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Theorem mpfrcl 19729
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)‘𝑅)
Assertion
Ref Expression
mpfrcl (𝑋𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))

Proof of Theorem mpfrcl
Dummy variables 𝑎 𝑏 𝑓 𝑔 𝑖 𝑟 𝑠 𝑤 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 4129 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2eleq2s 2910 . 2 (𝑋𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4 rneq 5559 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5 rn0 5585 . . . 4 ran ∅ = ∅
64, 5syl6eq 2863 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
76necon3i 3017 . 2 (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8 fveq1 6410 . . . . . . 7 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9 0fv 6450 . . . . . . 7 (∅‘𝑅) = ∅
108, 9syl6eq 2863 . . . . . 6 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
1110necon3i 3017 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12 reldmevls 19728 . . . . . . . 8 Rel dom evalSub
1312ovprc1 6915 . . . . . . 7 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
1413necon1ai 3012 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15 n0 4139 . . . . . . 7 ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16 df-evls 19717 . . . . . . . . . 10 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)))))))
1716elmpt2cl2 7111 . . . . . . . . 9 (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
1918exlimiv 2021 . . . . . . 7 (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
2015, 19sylbi 208 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
2114, 20jcai 508 . . . . 5 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6424 . . . . . . . . . . . . 13 (Base‘𝑠) ∈ V
24 nfcv 2955 . . . . . . . . . . . . . 14 𝑏(SubRing‘𝑠)
25 nfcsb1v 3751 . . . . . . . . . . . . . 14 𝑏(Base‘𝑠) / 𝑏(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)))))
2624, 25nfmpt 4947 . . . . . . . . . . . . 13 𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ (Base‘𝑠) / 𝑏(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))))))
27 csbeq1a 3744 . . . . . . . . . . . . . 14 (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))))) = (Base‘𝑠) / 𝑏(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))))))
2827mpteq2dv 4946 . . . . . . . . . . . . 13 (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) ↦ (Base‘𝑠) / 𝑏(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)))))))
2923, 26, 28csbief 3760 . . . . . . . . . . . 12 (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) ↦ (Base‘𝑠) / 𝑏(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))))))
30 fveq2 6411 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
3130adantl 469 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32 fveq2 6411 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
3332adantl 469 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
3433csbeq1d 3742 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) / 𝑏(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(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))))))
35 id 22 . . . . . . . . . . . . . . . . . 18 (𝑖 = 𝐼𝑖 = 𝐼)
36 oveq1 6884 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
3735, 36oveqan12d 6896 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
3837csbeq1d 3742 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝐼𝑠 = 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))))))
39 id 22 . . . . . . . . . . . . . . . . . . . 20 (𝑠 = 𝑆𝑠 = 𝑆)
40 oveq2 6885 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑏𝑚 𝑖) = (𝑏𝑚 𝐼))
4139, 40oveqan12rd 6897 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑠s (𝑏𝑚 𝑖)) = (𝑆s (𝑏𝑚 𝐼)))
4241oveq2d 6893 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑤 RingHom (𝑠s (𝑏𝑚 𝑖))) = (𝑤 RingHom (𝑆s (𝑏𝑚 𝐼))))
4340adantr 468 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑏𝑚 𝑖) = (𝑏𝑚 𝐼))
4443xpeq1d 5346 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑏𝑚 𝑖) × {𝑥}) = ((𝑏𝑚 𝐼) × {𝑥}))
4544mpteq2dv 4946 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥})))
4645eqeq2d 2823 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥}))))
4735, 36oveqan12d 6896 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
4847coeq2d 5493 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
49 simpl 470 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → 𝑖 = 𝐼)
5043mpteq1d 4939 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥)))
5149, 50mpteq12dv 4934 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥))))
