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Theorem evlsval 21548
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)β€˜π‘…)
evlsval.w π‘Š = (𝐼 mPoly π‘ˆ)
evlsval.v 𝑉 = (𝐼 mVar π‘ˆ)
evlsval.u π‘ˆ = (𝑆 β†Ύs 𝑅)
evlsval.t 𝑇 = (𝑆 ↑s (𝐡 ↑m 𝐼))
evlsval.b 𝐡 = (Baseβ€˜π‘†)
evlsval.a 𝐴 = (algScβ€˜π‘Š)
evlsval.x 𝑋 = (π‘₯ ∈ 𝑅 ↦ ((𝐡 ↑m 𝐼) Γ— {π‘₯}))
evlsval.y π‘Œ = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ (𝐡 ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
Assertion
Ref Expression
evlsval ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)) β†’ 𝑄 = (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)))
Distinct variable groups:   𝑓,𝐼,𝑔,π‘₯   𝑅,𝑓,π‘₯   𝑆,𝑓,𝑔,π‘₯   𝑇,𝑓   𝑓,π‘Š
Allowed substitution hints:   𝐴(π‘₯,𝑓,𝑔)   𝐡(π‘₯,𝑓,𝑔)   𝑄(π‘₯,𝑓,𝑔)   𝑅(𝑔)   𝑇(π‘₯,𝑔)   π‘ˆ(π‘₯,𝑓,𝑔)   𝑉(π‘₯,𝑓,𝑔)   π‘Š(π‘₯,𝑔)   𝑋(π‘₯,𝑓,𝑔)   π‘Œ(π‘₯,𝑓,𝑔)   𝑍(π‘₯,𝑓,𝑔)
Proof of Theorem evlsval
Dummy variables 𝑏 𝑖 π‘Ÿ 𝑠 𝑀 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 evlsval.q . . . 4 𝑄 = ((𝐼 evalSub 𝑆)β€˜π‘…)
2 elex 3477 . . . . 5 (𝐼 ∈ 𝑍 β†’ 𝐼 ∈ V)
3 fveq2 6862 . . . . . . . . . 10 (𝑠 = 𝑆 β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
43adantl 482 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
54csbeq1d 3877 . . . . . . . 8 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘ ) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))) = ⦋(Baseβ€˜π‘†) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))))
6 fvex 6875 . . . . . . . . . 10 (Baseβ€˜π‘†) ∈ V
76a1i 11 . . . . . . . . 9 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (Baseβ€˜π‘†) ∈ V)
8 simplr 767 . . . . . . . . . . 11 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ 𝑠 = 𝑆)
98fveq2d 6866 . . . . . . . . . 10 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ (SubRingβ€˜π‘ ) = (SubRingβ€˜π‘†))
10 simpll 765 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ 𝑖 = 𝐼)
11 oveq1 7384 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 β†’ (𝑠 β†Ύs π‘Ÿ) = (𝑆 β†Ύs π‘Ÿ))
1211ad2antlr 725 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ (𝑠 β†Ύs π‘Ÿ) = (𝑆 β†Ύs π‘Ÿ))
1310, 12oveq12d 7395 . . . . . . . . . . . 12 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ (𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))
1413csbeq1d 3877 . . . . . . . . . . 11 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))) = ⦋(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))))
15 ovexd 7412 . . . . . . . . . . . 12 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) ∈ V)
16 simprr 771 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))
17 simplr 767 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ 𝑠 = 𝑆)
18 simprl 769 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ 𝑏 = (Baseβ€˜π‘†))
19 simpll 765 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ 𝑖 = 𝐼)
2018, 19oveq12d 7395 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑏 ↑m 𝑖) = ((Baseβ€˜π‘†) ↑m 𝐼))
2117, 20oveq12d 7395 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑠 ↑s (𝑏 ↑m 𝑖)) = (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))
2216, 21oveq12d 7395 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖))) = ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))))
2316fveq2d 6866 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (algScβ€˜π‘€) = (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ))))
2423coeq2d 5838 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑓 ∘ (algScβ€˜π‘€)) = (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))))
2520xpeq1d 5682 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ ((𝑏 ↑m 𝑖) Γ— {π‘₯}) = (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯}))
2625mpteq2dv 5227 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})))
2724, 26eqeq12d 2747 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ ((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ↔ (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯}))))
2817oveq1d 7392 . . . . . . . . . . . . . . . . . 18 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑠 β†Ύs π‘Ÿ) = (𝑆 β†Ύs π‘Ÿ))
2919, 28oveq12d 7395 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ)) = (𝐼 mVar (𝑆 β†Ύs π‘Ÿ)))
3029coeq2d 5838 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))))
3120mpteq1d 5220 . . . . . . . . . . . . . . . . 17 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)) = (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
3219, 31mpteq12dv 5216 . . . . . . . . . . . . . . . 16 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))
3330, 32eqeq12d 2747 . . . . . . . . . . . . . . 15 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ ((𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))
