Theorem evls1sca 20048

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 7833	. . . . . . 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 2825	. . . . . . 7 ⊢ ((1o evalSub 𝑆)‘𝑅) = ((1o evalSub 𝑆)‘𝑅)
6		eqid 2825	. . . . . . 7 ⊢ (1o mPoly 𝑈) = (1o mPoly 𝑈)
7		evls1sca.u	. . . . . . 7 ⊢ 𝑈 = (𝑆 ↾s 𝑅)
8		eqid 2825	. . . . . . 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 19881	. . . . . 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 1496	. . . . 5 ⊢ (𝜑 → ((1o evalSub 𝑆)‘𝑅) ∈ ((1o mPoly 𝑈) RingHom (𝑆 ↑s (𝐵 ↑𝑚 1o))))
12		eqid 2825	. . . . . 6 ⊢ (Base‘(1o mPoly 𝑈)) = (Base‘(1o mPoly 𝑈))
13		eqid 2825	. . . . . 6 ⊢ (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o))) = (Base‘(𝑆 ↑s (𝐵 ↑𝑚 1o)))
14	12, 13	rhmf 19082	. . . . 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 2825	. . . . . . 7 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊)
18	7	subrgring 19139	. . . . . . . . 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 19978	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ Ring)
22	19, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ Ring)
23	20	ply1lmod 19982	. . . . . . . 8 ⊢ (𝑈 ∈ Ring → 𝑊 ∈ LMod)
24	19, 23	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑊 ∈ LMod)
25		eqid 2825	. . . . . . 7 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
26		eqid 2825	. . . . . . 7 ⊢ (Base‘𝑊) = (Base‘𝑊)
27	16, 17, 22, 24, 25, 26	asclf 19698	. . . . . 6 ⊢ (𝜑 → 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊))
28	9	subrgss 19137	. . . . . . . . . 10 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ⊆ 𝐵)
29	4, 28	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑅 ⊆ 𝐵)
30	7, 9	ressbas2 16294	. . . . . . . . 9 ⊢ (𝑅 ⊆ 𝐵 → 𝑅 = (Base‘𝑈))
31	29, 30	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝑅 = (Base‘𝑈))
32	20	ply1sca 19983	. . . . . . . . . 10 ⊢ (𝑈 ∈ Ring → 𝑈 = (Scalar‘𝑊))
33	19, 32	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 = (Scalar‘𝑊))
34	33	fveq2d 6437	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑈) = (Base‘(Scalar‘𝑊)))
35	31, 34	eqtrd 2861	. . . . . . 7 ⊢ (𝜑 → 𝑅 = (Base‘(Scalar‘𝑊)))
36		eqid 2825	. . . . . . . . . 10 ⊢ (PwSer1‘𝑈) = (PwSer1‘𝑈)
37	20, 36, 26	ply1bas 19925	. . . . . . . . 9 ⊢ (Base‘𝑊) = (Base‘(1o mPoly 𝑈))
38	37	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝑊) = (Base‘(1o mPoly 𝑈)))
39	38	eqcomd 2831	. . . . . . 7 ⊢ (𝜑 → (Base‘(1o mPoly 𝑈)) = (Base‘𝑊))
40	35, 39	feq23d 6273	. . . . . 6 ⊢ (𝜑 → (𝐴:𝑅⟶(Base‘(1o mPoly 𝑈)) ↔ 𝐴:(Base‘(Scalar‘𝑊))⟶(Base‘𝑊)))
41	27, 40	mpbird 249	. . . . 5 ⊢ (𝜑 → 𝐴:𝑅⟶(Base‘(1o mPoly 𝑈)))
42		evls1sca.x	. . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝑅)
43	41, 42	ffvelrnd 6609	. . . 4 ⊢ (𝜑 → (𝐴‘𝑋) ∈ (Base‘(1o mPoly 𝑈)))
44		fvco3 6522	. . . 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 581	. . 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 2825	. . . . . . . . 9 ⊢ (algSc‘𝑊) = (algSc‘𝑊)
48	20, 47	ply1ascl 19988	. . . . . . . 8 ⊢ (algSc‘𝑊) = (algSc‘(1o mPoly 𝑈))
49	46, 48	syl6eq 2877	. . . . . . 7 ⊢ (𝜑 → 𝐴 = (algSc‘(1o mPoly 𝑈)))
50	49	fveq1d 6435	. . . . . 6 ⊢ (𝜑 → (𝐴‘𝑋) = ((algSc‘(1o mPoly 𝑈))‘𝑋))
