Theorem evls1sca 19688

Description: Univariate polynomial evaluation maps scalars to constant functions. (Contributed by AV, 8-Sep-2019.)

Hypotheses

Ref	Expression
evls1sca.q	⊢ 𝑄 = (𝑆 evalSub1 𝑅)
evls1sca.w	⊢ 𝑊 = (Poly1‘𝑈)
evls1sca.u	⊢ 𝑈 = (𝑆 ↾s 𝑅)
evls1sca.b	⊢ 𝐵 = (Base‘𝑆)
evls1sca.a	⊢ 𝐴 = (algSc‘𝑊)
evls1sca.s	⊢ (𝜑 → 𝑆 ∈ CRing)
evls1sca.r	⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
evls1sca.x	⊢ (𝜑 → 𝑋 ∈ 𝑅)

Assertion

Ref	Expression
evls1sca	⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Proof of Theorem evls1sca

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1on 7567	. . . . . . 7 ⊢ 1𝑜 ∈ On
2	1	a1i 11	. . . . . 6 ⊢ (𝜑 → 1𝑜 ∈ On)
3		evls1sca.s	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ CRing)
4		evls1sca.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ (SubRing‘𝑆))
5		eqid 2622	. . . . . . 7 ⊢ ((1𝑜 evalSub 𝑆)‘𝑅) = ((1𝑜 evalSub 𝑆)‘𝑅)
6		eqid 2622	. . . . . . 7 ⊢ (1𝑜 mPoly 𝑈) = (1𝑜 mPoly 𝑈)
7		evls1sca.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
8		eqid 2622	. . . . . . 7 ⊢ (𝑆 ↑s (𝐵 ↑𝑚 1𝑜)) = (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))
9		evls1sca.b	. . . . . . 7 ⊢ 𝐵 = (Base‘𝑆)
10	5, 6, 7, 8, 9	evlsrhm 19521	. . . . . 6 ⊢ ((1𝑜 ∈ On ∧ 𝑆 ∈ CRing ∧ 𝑅 ∈ (SubRing‘𝑆)) → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
11	2, 3, 4, 10	syl3anc 1326	. . . . 5 ⊢ (𝜑 → ((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
12		eqid 2622	. . . . . 6 ⊢ (Base‘(1𝑜 mPoly 𝑈)) = (Base‘(1𝑜 mPoly 𝑈))
13		eqid 2622	. . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜)))
14	12, 13	rhmf 18726	. . . . 5 ⊢ (((1𝑜 evalSub 𝑆)‘𝑅) ∈ ((1𝑜 mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) → ((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
15	11, 14	syl 17	. . . 4 ⊢ (𝜑 → ((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))))
16		evls1sca.a	. . . . . . 7 ⊢ 𝐴 = (algSc‘𝑊)
17		eqid 2622	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
18	7	subrgring 18783	. . . . . . . . 9 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑈 ∈ Ring)
19	4, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ Ring)
20		evls1sca.w	. . . . . . . . 9 ⊢ 𝑊 = (Poly1‘𝑈)
21	20	ply1ring 19618	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
23	20	ply1lmod 19622	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ LMod)
24	19, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
25		eqid 2622	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26		eqid 2622	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
27	16, 17, 22, 24, 25, 26	asclf 19337	. . . . . 6 ⊢ (𝜑 → 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
28	9	subrgss 18781	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
29	4, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝐵)
30	7, 9	ressbas2 15931	. . . . . . . . 9 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
32	20	ply1sca 19623	. . . . . . . . . 10 ⊢ (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
33	19, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑊))
34	33	fveq2d 6195	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
35	31, 34	eqtrd 2656	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Base‘(Scalar‘𝑊)))
36		eqid 2622	. . . . . . . . . 10 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
37	20, 36, 26	ply1bas 19565	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈))
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑊) = (Base‘(1𝑜 mPoly 𝑈)))
39	38	eqcomd 2628	. . . . . . 7 ⊢ (𝜑 → (Base‘(1𝑜 mPoly 𝑈)) = (Base‘𝑊))
40	35, 39	feq23d 6040	. . . . . 6 ⊢ (𝜑 → (𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)) ↔ 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)))
