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Theorem mpfrcl 21495
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 4294 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2eleq2s 2856 . 2 (𝑋𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4 rneq 5891 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5 rn0 5881 . . . 4 ran ∅ = ∅
64, 5eqtrdi 2792 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
76necon3i 2976 . 2 (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8 fveq1 6841 . . . . . . 7 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9 0fv 6886 . . . . . . 7 (∅‘𝑅) = ∅
108, 9eqtrdi 2792 . . . . . 6 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
1110necon3i 2976 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12 reldmevls 21494 . . . . . . . 8 Rel dom evalSub
1312ovprc1 7396 . . . . . . 7 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
1413necon1ai 2971 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15 n0 4306 . . . . . . 7 ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16 df-evls 21482 . . . . . . . . . 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)))))))
1716elmpocl2 7597 . . . . . . . . 9 (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
1918exlimiv 1933 . . . . . . 7 (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
2015, 19sylbi 216 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
2114, 20jcai 517 . . . . 5 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6855 . . . . . . . . . . . . 13 (Base‘𝑠) ∈ V
24 nfcv 2907 . . . . . . . . . . . . . 14 𝑏(SubRing‘𝑠)
25 nfcsb1v 3880 . . . . . . . . . . . . . 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 5212 . . . . . . . . . . . . 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 3869 . . . . . . . . . . . . . 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 5207 . . . . . . . . . . . . 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 3890 . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
3130adantl 482 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32 fveq2 6842 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
3332adantl 482 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
3433csbeq1d 3859 . . . . . . . . . . . . . 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 7364 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
3735, 36oveqan12d 7376 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
3837csbeq1d 3859 . . . . . . . . . . . . . . . 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 7365 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑏m 𝑖) = (𝑏m 𝐼))
4139, 40oveqan12rd 7377 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑠s (𝑏m 𝑖)) = (𝑆s (𝑏m 𝐼)))
4241oveq2d 7373 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑤 RingHom (𝑠s (𝑏m 𝑖))) = (𝑤 RingHom (𝑆s (𝑏m 𝐼))))
4340adantr 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑏m 𝑖) = (𝑏m 𝐼))
4443xpeq1d 5662 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑏m 𝑖) × {𝑥}) = ((𝑏m 𝐼) × {𝑥}))
4544mpteq2dv 5207 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) = (𝑥𝑟 ↦ ((𝑏m 𝐼) × {𝑥})))
4645eqeq2d 2747 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝐼) × {𝑥}))))
4735, 36oveqan12d 7376 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
4847coeq2d 5818 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
49 simpl 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → 𝑖 = 𝐼)
5043mpteq1d 5200 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝑏m 𝐼) ↦ (𝑔𝑥)))
5149, 50mpteq12dv 5196 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏m 𝐼) ↦ (𝑔𝑥))))
5248, 51eqeq12d 2752 . . . . . . . . . . . . . . . . . . 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 631 . . . . . . . . . . . . . . . . . 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 7316 . . . . . . . . . . . . . . . . 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 3862 . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . 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 3862 . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . 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 5196 . . . . . . . . . . . 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, 59eqtrid 2788 . . . . . . . . . . 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 6855 . . . . . . . . . . . 12 (SubRing‘𝑆) ∈ V
6261mptex 7173 . . . . . . . . . . 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, 62ovmpoa 7510 . . . . . . . . . 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 5861 . . . . . . . . 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 2736 . . . . . . . . . 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 6193 . . . . . . . . 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, 66eqsstrdi 3998 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
6867ssneld 3946 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6877 . . . . . . 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 2960 . . . . 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 517 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74 df-3an 1089 . . 3 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
7573, 74sylibr 233 . 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wne 2943  Vcvv 3445  csb 3855  c0 4282  {csn 4586  cmpt 5188   × cxp 5631  dom cdm 5633  ran crn 5634  ccom 5637  cfv 6496  crio 7312  (class class class)co 7357  m cmap 8765  Basecbs 17083  s cress 17112  s cpws 17328  CRingccrg 19965   RingHom crh 20143  SubRingcsubrg 20218  algSccascl 21258   mVar cmvr 21307   mPoly cmpl 21308   evalSub ces 21480
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pr 5384
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-evls 21482
This theorem is referenced by:  mpff  21514  mpfaddcl  21515  mpfmulcl  21516  mpfind  21517  pf1rcl  21715  mpfpf1  21717
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