Theorem mpfrcl 22196

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 4296	. . 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 2883	. 2 ⊢ (𝑋 ∈ 𝑄 → ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
4		rneq 5917	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ran ∅)
5		rn0 5907	. . . 4 ⊢ ran ∅ = ∅
6	4, 5	eqtrdi 2816	. . 3 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) = ∅ → ran ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
7	6	necon3i 2992	. 2 ⊢ (ran ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅)
8		fveq1 6870	. . . . . . 7 ⊢ ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = (∅‘𝑅))
9		0fv 6912	. . . . . . 7 ⊢ (∅‘𝑅) = ∅
10	8, 9	eqtrdi 2816	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) = ∅ → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
11	10	necon3i 2992	. . . . 5 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 evalSub 𝑆) ≠ ∅)
12		reldmevls 22195	. . . . . . . 8 ⊢ Rel dom evalSub
13	12	ovprc1 7439	. . . . . . 7 ⊢ (¬ 𝐼 ∈ V → (𝐼 evalSub 𝑆) = ∅)
14	13	necon1ai 2987	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → 𝐼 ∈ V)
15		n0 4308	. . . . . . 7 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ ↔ ∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆))
16		df-evls 22185	. . . . . . . . . 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 7643	. . . . . . . . 9 ⊢ (𝑎 ∈ (𝐼 evalSub 𝑆) → 𝑆 ∈ CRing)
18	17	a1d 26	. . . . . . . 8 ⊢ (𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
19	18	exlimiv 1953	. . . . . . 7 ⊢ (∃𝑎 𝑎 ∈ (𝐼 evalSub 𝑆) → (𝐼 ∈ V → 𝑆 ∈ CRing))
20	15, 19	sylbi 220	. . . . . 6 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V → 𝑆 ∈ CRing))
21	14, 20	jcai 525	. . . . 5 ⊢ ((𝐼 evalSub 𝑆) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
22	11, 21	syl 18	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing))
23		fvex 6884	. . . . . . . . . . . . 13 ⊢ (Base‘𝑠) ∈ V
24		nfcv 2927	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑏(SubRing‘𝑠)
25		nfcsb1v 3879	. . . . . . . . . . . . . 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 5203	. . . . . . . . . . . . 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 3869	. . . . . . . . . . . . . 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 5199	. . . . . . . . . . . . 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 3889	. . . . . . . . . . . 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 6871	. . . . . . . . . . . . . 14 ⊢ (𝑠 = 𝑆 → (SubRing‘𝑠) = (SubRing‘𝑆))
31	30	adantl 486	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (SubRing‘𝑠) = (SubRing‘𝑆))
32		fveq2 6871	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = 𝑆 → (Base‘𝑠) = (Base‘𝑆))
33	32	adantl 486	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (Base‘𝑠) = (Base‘𝑆))
34	33	csbeq1d 3859	. . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖 = 𝐼 → 𝑖 = 𝐼)
36		oveq1 7407	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑠 = 𝑆 → (𝑠 ↾s 𝑟) = (𝑆 ↾s 𝑟))
37	35, 36	oveqan12d 7419	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑖 mPoly (𝑠 ↾s 𝑟)) = (𝐼 mPoly (𝑆 ↾s 𝑟)))
38	37	csbeq1d 3859	. . . . . . . . . . . . . . . 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 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 = 𝑆 → 𝑠 = 𝑆)
40		oveq2 7408	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑖 = 𝐼 → (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
41	39, 40	oveqan12rd 7420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑠 ↑s (𝑏 ↑m 𝑖)) = (𝑆 ↑s (𝑏 ↑m 𝐼)))
42	41	oveq2d 7416	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑤 RingHom (𝑠 ↑s (𝑏 ↑m 𝑖))) = (𝑤 RingHom (𝑆 ↑s (𝑏 ↑m 𝐼))))