51	50	fveq2d 6437	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋)) = (((1o evalSub 𝑆)‘𝑅)‘((algSc‘(1o mPoly 𝑈))‘𝑋)))
52		eqid 2825	. . . . . 6 ⊢ (algSc‘(1o mPoly 𝑈)) = (algSc‘(1o mPoly 𝑈))
53	5, 6, 7, 9, 52, 2, 3, 4, 42	evlssca 19882	. . . . 5 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘((algSc‘(1o mPoly 𝑈))‘𝑋)) = ((𝐵 ↑𝑚 1o) × {𝑋}))
54	51, 53	eqtrd 2861	. . . 4 ⊢ (𝜑 → (((1o evalSub 𝑆)‘𝑅)‘(𝐴‘𝑋)) = ((𝐵 ↑𝑚 1o) × {𝑋}))
55	54	fveq2d 6437	. . 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 2826	. . . . 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 5512	. . . . . 6 ⊢ (𝑥 = ((𝐵 ↑𝑚 1o) × {𝑋}) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
58	57	adantl 475	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = ((𝐵 ↑𝑚 1o) × {𝑋})) → (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))
59	29, 42	sseldd 3828	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
60		fconst6g 6331	. . . . . . 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	9	fvexi 6447	. . . . . . . 8 ⊢ 𝐵 ∈ V
63	62	a1i 11	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ V)
64		ovex 6937	. . . . . . . 8 ⊢ (𝐵 ↑𝑚 1o) ∈ V
65	64	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝐵 ↑𝑚 1o) ∈ V)
66	63, 65	elmapd 8136	. . . . . 6 ⊢ (𝜑 → (((𝐵 ↑𝑚 1o) × {𝑋}) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↔ ((𝐵 ↑𝑚 1o) × {𝑋}):(𝐵 ↑𝑚 1o)⟶𝐵))
67	61, 66	mpbird 249	. . . . 5 ⊢ (𝜑 → ((𝐵 ↑𝑚 1o) × {𝑋}) ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)))
68		snex 5129	. . . . . . . 8 ⊢ {𝑋} ∈ V
69	64, 68	xpex 7223	. . . . . . 7 ⊢ ((𝐵 ↑𝑚 1o) × {𝑋}) ∈ V
70	69	a1i 11	. . . . . 6 ⊢ (𝜑 → ((𝐵 ↑𝑚 1o) × {𝑋}) ∈ V)
71		mptexg 6740	. . . . . . 7 ⊢ (𝐵 ∈ V → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) ∈ V)
72	63, 71	syl 17	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) ∈ V)
73		coexg 7379	. . . . . 6 ⊢ ((((𝐵 ↑𝑚 1o) × {𝑋}) ∈ V ∧ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) ∈ V) → (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ V)
74	70, 72, 73	syl2anc 581	. . . . 5 ⊢ (𝜑 → (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) ∈ V)
75	56, 58, 67, 74	fvmptd 6535	. . . 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}))))
76		fconst6g 6331	. . . . . . 7 ⊢ (𝑦 ∈ 𝐵 → (1o × {𝑦}):1o⟶𝐵)
77	76	adantl 475	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1o × {𝑦}):1o⟶𝐵)
78	62, 1	pm3.2i 464	. . . . . . . 8 ⊢ (𝐵 ∈ V ∧ 1o ∈ On)
79	78	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (𝐵 ∈ V ∧ 1o ∈ On))
80		elmapg 8135	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 1o ∈ On) → ((1o × {𝑦}) ∈ (𝐵 ↑𝑚 1o) ↔ (1o × {𝑦}):1o⟶𝐵))
81	79, 80	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → ((1o × {𝑦}) ∈ (𝐵 ↑𝑚 1o) ↔ (1o × {𝑦}):1o⟶𝐵))
82	77, 81	mpbird 249	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (1o × {𝑦}) ∈ (𝐵 ↑𝑚 1o))
83		eqidd 2826	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})) = (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))
84		fconstmpt 5398	. . . . . 6 ⊢ ((𝐵 ↑𝑚 1o) × {𝑋}) = (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ 𝑋)
