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Theorem mpfrcl 19339
 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 3880 . . 3 (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2 mpfrcl.q . . 3 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
31, 2eleq2s 2706 . 2 (𝑋𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4 rneq 5272 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5 rn0 5298 . . . 4 ran ∅ = ∅
64, 5syl6eq 2660 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
76necon3i 2814 . 2 (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8 fveq1 6102 . . . . . . 7 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9 0fv 6137 . . . . . . 7 (∅‘𝑅) = ∅
108, 9syl6eq 2660 . . . . . 6 ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
1110necon3i 2814 . . . . 5 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12 reldmevls 19338 . . . . . . . 8 Rel dom evalSub
1312ovprc1 6582 . . . . . . 7 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
1413necon1ai 2809 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15 n0 3890 . . . . . . 7 ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16 df-evls 19327 . . . . . . . . . 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 6776 . . . . . . . . 9 (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
1817a1d 25 . . . . . . . 8 (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
1918exlimiv 1845 . . . . . . 7 (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
2015, 19sylbi 206 . . . . . 6 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
2114, 20jcai 557 . . . . 5 ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
2211, 21syl 17 . . . 4 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23 fvex 6113 . . . . . . . . . . . . 13 (Base‘𝑠) ∈ V
24 nfcv 2751 . . . . . . . . . . . . . 14 𝑏(SubRing‘𝑠)
25 nfcsb1v 3515 . . . . . . . . . . . . . 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 4674 . . . . . . . . . . . . 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 3508 . . . . . . . . . . . . . 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 4673 . . . . . . . . . . . . 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 3524 . . . . . . . . . . . 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 6103 . . . . . . . . . . . . . 14 (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
3130adantl 481 . . . . . . . . . . . . 13 ((𝑖 = 𝐼𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32 fveq2 6103 . . . . . . . . . . . . . . . 16 (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
3332adantl 481 . . . . . . . . . . . . . . 15 ((𝑖 = 𝐼𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
3433csbeq1d 3506 . . . . . . . . . . . . . 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 6556 . . . . . . . . . . . . . . . . . 18 (𝑠 = 𝑆 → (𝑠s 𝑟) = (𝑆s 𝑟))
3735, 36oveqan12d 6568 . . . . . . . . . . . . . . . . 17 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mPoly (𝑠s 𝑟)) = (𝐼 mPoly (𝑆s 𝑟)))
3837csbeq1d 3506 . . . . . . . . . . . . . . . 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 6557 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝐼 → (𝑏𝑚 𝑖) = (𝑏𝑚 𝐼))
4139, 40oveqan12rd 6569 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑠s (𝑏𝑚 𝑖)) = (𝑆s (𝑏𝑚 𝐼)))
4241oveq2d 6565 . . . . . . . . . . . . . . . . . 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 5062 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑏𝑚 𝑖) × {𝑥}) = ((𝑏𝑚 𝐼) × {𝑥}))
4544mpteq2dv 4673 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥})))
4645eqeq2d 2620 . . . . . . . . . . . . . . . . . . 19 ((𝑖 = 𝐼𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥𝑟 ↦ ((𝑏𝑚 𝐼) × {𝑥}))))
4735, 36oveqan12d 6568 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑖 mVar (𝑠s 𝑟)) = (𝐼 mVar (𝑆s 𝑟)))
4847coeq2d 5206 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆s 𝑟))))
49 simpl 472 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → 𝑖 = 𝐼)
5043mpteq1d 4666 . . . . . . . . . . . . . . . . . . . . 21 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥)) = (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥)))
5149, 50mpteq12dv 4663 . . . . . . . . . . . . . . . . . . . 20 ((𝑖 = 𝐼𝑠 = 𝑆) → (𝑥𝑖 ↦ (𝑔 ∈ (𝑏𝑚 𝑖) ↦ (𝑔𝑥))) = (𝑥𝐼 ↦ (𝑔 ∈ (𝑏𝑚 𝐼) ↦ (𝑔𝑥))))
5248, 51eqeq12d 2625 . . . . . . . . . . . . . . . . . . 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 743 . . . . . . . . . . . . . . . . . 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 6514 . . . . . . . . . . . . . . . . 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 3944 . . . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . . . 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 3944 . . . . . . . . . . . . . 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 2644 . . . . . . . . . . . . 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 4663 . . . . . . . . . . . 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 2656 . . . . . . . . . . 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 6113 . . . . . . . . . . . 12 (SubRing‘𝑆) ∈ V
6261mptex 6390 . . . . . . . . . . 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 6689 . . . . . . . . . 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 5248 . . . . . . . . 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 2610 . . . . . . . . . 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 5548 . . . . . . . . 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 3618 . . . . . . . 8 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
6867ssneld 3570 . . . . . . 7 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69 ndmfv 6128 . . . . . . 7 𝑅 ∈ dom (𝐼 evalSub 𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
7068, 69syl6 34 . . . . . 6 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅))
7170necon1ad 2799 . . . . 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 557 . . 3 (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74 df-3an 1033 . . 3 ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
7573, 74sylibr 223 . 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 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  Vcvv 3173  ⦋csb 3499  ∅c0 3874  {csn 4125   ↦ cmpt 4643   × cxp 5036  dom cdm 5038  ran crn 5039   ∘ ccom 5042  ‘cfv 5804  ℩crio 6510  (class class class)co 6549   ↑𝑚 cmap 7744  Basecbs 15695   ↾s cress 15696   ↑s cpws 15930  CRingccrg 18371   RingHom crh 18535  SubRingcsubrg 18599  algSccascl 19132   mVar cmvr 19173   mPoly cmpl 19174   evalSub ces 19325 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-evls 19327 This theorem is referenced by:  mpff  19354  mpfaddcl  19355  mpfmulcl  19356  mpfind  19357  pf1rcl  19534  mpfpf1  19536
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