5248, 51eqeq12d 2828 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥)))))
5346, 52anbi12d 618 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼𝑠 = 𝑆) → (((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)))) ↔ ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥))))))
5442, 53riotaeqbidv 6841 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∈ (𝑤 RingHom (𝑠s (𝑏𝑚 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))))) = (𝑓 ∈ (𝑤 RingHom (𝑆s (𝑏𝑚 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥))))))
5554csbeq2dv 4196 . . . . . . . . . . . . . . . 16 ((𝑖 = 𝐼𝑠 = 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))))))
5638, 55eqtrd 2847 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 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))))))
5756csbeq2dv 4196 . . . . . . . . . . . . . 14 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑆) / 𝑏(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(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))))))
5834, 57eqtrd 2847 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) / 𝑏(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(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))))))
5931, 58mpteq12dv 4934 . . . . . . . . . . . 12 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑟 ∈ (SubRing‘𝑠) ↦ (Base‘𝑠) / 𝑏(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) ↦ (Base‘𝑆) / 𝑏(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)))))))
6029, 59syl5eq 2859 . . . . . . . . . . 11 ((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) ↦ (Base‘𝑆) / 𝑏(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)))))))
61 fvex 6424 . . . . . . . . . . . 12 (SubRing‘𝑆) ∈ V
6261mptex 6714 . . . . . . . . . . 11 (𝑟 ∈ (SubRing‘𝑆) ↦ (Base‘𝑆) / 𝑏(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)))))) ∈ V
6360, 16, 62ovmpt2a 7024 . . . . . . . . . 10 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (𝐼 evalSub 𝑆) = (𝑟 ∈ (SubRing‘𝑆) ↦ (Base‘𝑆) / 𝑏(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)))))))
6463dmeqd 5534 . . . . . . . . 9 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) = dom (𝑟 ∈ (SubRing‘𝑆) ↦ (Base‘𝑆) / 𝑏(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)))))))
65 eqid 2813 . . . . . . . . . 10 (𝑟 ∈ (SubRing‘𝑆) ↦ (Base‘𝑆) / 𝑏(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) ↦ (Base‘𝑆) / 𝑏(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))))))
6665dmmptss 5852 . . . . . . . . 9 dom (𝑟 ∈ (SubRing‘𝑆) ↦ (Base‘𝑆) / 𝑏(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)))))) ⊆ (SubRing‘𝑆)
6764, 66syl6eqss 3859 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
6867ssneld 3807 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6441 . . . . . . 7 𝑅 ∈ dom (𝐼 evalSub 𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
7068, 69syl6 35 . . . . . 6 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅))
7170necon1ad 3002 . . . . 5 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → 𝑅 ∈ (SubRing‘𝑆)))
7271com12 32 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → 𝑅 ∈ (SubRing‘𝑆)))
7322, 72jcai 508 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74 df-3an 1102 . . 3 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
7573, 74sylibr 225 . 2 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
763, 7, 753syl 18 1 (𝑋𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  w3a 1100   = wceq 1637  wex 1859  wcel 2157  wne 2985  Vcvv 3398  csb 3735  c0 4123  {csn 4377  cmpt 4930   × cxp 5316  dom cdm 5318  ran crn 5319  ccom 5322  cfv 6104  crio 6837  (class class class)co 6877  𝑚 cmap 8095  Basecbs 16071  s cress 16072  s cpws 16315  CRingccrg 18753   RingHom crh 18919  SubRingcsubrg 18983  algSccascl 19523   mVar cmvr 19564   mPoly cmpl 19565   evalSub ces 19715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791  ax-rep 4971  ax-sep 4982  ax-nul 4990  ax-pow 5042  ax-pr 5103
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2638  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ne 2986  df-ral 3108  df-rex 3109  df-reu 3110  df-rab 3112  df-v 3400  df-sbc 3641  df-csb 3736  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-mpt 4931  df-id 5226  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-iota 6067  df-fun 6106  df-fn 6107  df-f 6108  df-f1 6109  df-fo 6110  df-f1o 6111  df-fv 6112  df-riota 6838  df-ov 6880  df-oprab 6881  df-mpt2 6882  df-evls 19717
This theorem is referenced by:  mpff  19744  mpfaddcl  19745  mpfmulcl  19746  mpfind  19747  pf1rcl  19924  mpfpf1  19926
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