MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpfrcl Structured version   Visualization version   GIF version

Theorem mpfrcl 22024
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 4330 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2eleq2s 2846 . 2 (𝑋𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4 rneq 5932 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5 rn0 5922 . . . 4 ran ∅ = ∅
64, 5eqtrdi 2783 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
76necon3i 2968 . 2 (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8 fveq1 6890 . . . . . . 7 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9 0fv 6935 . . . . . . 7 (∅‘𝑅) = ∅
108, 9eqtrdi 2783 . . . . . 6 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
1110necon3i 2968 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12 reldmevls 22023 . . . . . . . 8 Rel dom evalSub
1312ovprc1 7453 . . . . . . 7 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
1413necon1ai 2963 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15 n0 4342 . . . . . . 7 ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16 df-evls 22011 . . . . . . . . . 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 7658 . . . . . . . . 9 (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
1918exlimiv 1926 . . . . . . 7 (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
2015, 19sylbi 216 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
2114, 20jcai 516 . . . . 5 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6904 . . . . . . . . . . . . 13 (Base‘𝑠) ∈ V
24 nfcv 2898 . . . . . . . . . . . . . 14 𝑏(SubRing‘𝑠)
25 nfcsb1v 3914 . . . . . . . . . . . . . 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 5249 . . . . . . . . . . . . 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 3903 . . . . . . . . . . . . . 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 5244 . . . . . . . . . . . . 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 3924 . . . . . . . . . . . 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 6891 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
3130adantl 481 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32 fveq2 6891 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
3332adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
3433csbeq1d 3893 . . . . . . . . . . . . . 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 7421 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
3735, 36oveqan12d 7433 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
3837csbeq1d 3893 . . . . . . . . . . . . . . . 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 7422 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑏m 𝑖) = (𝑏m 𝐼))
4139, 40oveqan12rd 7434 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑠s (𝑏m 𝑖)) = (𝑆s (𝑏m 𝐼)))
4241oveq2d 7430 . . . . . . . . . . . . . . . . . 18 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑤 RingHom (𝑠s (𝑏m 𝑖))) = (𝑤 RingHom (𝑆s (𝑏m 𝐼))))
4340adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑏m 𝑖) = (𝑏m 𝐼))
4443xpeq1d 5701 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑏m 𝑖) × {𝑥}) = ((𝑏m 𝐼) × {𝑥}))
4544mpteq2dv 5244 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) = (𝑥𝑟 ↦ ((𝑏m 𝐼) × {𝑥})))
4645eqeq2d 2738 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏m 𝐼) × {𝑥}))))
4735, 36oveqan12d 7433 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
4847coeq2d 5859 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
49 simpl 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → 𝑖 = 𝐼)
5043mpteq1d 5237 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝑏m 𝐼) ↦ (𝑔𝑥)))
5149, 50mpteq12dv 5233 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏m 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏m 𝐼) ↦ (𝑔𝑥))))
5248, 51eqeq12d 2743 . . . . . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . . . . . 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 7373 . . . . . . . . . . . . . . . . 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 3896 . . . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . . . 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 3896 . . . . . . . . . . . . . 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 2767 . . . . . . . . . . . . 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 5233 . . . . . . . . . . . 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 2779 . . . . . . . . . . 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 6904 . . . . . . . . . . . 12 (SubRing‘𝑆) ∈ V
6261mptex 7229 . . . . . . . . . . 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 7570 . . . . . . . . . 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 5902 . . . . . . . . 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 2727 . . . . . . . . . 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 6239 . . . . . . . . 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 4032 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
6867ssneld 3980 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6926 . . . . . . 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 2952 . . . . 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 516 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74 df-3an 1087 . . 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 395  w3a 1085   = wceq 1534  wex 1774  wcel 2099  wne 2935  Vcvv 3469  csb 3889  c0 4318  {csn 4624  cmpt 5225   × cxp 5670  dom cdm 5672  ran crn 5673  ccom 5676  cfv 6542  crio 7369  (class class class)co 7414  m cmap 8838  Basecbs 17173  s cress 17202  s cpws 17421  CRingccrg 20167   RingHom crh 20401  SubRingcsubrg 20499  algSccascl 21779   mVar cmvr 21831   mPoly cmpl 21832   evalSub ces 22009
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-evls 22011
This theorem is referenced by:  mpff  22043  mpfaddcl  22044  mpfmulcl  22045  mpfind  22046  pf1rcl  22261  mpfpf1  22263
  Copyright terms: Public domain W3C validator