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Theorem mpfrcl 19566
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 3954 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2eleq2s 2748 . 2 (𝑋𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4 rneq 5383 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5 rn0 5409 . . . 4 ran ∅ = ∅
64, 5syl6eq 2701 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
76necon3i 2855 . 2 (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8 fveq1 6228 . . . . . . 7 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9 0fv 6265 . . . . . . 7 (∅‘𝑅) = ∅
108, 9syl6eq 2701 . . . . . 6 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
1110necon3i 2855 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12 reldmevls 19565 . . . . . . . 8 Rel dom evalSub
1312ovprc1 6724 . . . . . . 7 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
1413necon1ai 2850 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15 n0 3964 . . . . . . 7 ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16 df-evls 19554 . . . . . . . . . 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 6920 . . . . . . . . 9 (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
1918exlimiv 1898 . . . . . . 7 (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
2015, 19sylbi 207 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
2114, 20jcai 558 . . . . 5 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6239 . . . . . . . . . . . . 13 (Base‘𝑠) ∈ V
24 nfcv 2793 . . . . . . . . . . . . . 14 𝑏(SubRing‘𝑠)
25 nfcsb1v 3582 . . . . . . . . . . . . . 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 4779 . . . . . . . . . . . . 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 3575 . . . . . . . . . . . . . 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 4778 . . . . . . . . . . . . 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 3591 . . . . . . . . . . . 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 6229 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
3130adantl 481 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32 fveq2 6229 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
3332adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
3433csbeq1d 3573 . . . . . . . . . . . . . 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 6697 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
3735, 36oveqan12d 6709 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
3837csbeq1d 3573 . . . . . . . . . . . . . . . 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 6698 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑏𝑚 𝑖) = (𝑏𝑚 𝐼))
4139, 40oveqan12rd 6710 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑠s (𝑏𝑚 𝑖)) = (𝑆s (𝑏𝑚 𝐼)))
4241oveq2d 6706 . . . . . . . . . . . . . . . . . 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 5172 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑏𝑚 𝑖) × {𝑥}) = ((𝑏𝑚 𝐼) × {𝑥}))
4544mpteq2dv 4778 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥})))
4645eqeq2d 2661 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥}))))
4735, 36oveqan12d 6709 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
4847coeq2d 5317 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
49 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → 𝑖 = 𝐼)
5043mpteq1d 4771 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥)))
5149, 50mpteq12dv 4766 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥))))
5248, 51eqeq12d 2666 . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . 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 6654 . . . . . . . . . . . . . . . . 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 4025 . . . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . . . 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 4025 . . . . . . . . . . . . . 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 2685 . . . . . . . . . . . . 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 4766 . . . . . . . . . . . 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 2697 . . . . . . . . . . 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 6239 . . . . . . . . . . . 12 (SubRing‘𝑆) ∈ V
6261mptex 6527 . . . . . . . . . . 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 6833 . . . . . . . . . 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 5358 . . . . . . . . 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 2651 . . . . . . . . . 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 5669 . . . . . . . . 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 3688 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
6867ssneld 3638 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6256 . . . . . . 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 2840 . . . . 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 558 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74 df-3an 1056 . . 3 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
7573, 74sylibr 224 . 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  Vcvv 3231  csb 3566  c0 3948  {csn 4210  cmpt 4762   × cxp 5141  dom cdm 5143  ran crn 5144  ccom 5147  cfv 5926  crio 6650  (class class class)co 6690  𝑚 cmap 7899  Basecbs 15904  s cress 15905  s cpws 16154  CRingccrg 18594   RingHom crh 18760  SubRingcsubrg 18824  algSccascl 19359   mVar cmvr 19400   mPoly cmpl 19401   evalSub ces 19552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-evls 19554
This theorem is referenced by:  mpff  19581  mpfaddcl  19582  mpfmulcl  19583  mpfind  19584  pf1rcl  19761  mpfpf1  19763
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