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Theorem evlsval 19438
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 3198 . . . . 5 (𝐼𝑍𝐼 ∈ V)
3 fveq2 6148 . . . . . . . . . 10 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
43adantl 482 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
54csbeq1d 3521 . . . . . . . 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 6158 . . . . . . . . . 10 (Base‘𝑆) ∈ V
76a1i 11 . . . . . . . . 9 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑆) ∈ V)
8 simplr 791 . . . . . . . . . . 11 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑠 = 𝑆)
98fveq2d 6152 . . . . . . . . . 10 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (SubRing‘𝑠) = (SubRing‘𝑆))
10 simpll 789 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → 𝑖 = 𝐼)
11 oveq1 6611 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
1211ad2antlr 762 . . . . . . . . . . . . 13 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑠s 𝑟) = (𝑆s 𝑟))
1310, 12oveq12d 6622 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
1413csbeq1d 3521 . . . . . . . . . . 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 ovex 6632 . . . . . . . . . . . . 13 (𝐼 mPoly (𝑆s 𝑟)) ∈ V
1615a1i 11 . . . . . . . . . . . 12 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ 𝑏 = (Base‘𝑆)) → (𝐼 mPoly (𝑆s 𝑟)) ∈ V)
17 simprr 795 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))
18 simplr 791 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑠 = 𝑆)
19 simprl 793 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑏 = (Base‘𝑆))
20 simpll 789 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → 𝑖 = 𝐼)
2119, 20oveq12d 6622 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑏𝑚 𝑖) = ((Base‘𝑆) ↑𝑚 𝐼))
2218, 21oveq12d 6622 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s (𝑏𝑚 𝑖)) = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼)))
2317, 22oveq12d 6622 . . . . . . . . . . . . . 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))))
2417fveq2d 6152 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (algSc‘𝑤) = (algSc‘(𝐼 mPoly (𝑆s 𝑟))))
2524coeq2d 5244 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (algSc‘𝑤)) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))))
2621xpeq1d 5098 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → ((𝑏𝑚 𝑖) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
2726mpteq2dv 4705 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) = (𝑥𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
2825, 27eqeq12d 2636 . . . . . . . . . . . . . . 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}))))
2918oveq1d 6619 . . . . . . . . . . . . . . . . . 18 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑠s 𝑟) = (𝑆s 𝑟))
3020, 29oveq12d 6622 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
3130coeq2d 5244 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
3221mpteq1d 4698 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
3320, 32mpteq12dv 4693 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼𝑠 = 𝑆) ∧ (𝑏 = (Base‘𝑆) ∧ 𝑤 = (𝐼 mPoly (𝑆s 𝑟)))) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥))))
3431, 33eqeq12d 2636 . . . . . . . . . . . . . . 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)))))
3528, 34anbi12d 746 . . . . . . . . . . . . . 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))))))
3623, 35riotaeqbidv 6568 . . . . . . . . . . . . 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))))))
3736anassrs 679 . . . . . . . . . . . 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))))))
3816, 37csbied 3541 . . . . . . . . . . 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))))))
3914, 38eqtrd 2655 . . . . . . . . . 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))))))
409, 39mpteq12dv 4693 . . . . . . . . 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)))))))
417, 40csbied 3541 . . . . . . . 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)))))))
425, 41eqtrd 2655 . . . . . . 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)))))))
43 df-evls 19425 . . . . . . 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)))))))
44 fvex 6158 . . . . . . . 8 (SubRing‘𝑆) ∈ V
4544mptex 6440 . . . . . . 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
4642, 43, 45ovmpt2a 6744 . . . . . 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)))))))
4746fveq1d 6150 . . . . 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))
482, 47sylan 488 . . . 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))
491, 48syl5eq 2667 . . 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))
50493adant3 1079 . 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))
51 oveq2 6612 . . . . . . . 8 (𝑟 = 𝑅 → (𝑆s 𝑟) = (𝑆s 𝑅))
5251oveq2d 6620 . . . . . . 7 (𝑟 = 𝑅 → (𝐼 mPoly (𝑆s 𝑟)) = (𝐼 mPoly (𝑆s 𝑅)))
5352oveq1d 6619 . . . . . 6 (𝑟 = 𝑅 → ((𝐼 mPoly (𝑆s 𝑟)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))))
