . . . . 9 𝑋 = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥}))
8479, 83eqeq12i 2773 . . . . . . . 8 ((𝑓𝐴) = 𝑋 ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})))
85 evlsval.v . . . . . . . . . . 11 𝑉 = (𝐼 mVar 𝑈)
8666oveq2i 7167 . . . . . . . . . . 11 (𝐼 mVar 𝑈) = (𝐼 mVar (𝑆s 𝑅))
8785, 86eqtri 2781 . . . . . . . . . 10 𝑉 = (𝐼 mVar (𝑆s 𝑅))
8887coeq2i 5706 . . . . . . . . 9 (𝑓𝑉) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅)))
89 evlsval.y . . . . . . . . . 10 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)))
90 eqid 2758 . . . . . . . . . . . 12 (𝑔𝑥) = (𝑔𝑥)
9171, 90mpteq12i 5129 . . . . . . . . . . 11 (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))
9291mpteq2i 5128 . . . . . . . . . 10 (𝑥𝐼 ↦ (𝑔 ∈ (𝐵m 𝐼) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))
9389, 92eqtri 2781 . . . . . . . . 9 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))
9488, 93eqeq12i 2773 . . . . . . . 8 ((𝑓𝑉) = 𝑌 ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))
9584, 94anbi12i 629 . . . . . . 7 (((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))
9695a1i 11 . . . . . 6 (⊤ → (((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌) ↔ ((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
9775, 96riotaeqbidv 7117 . . . . 5 (⊤ → (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))
9897mptru 1545 . . . 4 (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)) = (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥)))))
9964, 98eqtr4di 2811 . . 3 (𝑅 ∈ (SubRing‘𝑆) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅) = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)))
100993ad2ant3 1132 . 2 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((𝑟 ∈ (SubRing‘𝑆) ↦ (𝑓 ∈ ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑m 𝐼)))((𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑m 𝐼) ↦ (𝑔𝑥))))))‘𝑅) = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)))
10149, 100eqtrd 2793 1 ((𝐼𝑍𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → 𝑄 = (𝑓 ∈ (𝑊 RingHom 𝑇)((𝑓𝐴) = 𝑋 ∧ (𝑓𝑉) = 𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ⊤wtru 1539   ∈ wcel 2111  Vcvv 3409  ⦋csb 3807  {csn 4525   ↦ cmpt 5116   × cxp 5526   ∘ ccom 5532  ‘cfv 6340  ℩crio 7113  (class class class)co 7156   ↑m cmap 8422  Basecbs 16555   ↾s cress 16556   ↑s cpws 16792  CRingccrg 19380   RingHom crh 19549  SubRingcsubrg 19613  algSccascl 20631   mVar cmvr 20681   mPoly cmpl 20682   evalSub ces 20847 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pr 5302 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-riota 7114  df-ov 7159  df-oprab 7160  df-mpo 7161  df-evls 20849 This theorem is referenced by:  evlsval2  20864  evlsval3  39816
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