3427, 33anbi12d 631 . . . . . . . . . . . . . 14 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))) ↔ ((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
3522, 34riotaeqbidv 7336 . . . . . . . . . . . . 13 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ (𝑏 = (Baseβ€˜π‘†) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) β†’ (℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
3635anassrs 468 . . . . . . . . . . . 12 ((((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) ∧ 𝑀 = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ))) β†’ (℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
3715, 36csbied 3911 . . . . . . . . . . 11 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ ⦋(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
3814, 37eqtrd 2771 . . . . . . . . . 10 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
399, 38mpteq12dv 5216 . . . . . . . . 9 (((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) ∧ 𝑏 = (Baseβ€˜π‘†)) β†’ (π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))))
407, 39csbied 3911 . . . . . . . 8 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘†) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))))
415, 40eqtrd 2771 . . . . . . 7 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘ ) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))))
42 df-evls 21534 . . . . . . 7 evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œ(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) / π‘€β¦Œ(℩𝑓 ∈ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)))))))
43 fvex 6875 . . . . . . . 8 (SubRingβ€˜π‘†) ∈ V
4443mptex 7193 . . . . . . 7 (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))) ∈ V
4541, 42, 44ovmpoa 7530 . . . . . 6 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ (𝐼 evalSub 𝑆) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))))
4645fveq1d 6864 . . . . 5 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…))
472, 46sylan 580 . . . 4 ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…))
481, 47eqtrid 2783 . . 3 ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing) β†’ 𝑄 = ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…))
49483adant3 1132 . 2 ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)) β†’ 𝑄 = ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…))
50 oveq2 7385 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑆 β†Ύs π‘Ÿ) = (𝑆 β†Ύs 𝑅))
5150oveq2d 7393 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) = (𝐼 mPoly (𝑆 β†Ύs 𝑅)))
5251oveq1d 7392 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))) = ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))))
5351fveq2d 6866 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ))) = (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅))))
5453coeq2d 5838 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))))
55 mpteq1 5218 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})))
5654, 55eqeq12d 2747 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ↔ (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯}))))
5750oveq2d 7393 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ)) = (𝐼 mVar (𝑆 β†Ύs 𝑅)))
5857coeq2d 5838 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))))
5958eqeq1d 2733 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))
6056, 59anbi12d 631 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))) ↔ ((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
6152, 60riotaeqbidv 7336 . . . . 5 (π‘Ÿ = 𝑅 β†’ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
62 eqid 2731 . . . . 5 (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
63 riotaex 7337 . . . . 5 (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))) ∈ V
6461, 62, 63fvmpt 6968 . . . 4 (𝑅 ∈ (SubRingβ€˜π‘†) β†’ ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
65 evlsval.w . . . . . . . . 9 π‘Š = (𝐼 mPoly π‘ˆ)
66 evlsval.u . . . . . . . . . 10 π‘ˆ = (𝑆 β†Ύs 𝑅)
6766oveq2i 7388 . . . . . . . . 9 (𝐼 mPoly π‘ˆ) = (𝐼 mPoly (𝑆 β†Ύs 𝑅))
6865, 67eqtri 2759 . . . . . . . 8 π‘Š = (𝐼 mPoly (𝑆 β†Ύs 𝑅))
69 evlsval.t . . . . . . . . 9 𝑇 = (𝑆 ↑s (𝐡 ↑m 𝐼))
70 evlsval.b . . . . . . . . . . 11 𝐡 = (Baseβ€˜π‘†)
7170oveq1i 7387 . . . . . . . . . 10 (𝐡 ↑m 𝐼) = ((Baseβ€˜π‘†) ↑m 𝐼)
7271oveq2i 7388 . . . . . . . . 9 (𝑆 ↑s (𝐡 ↑m 𝐼)) = (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))
7369, 72eqtri 2759 . . . . . . . 8 𝑇 = (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))
7468, 73oveq12i 7389 . . . . . . 7 (π‘Š RingHom 𝑇) = ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))