5452fveq2d 6152 . . . . . . . . 9 (𝑟 = 𝑅 → (algSc‘(𝐼 mPoly (𝑆s 𝑟))) = (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
5554coeq2d 5244 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑟)))) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))))
56 mpteq1 4697 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥𝑟 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
5755, 56eqeq12d 2636 . . . . . . 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}))))
5851oveq2d 6620 . . . . . . . . 9 (𝑟 = 𝑅 → (𝐼 mVar (𝑆s 𝑟)) = (𝐼 mVar (𝑆s 𝑅)))
5958coeq2d 5244 . . . . . . . 8 (𝑟 = 𝑅 → (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))))
6059eqeq1d 2623 . . . . . . 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)))))
6157, 60anbi12d 746 . . . . . 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))))))
6253, 61riotaeqbidv 6568 . . . . 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))))))
63 eqid 2621 . . . . 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))))))
64 riotaex 6569 . . . . 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
6562, 63, 64fvmpt 6239 . . . 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))))))
66 evlsval.w . . . . . . . . 9 𝑊 = (𝐼 mPoly 𝑈)
67 evlsval.u . . . . . . . . . 10 𝑈 = (𝑆s 𝑅)
6867oveq2i 6615 . . . . . . . . 9 (𝐼 mPoly 𝑈) = (𝐼 mPoly (𝑆s 𝑅))
6966, 68eqtri 2643 . . . . . . . 8 𝑊 = (𝐼 mPoly (𝑆s 𝑅))
70 evlsval.t . . . . . . . . 9 𝑇 = (𝑆s (𝐵𝑚 𝐼))
71 evlsval.b . . . . . . . . . . 11 𝐵 = (Base‘𝑆)
7271oveq1i 6614 . . . . . . . . . 10 (𝐵𝑚 𝐼) = ((Base‘𝑆) ↑𝑚 𝐼)
7372oveq2i 6615 . . . . . . . . 9 (𝑆s (𝐵𝑚 𝐼)) = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))
7470, 73eqtri 2643 . . . . . . . 8 𝑇 = (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))
7569, 74oveq12i 6616 . . . . . . 7 (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼)))
7675a1i 11 . . . . . 6 (⊤ → (𝑊 RingHom 𝑇) = ((𝐼 mPoly (𝑆s 𝑅)) RingHom (𝑆s ((Base‘𝑆) ↑𝑚 𝐼))))
77 evlsval.a . . . . . . . . . . 11 𝐴 = (algSc‘𝑊)
7869fveq2i 6151 . . . . . . . . . . 11 (algSc‘𝑊) = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
7977, 78eqtri 2643 . . . . . . . . . 10 𝐴 = (algSc‘(𝐼 mPoly (𝑆s 𝑅)))
8079coeq2i 5242 . . . . . . . . 9 (𝑓𝐴) = (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅))))
81 evlsval.x . . . . . . . . . 10 𝑋 = (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥}))
8272xpeq1i 5095 . . . . . . . . . . 11 ((𝐵𝑚 𝐼) × {𝑥}) = (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})
8382mpteq2i 4701 . . . . . . . . . 10 (𝑥𝑅 ↦ ((𝐵𝑚 𝐼) × {𝑥})) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
8481, 83eqtri 2643 . . . . . . . . 9 𝑋 = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥}))
8580, 84eqeq12i 2635 . . . . . . . 8 ((𝑓𝐴) = 𝑋 ↔ (𝑓 ∘ (algSc‘(𝐼 mPoly (𝑆s 𝑅)))) = (𝑥𝑅 ↦ (((Base‘𝑆) ↑𝑚 𝐼) × {𝑥})))
86 evlsval.v . . . . . . . . . . 11 𝑉 = (𝐼 mVar 𝑈)
8767oveq2i 6615 . . . . . . . . . . 11 (𝐼 mVar 𝑈) = (𝐼 mVar (𝑆s 𝑅))
8886, 87eqtri 2643 . . . . . . . . . 10 𝑉 = (𝐼 mVar (𝑆s 𝑅))
8988coeq2i 5242 . . . . . . . . 9 (𝑓𝑉) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅)))
90 evlsval.y . . . . . . . . . 10 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)))
91 eqid 2621 . . . . . . . . . . . 12 (𝑔𝑥) = (𝑔𝑥)
9272, 91mpteq12i 4702 . . . . . . . . . . 11 (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥)) = (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥))
9392mpteq2i 4701 . . . . . . . . . 10 (𝑥𝐼 ↦ (𝑔 ∈ (𝐵𝑚 𝐼) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
9490, 93eqtri 2643 . . . . . . . . 9 𝑌 = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥)))
9589, 94eqeq12i 2635 . . . . . . . 8 ((𝑓𝑉) = 𝑌 ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑅))) = (𝑥𝐼 ↦ (𝑔 ∈ ((Base‘𝑆) ↑𝑚 𝐼) ↦ (𝑔𝑥))))
9685, 95anbi12i 732 . . . . . . 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)))))
9796a1i 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))))))
9876, 97riotaeqbidv 6568 . . . . 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))))))
9998trud 1490 . . . 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)))))
10065, 99syl6eqr 2673 . . 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)))
1011003ad2ant3 1082 . 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)))
10250, 101eqtrd 2655 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 384  w3a 1036   = wceq 1480  wtru 1481  wcel 1987  Vcvv 3186  csb 3514  {csn 4148  cmpt 4673   × cxp 5072  ccom 5078  cfv 5847  crio 6564  (class class class)co 6604  𝑚 cmap 7802  Basecbs 15781  s cress 15782  s cpws 16028  CRingccrg 18469   RingHom crh 18633  SubRingcsubrg 18697  algSccascl 19230   mVar cmvr 19271   mPoly cmpl 19272   evalSub ces 19423
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-evls 19425
This theorem is referenced by:  evlsval2  19439
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