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Theorem mpfrcl 21869
Description: Reverse closure for the set of polynomial functions. (Contributed by Stefan O'Rear, 19-Mar-2015.)
Hypothesis
Ref Expression
mpfrcl.q 𝑄 = ran ((𝐼 evalSub 𝑆)β€˜π‘…)
Assertion
Ref Expression
mpfrcl (𝑋 ∈ 𝑄 β†’ (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)))
Proof of Theorem mpfrcl
Dummy variables π‘Ž 𝑏 𝑓 𝑔 𝑖 π‘Ÿ 𝑠 𝑀 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 4335 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)β€˜π‘…) β†’ ran ((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ…)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)β€˜π‘…)
31, 2eleq2s 2849 . 2 (𝑋 ∈ 𝑄 β†’ ran ((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ…)
4 rneq 5936 . . . 4 (((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ… β†’ ran ((𝐼 evalSub 𝑆)β€˜π‘…) = ran βˆ…)
5 rn0 5926 . . . 4 ran βˆ… = βˆ…
64, 5eqtrdi 2786 . . 3 (((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ… β†’ ran ((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ…)
76necon3i 2971 . 2 (ran ((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ…)
8 fveq1 6891 . . . . . . 7 ((𝐼 evalSub 𝑆) = βˆ… β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = (βˆ…β€˜π‘…))
9 0fv 6936 . . . . . . 7 (βˆ…β€˜π‘…) = βˆ…
108, 9eqtrdi 2786 . . . . . 6 ((𝐼 evalSub 𝑆) = βˆ… β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ…)
1110necon3i 2971 . . . . 5 (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ (𝐼 evalSub 𝑆) β‰  βˆ…)
12 reldmevls 21868 . . . . . . . 8 Rel dom evalSub
1312ovprc1 7452 . . . . . . 7 (Β¬ 𝐼 ∈ V β†’ (𝐼 evalSub 𝑆) = βˆ…)
1413necon1ai 2966 . . . . . 6 ((𝐼 evalSub 𝑆) β‰  βˆ… β†’ 𝐼 ∈ V)
15 n0 4347 . . . . . . 7 ((𝐼 evalSub 𝑆) β‰  βˆ… ↔ βˆƒπ‘Ž π‘Ž ∈ (𝐼 evalSub 𝑆))
16 df-evls 21856 . . . . . . . . . 10 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‘₯)))))))
1716elmpocl2 7654 . . . . . . . . 9 (π‘Ž ∈ (𝐼 evalSub 𝑆) β†’ 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (π‘Ž ∈ (𝐼 evalSub 𝑆) β†’ (𝐼 ∈ V β†’ 𝑆 ∈ CRing))
1918exlimiv 1931 . . . . . . 7 (βˆƒπ‘Ž π‘Ž ∈ (𝐼 evalSub 𝑆) β†’ (𝐼 ∈ V β†’ 𝑆 ∈ CRing))
2015, 19sylbi 216 . . . . . 6 ((𝐼 evalSub 𝑆) β‰  βˆ… β†’ (𝐼 ∈ V β†’ 𝑆 ∈ CRing))
2114, 20jcai 515 . . . . 5 ((𝐼 evalSub 𝑆) β‰  βˆ… β†’ (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6905 . . . . . . . . . . . . 13 (Baseβ€˜π‘ ) ∈ V
24 nfcv 2901 . . . . . . . . . . . . . 14 Ⅎ𝑏(SubRingβ€˜π‘ )
25 nfcsb1v 3919 . . . . . . . . . . . . . 14 Ⅎ𝑏⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘₯)))))
2624, 25nfmpt 5256 . . . . . . . . . . . . 13 Ⅎ𝑏(π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘₯))))))
27 csbeq1a 3908 . . . . . . . . . . . . . 14 (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‘₯))))) = ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘₯))))))
2827mpteq2dv 5251 . . . . . . . . . . . . 13 (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‘ ) ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘₯)))))))
2923, 26, 28csbief 3929 . . . . . . . . . . . 12 ⦋(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‘ ) ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘₯))))))
30 fveq2 6892 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 β†’ (SubRingβ€˜π‘ ) = (SubRingβ€˜π‘†))
3130adantl 480 . . . . . . . . . . . . 13 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (SubRingβ€˜π‘ ) = (SubRingβ€˜π‘†))
32 fveq2 6892 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
3332adantl 480 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (Baseβ€˜π‘ ) = (Baseβ€˜π‘†))
3433csbeq1d 3898 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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¦Œβ¦‹(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‘₯))))))
35 id 22 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼 β†’ 𝑖 = 𝐼)
36 oveq1 7420 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 β†’ (𝑠 β†Ύs π‘Ÿ) = (𝑆 β†Ύs π‘Ÿ))
3735, 36oveqan12d 7432 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑖 mPoly (𝑠 β†Ύs π‘Ÿ)) = (𝐼 mPoly (𝑆 β†Ύs π‘Ÿ)))
3837csbeq1d 3898 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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‘₯))))))
39 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑆 β†’ 𝑠 = 𝑆)
40 oveq2 7421 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 β†’ (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
4139, 40oveqan12rd 7433 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑠 ↑s (𝑏 ↑m 𝑖)) = (𝑆 ↑s (𝑏 ↑m 𝐼)))
4241oveq2d 7429 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑀 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖))) = (𝑀 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼))))
4340adantr 479 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
4443xpeq1d 5706 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ((𝑏 ↑m 𝑖) Γ— {π‘₯}) = ((𝑏 ↑m 𝐼) Γ— {π‘₯}))
4544mpteq2dv 5251 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝐼) Γ— {π‘₯})))
4645eqeq2d 2741 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ((𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝑖) Γ— {π‘₯})) ↔ (𝑓 ∘ (algScβ€˜π‘€)) = (π‘₯ ∈ π‘Ÿ ↦ ((𝑏 ↑m 𝐼) Γ— {π‘₯}))))