7574a1i 11 . . . . . 6 (⊀ β†’ (π‘Š RingHom 𝑇) = ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼))))
76 evlsval.a . . . . . . . . . . 11 𝐴 = (algScβ€˜π‘Š)
7768fveq2i 6865 . . . . . . . . . . 11 (algScβ€˜π‘Š) = (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))
7876, 77eqtri 2759 . . . . . . . . . 10 𝐴 = (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))
7978coeq2i 5836 . . . . . . . . 9 (𝑓 ∘ 𝐴) = (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅))))
80 evlsval.x . . . . . . . . . 10 𝑋 = (π‘₯ ∈ 𝑅 ↦ ((𝐡 ↑m 𝐼) Γ— {π‘₯}))
8171xpeq1i 5679 . . . . . . . . . . 11 ((𝐡 ↑m 𝐼) Γ— {π‘₯}) = (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})
8281mpteq2i 5230 . . . . . . . . . 10 (π‘₯ ∈ 𝑅 ↦ ((𝐡 ↑m 𝐼) Γ— {π‘₯})) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯}))
8380, 82eqtri 2759 . . . . . . . . 9 𝑋 = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯}))
8479, 83eqeq12i 2749 . . . . . . . 8 ((𝑓 ∘ 𝐴) = 𝑋 ↔ (𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})))
85 evlsval.v . . . . . . . . . . 11 𝑉 = (𝐼 mVar π‘ˆ)
8666oveq2i 7388 . . . . . . . . . . 11 (𝐼 mVar π‘ˆ) = (𝐼 mVar (𝑆 β†Ύs 𝑅))
8785, 86eqtri 2759 . . . . . . . . . 10 𝑉 = (𝐼 mVar (𝑆 β†Ύs 𝑅))
8887coeq2i 5836 . . . . . . . . 9 (𝑓 ∘ 𝑉) = (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅)))
89 evlsval.y . . . . . . . . . 10 π‘Œ = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ (𝐡 ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
90 eqid 2731 . . . . . . . . . . . 12 (π‘”β€˜π‘₯) = (π‘”β€˜π‘₯)
9171, 90mpteq12i 5231 . . . . . . . . . . 11 (𝑔 ∈ (𝐡 ↑m 𝐼) ↦ (π‘”β€˜π‘₯)) = (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))
9291mpteq2i 5230 . . . . . . . . . 10 (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ (𝐡 ↑m 𝐼) ↦ (π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
9389, 92eqtri 2759 . . . . . . . . 9 π‘Œ = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
9488, 93eqeq12i 2749 . . . . . . . 8 ((𝑓 ∘ 𝑉) = π‘Œ ↔ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))
9584, 94anbi12i 627 . . . . . . 7 (((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ) ↔ ((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))
9695a1i 11 . . . . . 6 (⊀ β†’ (((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ) ↔ ((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
9775, 96riotaeqbidv 7336 . . . . 5 (⊀ β†’ (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))
9897mptru 1548 . . . 4 (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)) = (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs 𝑅)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs 𝑅)))) = (π‘₯ ∈ 𝑅 ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs 𝑅))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))
9964, 98eqtr4di 2789 . . 3 (𝑅 ∈ (SubRingβ€˜π‘†) β†’ ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…) = (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)))
100993ad2ant3 1135 . 2 ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)) β†’ ((π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ (℩𝑓 ∈ ((𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)) RingHom (𝑆 ↑s ((Baseβ€˜π‘†) ↑m 𝐼)))((𝑓 ∘ (algScβ€˜(𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))) = (π‘₯ ∈ π‘Ÿ ↦ (((Baseβ€˜π‘†) ↑m 𝐼) Γ— {π‘₯})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ ((Baseβ€˜π‘†) ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))))β€˜π‘…) = (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)))
10149, 100eqtrd 2771 1 ((𝐼 ∈ 𝑍 ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)) β†’ 𝑄 = (℩𝑓 ∈ (π‘Š RingHom 𝑇)((𝑓 ∘ 𝐴) = 𝑋 ∧ (𝑓 ∘ 𝑉) = π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βŠ€wtru 1542   ∈ wcel 2106  Vcvv 3459  β¦‹csb 3873  {csn 4606   ↦ cmpt 5208   Γ— cxp 5651   ∘ ccom 5657  β€˜cfv 6516  β„©crio 7332  (class class class)co 7377   ↑m cmap 8787  Basecbs 17109   β†Ύs cress 17138   ↑s cpws 17357  CRingccrg 19994   RingHom crh 20174  SubRingcsubrg 20281  algSccascl 21310   mVar cmvr 21359   mPoly cmpl 21360   evalSub ces 21532
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5262  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3365  df-rab 3419  df-v 3461  df-sbc 3758  df-csb 3874  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4886  df-iun 4976  df-br 5126  df-opab 5188  df-mpt 5209  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-iota 6468  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-riota 7333  df-ov 7380  df-oprab 7381  df-mpo 7382  df-evls 21534
This theorem is referenced by:  evlsval2  21549  evlsval3  40835
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