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Theorem evlsval 19567
 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 3243 . . . . 5 (𝐼𝑍𝐼 ∈ V)
3 fveq2 6229 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
43adantl 481 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
54csbeq1d 3573 . . . . . . . 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 6239 . . . . . . . . . 10 (Base‘𝑆) ∈ V
76a1i 11 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑆) ∈ V)
8 simplr 807 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑠 = 𝑆)
98fveq2d 6233 . . . . . . . . . 10 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (SubRing‘𝑠) = (SubRing‘𝑆))
10 simpll 805 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑖 = 𝐼)
11 oveq1 6697 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
1211ad2antlr 763 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑠s 𝑟) = (𝑆s 𝑟))
1310, 12oveq12d 6708 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
1413csbeq1d 3573 . . . . . . . . . . 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 6720 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝐼 mPoly (𝑆s 𝑟)) ∈ V)
16 simprr 811 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))
17 simplr 807 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑠 = 𝑆)
18 simprl 809 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑏 = (Base‘𝑆))
19 simpll 805 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑖 = 𝐼)
2018, 19oveq12d 6708 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑏𝑚 𝑖) = ((Base‘𝑆) ↑𝑚 𝐼))
2117, 20oveq12d 6708 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s (𝑏𝑚 𝑖)) = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼)))
2216, 21oveq12d 6708 . . . . . . . . . . . . . 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 6233 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (algSc‘𝑤) = (algSc‘(𝐼 mPoly (𝑆s 𝑟))))
2423coeq2d 5317 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (algSc‘𝑤)) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))))
2520xpeq1d 5172 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → ((𝑏𝑚 𝑖) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
2625mpteq2dv 4778 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
2724, 26eqeq12d 2666 . . . . . . . . . . . . . . 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 6705 . . . . . . . . . . . . . . . . . 18 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s 𝑟) = (𝑆s 𝑟))
2919, 28oveq12d 6708 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
3029coeq2d 5317 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
3120mpteq1d 4771 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
3219, 31mpteq12dv 4766 . . . . . . . . . . . . . . . 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 2666 . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . 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 6654 . . . . . . . . . . . . 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 681 . . . . . . . . . . . 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 3593 . . . . . . . . . . 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 2685 . . . . . . . . . 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 4766 . . . . . . . . 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 3593 . . . . . . . 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 2685 . . . . . . 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 19554 . . . . . . 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 6239 . . . . . . . 8 (SubRing‘𝑆) ∈ V
4443mptex 6527 . . . . . . 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, 44ovmpt2a 6833 . . . . . 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 6231 . . . . 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 487 . . . 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 2697 . . 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 1101 . 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 6698 . . . . . . . 8 (𝑟 = 𝑅 → (𝑆s 𝑟) = (𝑆s 𝑅))
5150oveq2d 6706 . . . . . . 7 (𝑟 = 𝑅 → (𝐼 mPoly (𝑆s 𝑟)) = (𝐼 mPoly (𝑆s 𝑅)))
5251oveq1d 6705 . . . . . 6 (𝑟 = 𝑅 → ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))))
5351fveq2d 6233 . . . . . . . . 9 (𝑟 = 𝑅 → (algSc‘(𝐼 mPoly (𝑆s 𝑟))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
5453coeq2d 5317 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))))
55 mpteq1 4770 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
5654, 55eqeq12d 2666 . . . . . . 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 6706 . . . . . . . . 9 (𝑟 = 𝑅 → (𝐼 mVar (𝑆s 𝑟)) = (𝐼 mVar (𝑆s 𝑅)))
5857coeq2d 5317 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))))
5958eqeq1d 2653 . . . . . . 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 747 . . . . . 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 6654 . . . . 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 2651 . . . . 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 6655 . . . . 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 6321 . . . 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 6701 . . . . . . . . 9 (𝐼 mPoly 𝑈) = (𝐼 mPoly (𝑆s 𝑅))
6865, 67eqtri 2673 . . . . . . . 8 𝑊 = (𝐼 mPoly (𝑆s 𝑅))
69 evlsval.t . . . . . . . . 9 𝑇 = (𝑆s (𝐵𝑚 𝐼))
70 evlsval.b . . . . . . . . . . 11 𝐵 = (Base‘𝑆)
7170oveq1i 6700 . . . . . . . . . 10 (𝐵𝑚 𝐼) = ((Base‘𝑆) ↑𝑚 𝐼)
7271oveq2i 6701 . . . . . . . . 9 (𝑆s (𝐵𝑚 𝐼)) = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))
7369, 72eqtri 2673 . . . . . . . 8 𝑇 = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))
7468, 73oveq12i 6702 . . . . . . 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 6232 . . . . . . . . . . 11 (algSc‘𝑊) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
7876, 77eqtri 2673 . . . . . . . . . 10 𝐴 = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
7978coeq2i 5315 . . . . . . . . 9 (𝑓𝐴) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
80 evlsval.x . . . . . . . . . 10 𝑋 = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥}))
8171xpeq1i 5169 . . . . . . . . . . 11 ((𝐵𝑚 𝐼) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})
8281mpteq2i 4774 . . . . . . . . . 10 (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
8380, 82eqtri 2673 . . . . . . . . 9 𝑋 = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
8479, 83eqeq12i 2665 . . . . . . . 8 ((𝑓𝐴) = 𝑋 ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
85 evlsval.v . . . . . . . . . . 11 𝑉 = (𝐼 mVar 𝑈)
8666oveq2i 6701 . . . . . . . . . . 11 (𝐼 mVar 𝑈) = (𝐼 mVar (𝑆s 𝑅))
8785, 86eqtri 2673 . . . . . . . . . 10 𝑉 = (𝐼 mVar (𝑆s 𝑅))
8887coeq2i 5315 . . . . . . . . 9 (𝑓𝑉) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅)))
89 evlsval.y . . . . . . . . . 10 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)))
90 eqid 2651 . . . . . . . . . . . 12 (𝑔𝑥) = (𝑔𝑥)
9171, 90mpteq12i 4775 . . . . . . . . . . 11 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥))
9291mpteq2i 4774 . . . . . . . . . 10 (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
9389, 92eqtri 2673 . . . . . . . . 9 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
9488, 93eqeq12i 2665 . . . . . . . 8 ((𝑓𝑉) = 𝑌 ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥))))
9584, 94anbi12i 733 . . . . . . 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 6654 . . . . 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))))))
9897trud 1533 . . . 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, 98syl6eqr 2703 . . 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 1104 . 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 2685 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 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ⊤wtru 1524   ∈ wcel 2030  Vcvv 3231  ⦋csb 3566  {csn 4210   ↦ cmpt 4762   × cxp 5141   ∘ ccom 5147  ‘cfv 5926  ℩crio 6650  (class class class)co 6690   ↑𝑚 cmap 7899  Basecbs 15904   ↾s cress 15905   ↑s cpws 16154  CRingccrg 18594   RingHom crh 18760  SubRingcsubrg 18824  algSccascl 19359   mVar cmvr 19400   mPoly cmpl 19401   evalSub ces 19552 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-evls 19554 This theorem is referenced by:  evlsval2  19568
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