4735, 36oveqan12d 7432 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ)) = (𝐼 mVar (𝑆 β†Ύs π‘Ÿ)))
4847coeq2d 5863 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑓 ∘ (𝑖 mVar (𝑠 β†Ύs π‘Ÿ))) = (𝑓 ∘ (𝐼 mVar (𝑆 β†Ύs π‘Ÿ))))
49 simpl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ 𝑖 = 𝐼)
5043mpteq1d 5244 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯)) = (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))
5149, 50mpteq12dv 5240 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (π‘₯ ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (π‘”β€˜π‘₯))) = (π‘₯ ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (π‘”β€˜π‘₯))))
5248, 51eqeq12d 2746 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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 ∈ (𝑏 ↑m 𝐼) ↦ (π‘”β€˜π‘₯)))))
5346, 52anbi12d 629 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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€˜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‘₯))))))
5442, 53riotaeqbidv 7372 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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 ∈ (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‘₯))))))
5554csbeq2dv 3901 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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‘₯))))))
5638, 55eqtrd 2770 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) 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‘₯))))))
5756csbeq2dv 3901 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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¦Œβ¦‹(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‘₯))))))
5834, 57eqtrd 2770 . . . . . . . . . . . . 13 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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¦Œβ¦‹(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‘₯))))))
5931, 58mpteq12dv 5240 . . . . . . . . . . . 12 ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) β†’ (π‘Ÿ ∈ (SubRingβ€˜π‘ ) ↦ ⦋(Baseβ€˜π‘ ) / π‘β¦Œβ¦‹(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‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))))
6029, 59eqtrid 2782 . . . . . . . . . . 11 ((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‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))))
61 fvex 6905 . . . . . . . . . . . 12 (SubRingβ€˜π‘†) ∈ V
6261mptex 7228 . . . . . . . . . . 11 (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))) ∈ V
6360, 16, 62ovmpoa 7567 . . . . . . . . . 10 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ (𝐼 evalSub 𝑆) = (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))))
6463dmeqd 5906 . . . . . . . . 9 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ dom (𝐼 evalSub 𝑆) = dom (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))))
65 eqid 2730 . . . . . . . . . 10 (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯))))))
6665dmmptss 6241 . . . . . . . . 9 dom (π‘Ÿ ∈ (SubRingβ€˜π‘†) ↦ ⦋(Baseβ€˜π‘†) / π‘β¦Œβ¦‹(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‘₯)))))) βŠ† (SubRingβ€˜π‘†)
6764, 66eqsstrdi 4037 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ dom (𝐼 evalSub 𝑆) βŠ† (SubRingβ€˜π‘†))
6867ssneld 3985 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ (Β¬ 𝑅 ∈ (SubRingβ€˜π‘†) β†’ Β¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6927 . . . . . . 7 (Β¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆) β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ…)
7068, 69syl6 35 . . . . . 6 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ (Β¬ 𝑅 ∈ (SubRingβ€˜π‘†) β†’ ((𝐼 evalSub 𝑆)β€˜π‘…) = βˆ…))
7170necon1ad 2955 . . . . 5 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ 𝑅 ∈ (SubRingβ€˜π‘†)))
7271com12 32 . . . 4 (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) β†’ 𝑅 ∈ (SubRingβ€˜π‘†)))
7322, 72jcai 515 . . 3 (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRingβ€˜π‘†)))
74 df-3an 1087 . . 3 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRingβ€˜π‘†)))
7573, 74sylibr 233 . 2 (((𝐼 evalSub 𝑆)β€˜π‘…) β‰  βˆ… β†’ (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)))
763, 7, 753syl 18 1 (𝑋 ∈ 𝑄 β†’ (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRingβ€˜π‘†)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1085   = wceq 1539  βˆƒwex 1779   ∈ wcel 2104   β‰  wne 2938  Vcvv 3472  β¦‹csb 3894  βˆ…c0 4323  {csn 4629   ↦ cmpt 5232   Γ— cxp 5675  dom cdm 5677  ran crn 5678   ∘ ccom 5681  β€˜cfv 6544  β„©crio 7368  (class class class)co 7413   ↑m cmap 8824  Basecbs 17150   β†Ύs cress 17179   ↑s cpws 17398  CRingccrg 20130   RingHom crh 20362  SubRingcsubrg 20459  algSccascl 21628   mVar cmvr 21679   mPoly cmpl 21680   evalSub ces 21854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-evls 21856
This theorem is referenced by:  mpff  21888  mpfaddcl  21889  mpfmulcl  21890  mpfind  21891  pf1rcl  22090  mpfpf1  22092
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