43	40	adantr 485	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑏 ↑m 𝑖) = (𝑏 ↑m 𝐼))
44	43	xpeq1d 5681	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → ((𝑏 ↑m 𝑖) × {𝑥}) = ((𝑏 ↑m 𝐼) × {𝑥}))
45	44	mpteq2dv 5199	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝑖) × {𝑥})) = (𝑥 ∈ 𝑟 ↦ ((𝑏 ↑m 𝐼) × {𝑥})))
46	45	eqeq2d 2776	. . . . . . . . . . . . . . . . . . 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 7419	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑖 mVar (𝑠 ↾s 𝑟)) = (𝐼 mVar (𝑆 ↾s 𝑟)))
48	47	coeq2d 5839	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑓 ∘ (𝑖 mVar (𝑠 ↾s 𝑟))) = (𝑓 ∘ (𝐼 mVar (𝑆 ↾s 𝑟))))
49		simpl 487	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → 𝑖 = 𝐼)
50	43	mpteq1d 5195	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 = 𝐼 ∧ 𝑠 = 𝑆) → (𝑔 ∈ (𝑏 ↑m 𝑖) ↦ (𝑔‘𝑥)) = (𝑔 ∈ (𝑏 ↑m 𝐼) ↦ (𝑔‘𝑥)))
51	49, 50	mpteq12dv 5192	. . . . . . . . . . . . . . . . . . . 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 2781	. . . . . . . . . . . . . . . . . . 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 643	. . . . . . . . . . . . . . . . . 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 7360	. . . . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . . . 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 3862	. . . . . . . . . . . . . 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 2800	. . . . . . . . . . . . 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 5192	. . . . . . . . . . . 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 2812	. . . . . . . . . . 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 6884	. . . . . . . . . . . 12 ⊢ (SubRing‘𝑆) ∈ V
62	61	mptex 7211	. . . . . . . . . . 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 7555	. . . . . . . . . 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 5886	. . . . . . . . 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 2765	. . . . . . . . . 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 6232	. . . . . . . . 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 3983	. . . . . . . 8 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → dom (𝐼 evalSub 𝑆) ⊆ (SubRing‘𝑆))
68	67	ssneld 3941	. . . . . . 7 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆)))
69		ndmfv 6903	. . . . . . 7 ⊢ (¬ 𝑅 ∈ dom (𝐼 evalSub 𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅)
70	68, 69	syl6 36	. . . . . 6 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (¬ 𝑅 ∈ (SubRing‘𝑆) → ((𝐼 evalSub 𝑆)‘𝑅) = ∅))
71	70	necon1ad 2977	. . . . 5 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → 𝑅 ∈ (SubRing‘𝑆)))
72	71	com12 33	. . . 4 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) → 𝑅 ∈ (SubRing‘𝑆)))
73	22, 72	jcai 525	. . 3 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
74		df-3an 1103	. . 3 ⊢ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) ↔ ((𝐼 ∈ V ∧ 𝑆 ∈ CRing) ∧ 𝑅 ∈ (SubRing‘𝑆)))
75	73, 74	sylibr 237	. 2 ⊢ (((𝐼 evalSub 𝑆)‘𝑅) ≠ ∅ → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))
76	3, 7, 75	3syl 19	1 ⊢ (𝑋 ∈ 𝑄 → (𝐼 ∈ V ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  Vcvv 3457  ⦋csb 3855  ∅c0 4288  {csn 4585   ↦ cmpt 5186   × cxp 5650  dom cdm 5652  ran crn 5653   ∘ ccom 5656  ‘cfv 6525  ℩crio 7356  (class class class)co 7400   ↑_m cmap 8812  Basecbs 17259   ↾_s cress 17280   ↑_s cpws 17489  CRingccrg 20307   RingHom crh 20542  SubRingcsubrg 20645  algSccascl 21962   mVar cmvr 22015   mPoly cmpl 22016   evalSub ces 22183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-evls 22185

This theorem is referenced by:  mpff  22223  mpfaddcl  22224  mpfmulcl  22225  mpfind  22226  pf1rcl  22470  mpfpf1  22472