41	27, 40	mpbird 247	. . . . 5 ⊢ (𝜑 → 𝐴:𝑅⟶(Base‘(1𝑜 mPoly 𝑈)))
42		evls1sca.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑅)
43	41, 42	ffvelrnd 6360	. . . 4 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘(1𝑜 mPoly 𝑈)))
44		fvco3 6275	. . . 4 ⊢ ((((1𝑜 evalSub 𝑆)‘𝑅):(Base‘(1𝑜 mPoly 𝑈))⟶(Base‘(𝑆 ↑s (𝐵 ↑𝑚 1𝑜))) ∧ (𝐴‘𝑋) ∈ (Base‘(1𝑜 mPoly 𝑈))) → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋))))
45	15, 43, 44	syl2anc 693	. . 3 ⊢ (𝜑 → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋))))
46	16	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝐴 = (algSc‘𝑊))
47		eqid 2622	. . . . . . . . 9 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
48	20, 47	ply1ascl 19628	. . . . . . . 8 ⊢ (algSc‘𝑊) = (algSc‘(1𝑜 mPoly 𝑈))
49	46, 48	syl6eq 2672	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (algSc‘(1𝑜 mPoly 𝑈)))
50	49	fveq1d 6193	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = ((algSc‘(1𝑜 mPoly 𝑈))‘𝑋))
51	50	fveq2d 6195	. . . . 5 ⊢ (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋)) = (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)))
52		eqid 2622	. . . . . 6 ⊢ (algSc‘(1𝑜 mPoly 𝑈)) = (algSc‘(1𝑜 mPoly 𝑈))
53	5, 6, 7, 9, 52, 2, 3, 4, 42	evlssca 19522	. . . . 5 ⊢ (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘((algSc‘(1𝑜 mPoly 𝑈))‘𝑋)) = ((𝐵 ↑𝑚 1𝑜) × {𝑋}))
54	51, 53	eqtrd 2656	. . . 4 ⊢ (𝜑 → (((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 1𝑜) × {𝑋}))
55	54	fveq2d 6195	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘(((1𝑜 evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋))) = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵 ↑𝑚 1𝑜) × {𝑋})))
56		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) = (𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))))
57		coeq1 5279	. . . . . 6 ⊢ (𝑥 = ((𝐵 ↑𝑚 1𝑜) × {𝑋}) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
58	57	adantl 482	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = ((𝐵 ↑𝑚 1𝑜) × {𝑋})) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) = (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
59	29, 42	sseldd 3604	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
60		fconst6g 6094	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → ((𝐵 ↑𝑚 1𝑜) × {𝑋}):(𝐵 ↑𝑚 1𝑜)⟶𝐵)
61	59, 60	syl 17	. . . . . 6 ⊢ (𝜑 → ((𝐵 ↑𝑚 1𝑜) × {𝑋}):(𝐵 ↑𝑚 1𝑜)⟶𝐵)
62		fvex 6201	. . . . . . . . 9 ⊢ (Base‘𝑆) ∈ V
63	9, 62	eqeltri 2697	. . . . . . . 8 ⊢ 𝐵 ∈ V
64	63	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
65		ovex 6678	. . . . . . . 8 ⊢ (𝐵 ↑𝑚 1𝑜) ∈ V
66	65	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵 ↑𝑚 1𝑜) ∈ V)
67	64, 66	elmapd 7871	. . . . . 6 ⊢ (𝜑 → (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↔ ((𝐵 ↑𝑚 1𝑜) × {𝑋}):(𝐵 ↑𝑚 1𝑜)⟶𝐵))
68	61, 67	mpbird 247	. . . . 5 ⊢ (𝜑 → ((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)))
69		snex 4908	. . . . . . . 8 ⊢ {𝑋} ∈ V
70	65, 69	xpex 6962	. . . . . . 7 ⊢ ((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∈ V
71	70	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∈ V)
72		mptexg 6484	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
73	64, 72	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) ∈ V)
74		coexg 7117	. . . . . 6 ⊢ ((((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∈ V ∧ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) ∈ V) → (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
75	71, 73, 74	syl2anc 693	. . . . 5 ⊢ (𝜑 → (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) ∈ V)
