Theorem mpfrcl 22040

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	⊢ (𝑋 ∈ 𝑄 → (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.

Step	Hyp	Ref	Expression
1		ne0i 4293	. . 3 ⊢ (𝑋 ∈ ran ((𝐼 evalSub 𝑆)‘𝑅) → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
2		mpfrcl.q	. . 3 ⊢ 𝑄 = ran ((𝐼 evalSub 𝑆)‘𝑅)
3	1, 2	eleq2s 2854	. 2 ⊢ (𝑋 ∈ 𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4		rneq 5885	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5		rn0 5875	. . . 4 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2787	. . 3 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
7	6	necon3i 2964	. 2 ⊢ (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8		fveq1 6833	. . . . . . 7 ⊢ ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9		0fv 6875	. . . . . . 7 ⊢ (∅‘𝑅) = ∅
10	8, 9	eqtrdi 2787	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
11	10	necon3i 2964	. . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12		reldmevls 22039	. . . . . . . 8 ⊢ Rel dom evalSub
13	12	ovprc1 7397	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
14	13	necon1ai 2959	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15		n0 4305	. . . . . . 7 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16		df-evls 22029	. . . . . . . . . 10 ⊢ evalSub = (𝑖 ∈ V, 𝑠 ∈ CRing ↦ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))))
17	16	elmpocl2 7601	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
18	17	a1d 25	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
19	18	exlimiv 1931	. . . . . . 7 ⊢ (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
20	15, 19	sylbi 217	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
21	14, 20	jcai 516	. . . . 5 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
22	11, 21	syl 17	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23		fvex 6847	. . . . . . . . . . . . 13 ⊢ (Base‘𝑠) ∈ V
24		nfcv 2898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(SubRing‘𝑠)
25		nfcsb1v 3873	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))
26	24, 25	nfmpt 5196	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑏(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))
27		csbeq1a 3863	. . . . . . . . . . . . . 14 ⊢ (𝑏 = (Base‘𝑠) → ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))
28	27	mpteq2dv 5192	. . . . . . . . . . . . 13 ⊢ (𝑏 = (Base‘𝑠) → (𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))))
29	23, 26, 28	csbief 3883	. . . . . . . . . . . 12 ⊢ ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))
30		fveq2 6834	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
31	30	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32		fveq2 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
33	32	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
34	33	csbeq1d 3853	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(Base‘𝑆) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))
35		id 22	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
36		oveq1 7365	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑆 → (𝑠 ↾s 𝑟) = (𝑆 ↾s 𝑟))
37	35, 36	oveqan12d 7377	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑖 mPoly (𝑠 ↾s 𝑟)) = (𝐼 mPoly (𝑆 ↾s 𝑟)))
38	37	csbeq1d 3853	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))))
39		id 22	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
40		oveq2 7366	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝐼 → (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
41	39, 40	oveqan12rd 7378	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑠 ↑s (𝑏 ↑m 𝑖)) = (𝑆 ↑s (𝑏 ↑m 𝐼)))
42	41	oveq2d 7374	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖))) = (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼))))
43	40	adantr 480	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
44	43	xpeq1d 5653	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ((𝑏 ↑m 𝑖) × {𝑥}) = ((𝑏 ↑m 𝐼) × {𝑥}))
45	44	mpteq2dv 5192	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})))
46	45	eqeq2d 2747	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ↔ (𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥}))))
47	35, 36	oveqan12d 7377	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑖 mVar (𝑠 ↾s 𝑟)) = (𝐼 mVar (𝑆 ↾s 𝑟)))
48	47	coeq2d 5811	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))))
49		simpl 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → 𝑖 = 𝐼)
50	43	mpteq1d 5188	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))
51	49, 50	mpteq12dv 5185	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))
52	48, 51	eqeq12d 2752	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ((𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))) ↔ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))
53	46, 52	anbi12d 632	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))) ↔ ((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
54	42, 53	riotaeqbidv 7318	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = (℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
55	54	csbeq2dv 3856	. . . . . . . . . . . . . . . 16 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
56	38, 55	eqtrd 2771	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
57	56	csbeq2dv 3856	. . . . . . . . . . . . . 14 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑆) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
58	34, 57	eqtrd 2771	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥))))) = ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
59	31, 58	mpteq12dv 5185	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(Base‘𝑠) / 𝑏⦌⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))))
60	29, 59	eqtrid 2783	. . . . . . . . . . 11 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ⦋(Base‘𝑠) / 𝑏⦌(𝑟 ∈ (SubRing‘𝑠) ↦ ⦋(𝑖 mPoly (𝑠 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) ∧ (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑥 ∈ 𝑖 ↦ (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))))
61		fvex 6847	. . . . . . . . . . . 12 ⊢ (SubRing‘𝑆) ∈ V
62	61	mptex 7169	. . . . . . . . . . 11 ⊢ (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))) ∈ V
63	60, 16, 62	ovmpoa 7513	. . . . . . . . . 10 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (𝐼 evalSub 𝑆) = (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))))
64	63	dmeqd 5854	. . . . . . . . 9 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) = dom (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))))
65		eqid 2736	. . . . . . . . . 10 ⊢ (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))) = (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥))))))
66	65	dmmptss 6199	. . . . . . . . 9 ⊢ dom (𝑟 ∈ (SubRing‘𝑆) ↦ ⦋(Base‘𝑆) / 𝑏⦌⦋(𝐼 mPoly (𝑆 ↾s 𝑟)) / 𝑤⦌(℩𝑓 ∈ (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼)))((𝑓 ∘ (algSc‘𝑤)) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})) ∧ (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))) = (𝑥 ∈ 𝐼 ↦ (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))))) ⊆ (SubRing‘𝑆)
67	64, 66	eqsstrdi 3978	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
68	67	ssneld 3935	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69		ndmfv 6866	. . . . . . 7 ⊢ (¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
70	68, 69	syl6 35	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅))
71	70	necon1ad 2949	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → 𝑅 ∈ (SubRing‘𝑆)))
72	71	com12 32	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → 𝑅 ∈ (SubRing‘𝑆)))
73	22, 72	jcai 516	. . 3 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74		df-3an 1088	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
75	73, 74	sylibr 234	. 2 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
76	3, 7, 75	3syl 18	1 ⊢ (𝑋 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2113   ≠ wne 2932  Vcvv 3440  ⦋csb 3849  ∅c0 4285  {csn 4580   ↦ cmpt 5179   × cxp 5622  dom cdm 5624  ran crn 5625   ∘ ccom 5628  ‘cfv 6492  ℩crio 7314  (class class class)co 7358   ↑_m cmap 8763  Basecbs 17136   ↾_s cress 17157   ↑_s cpws 17366  CRingccrg 20169   RingHom crh 20405  SubRingcsubrg 20502  algSccascl 21807   mVar cmvr 21861   mPoly cmpl 21862   evalSub ces 22027

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-evls 22029

This theorem is referenced by:  mpff  22067  mpfaddcl  22068  mpfmulcl  22069  mpfind  22070  pf1rcl  22293  mpfpf1  22295