76	56, 58, 68, 75	fvmptd 6288	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵 ↑𝑚 1𝑜) × {𝑋})) = (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))
77		fconst6g 6094	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (1𝑜 × {𝑦}):1𝑜⟶𝐵)
78	77	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1𝑜 × {𝑦}):1𝑜⟶𝐵)
79	63, 1	pm3.2i 471	. . . . . . . 8 ⊢ (𝐵 ∈ V ∧ 1𝑜 ∈ On)
80	79	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐵 ∈ V ∧ 1𝑜 ∈ On))
81		elmapg 7870	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 1𝑜 ∈ On) → ((1𝑜 × {𝑦}) ∈ (𝐵 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑦}):1𝑜⟶𝐵))
82	80, 81	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1𝑜 × {𝑦}) ∈ (𝐵 ↑𝑚 1𝑜) ↔ (1𝑜 × {𝑦}):1𝑜⟶𝐵))
83	78, 82	mpbird 247	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1𝑜 × {𝑦}) ∈ (𝐵 ↑𝑚 1𝑜))
84		eqidd 2623	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))
85		fconstmpt 5163	. . . . . 6 ⊢ ((𝐵 ↑𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ 𝑋)
86	85	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐵 ↑𝑚 1𝑜) × {𝑋}) = (𝑧 ∈ (𝐵 ↑𝑚 1𝑜) ↦ 𝑋))
87		eqidd 2623	. . . . 5 ⊢ (𝑧 = (1𝑜 × {𝑦}) → 𝑋 = 𝑋)
88	83, 84, 86, 87	fmptco 6396	. . . 4 ⊢ (𝜑 → (((𝐵 ↑𝑚 1𝑜) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
89	76, 88	eqtrd 2656	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦}))))‘((𝐵 ↑𝑚 1𝑜) × {𝑋})) = (𝑦 ∈ 𝐵 ↦ 𝑋))
90	45, 55, 89	3eqtrd 2660	. 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)) = (𝑦 ∈ 𝐵 ↦ 𝑋))
91		elpwg 4166	. . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
92	28, 91	mpbird 247	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ 𝒫 𝐵)
93	4, 92	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝐵)
94		evls1sca.q	. . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
95		eqid 2622	. . . . 5 ⊢ (1𝑜 evalSub 𝑆) = (1𝑜 evalSub 𝑆)
96	94, 95, 9	evls1fval 19684	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
97	3, 93, 96	syl2anc 693	. . 3 ⊢ (𝜑 → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅)))
98	97	fveq1d 6193	. 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1𝑜)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1𝑜 × {𝑦})))) ∘ ((1𝑜 evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)))
99		fconstmpt 5163	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
100	99	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋))
101	90, 98, 100	3eqtr4d 2666	1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158  {csn 4177   ↦ cmpt 4729   × cxp 5112   ∘ ccom 5118  Oncon0 5723  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553   ↑_𝑚 cmap 7857  Basecbs 15857   ↾_s cress 15858  Scalarcsca 15944   ↑_s cpws 16107  Ringcrg 18547  CRingccrg 18548   RingHom crh 18712  SubRingcsubrg 18776  LModclmod 18863  algSccascl 19311   mPoly cmpl 19353   evalSub ces 19504  PwSer₁cps1 19545  Poly₁cpl1 19547   evalSub₁ ces1 19678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-ofr 6898  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-fzo 12466  df-seq 12802  df-hash 13118  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-0g 16102  df-gsum 16103  df-prds 16108  df-pws 16110  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-mhm 17335  df-submnd 17336  df-grp 17425  df-minusg 17426  df-sbg 17427  df-mulg 17541  df-subg 17591  df-ghm 17658  df-cntz 17750  df-cmn 18195  df-abl 18196  df-mgp 18490  df-ur 18502  df-srg 18506  df-ring 18549  df-cring 18550  df-rnghom 18715  df-subrg 18778  df-lmod 18865  df-lss 18933  df-lsp 18972  df-assa 19312  df-asp 19313  df-ascl 19314  df-psr 19356  df-mvr 19357  df-mpl 19358  df-opsr 19360  df-evls 19506  df-psr1 19550  df-ply1 19552  df-evls1 19680

This theorem is referenced by:  evls1scasrng  19703