85	84	a1i 11	. . . . 5 ⊢ (𝜑 → ((𝐵 ↑𝑚 1o) × {𝑋}) = (𝑧 ∈ (𝐵 ↑𝑚 1o) ↦ 𝑋))
86		eqidd 2826	. . . . 5 ⊢ (𝑧 = (1o × {𝑦}) → 𝑋 = 𝑋)
87	82, 83, 85, 86	fmptco 6646	. . . 4 ⊢ (𝜑 → (((𝐵 ↑𝑚 1o) × {𝑋}) ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))) = (𝑦 ∈ 𝐵 ↦ 𝑋))
88	75, 87	eqtrd 2861	. . 3 ⊢ (𝜑 → ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦}))))‘((𝐵 ↑𝑚 1o) × {𝑋})) = (𝑦 ∈ 𝐵 ↦ 𝑋))
89	45, 55, 88	3eqtrd 2865	. 2 ⊢ (𝜑 → (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)) = (𝑦 ∈ 𝐵 ↦ 𝑋))
90		elpwg 4386	. . . . . 6 ⊢ (𝑅 ∈ (SubRing‘𝑆) → (𝑅 ∈ 𝒫 𝐵 ↔ 𝑅 ⊆ 𝐵))
91	28, 90	mpbird 249	. . . . 5 ⊢ (𝑅 ∈ (SubRing‘𝑆) → 𝑅 ∈ 𝒫 𝐵)
92	4, 91	syl 17	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝒫 𝐵)
93		evls1sca.q	. . . . 5 ⊢ 𝑄 = (𝑆 evalSub1 𝑅)
94		eqid 2825	. . . . 5 ⊢ (1o evalSub 𝑆) = (1o evalSub 𝑆)
95	93, 94, 9	evls1fval 20044	. . . 4 ⊢ ((𝑆 ∈ CRing ∧ 𝑅 ∈ 𝒫 𝐵) → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
96	3, 92, 95	syl2anc 581	. . 3 ⊢ (𝜑 → 𝑄 = ((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅)))
97	96	fveq1d 6435	. 2 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (((𝑥 ∈ (𝐵 ↑𝑚 (𝐵 ↑𝑚 1o)) ↦ (𝑥 ∘ (𝑦 ∈ 𝐵 ↦ (1o × {𝑦})))) ∘ ((1o evalSub 𝑆)‘𝑅))‘(𝐴‘𝑋)))
98		fconstmpt 5398	. . 3 ⊢ (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋)
99	98	a1i 11	. 2 ⊢ (𝜑 → (𝐵 × {𝑋}) = (𝑦 ∈ 𝐵 ↦ 𝑋))
100	89, 97, 99	3eqtr4d 2871	1 ⊢ (𝜑 → (𝑄‘(𝐴‘𝑋)) = (𝐵 × {𝑋}))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166  Vcvv 3414   ⊆ wss 3798  𝒫 cpw 4378  {csn 4397   ↦ cmpt 4952   × cxp 5340   ∘ ccom 5346  Oncon0 5963  ⟶wf 6119  ‘cfv 6123  (class class class)co 6905  1_oc1o 7819   ↑_𝑚 cmap 8122  Basecbs 16222   ↾_s cress 16223  Scalarcsca 16308   ↑_s cpws 16460  Ringcrg 18901  CRingccrg 18902   RingHom crh 19068  SubRingcsubrg 19132  LModclmod 19219  algSccascl 19672   mPoly cmpl 19714   evalSub ces 19864  PwSer₁cps1 19905  Poly₁cpl1 19907   evalSub₁ ces1 20038

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209  ax-inf2 8815  ax-cnex 10308  ax-resscn 10309  ax-1cn 10310  ax-icn 10311  ax-addcl 10312  ax-addrcl 10313  ax-mulcl 10314  ax-mulrcl 10315  ax-mulcom 10316  ax-addass 10317  ax-mulass 10318  ax-distr 10319  ax-i2m1 10320  ax-1ne0 10321  ax-1rid 10322  ax-rnegex 10323  ax-rrecex 10324  ax-cnre 10325  ax-pre-lttri 10326  ax-pre-lttrn 10327  ax-pre-ltadd 10328  ax-pre-mulgt0 10329

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3or 1114  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-pss 3814  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4659  df-int 4698  df-iun 4742  df-iin 4743  df-br 4874  df-opab 4936  df-mpt 4953  df-tr 4976  df-id 5250  df-eprel 5255  df-po 5263  df-so 5264  df-fr 5301  df-se 5302  df-we 5303  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-pred 5920  df-ord 5966  df-on 5967  df-lim 5968  df-suc 5969  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-isom 6132  df-riota 6866  df-ov 6908  df-oprab 6909  df-mpt2 6910  df-of 7157  df-ofr 7158  df-om 7327  df-1st 7428  df-2nd 7429  df-supp 7560  df-wrecs 7672  df-recs 7734  df-rdg 7772  df-1o 7826  df-2o 7827  df-oadd 7830  df-er 8009  df-map 8124  df-pm 8125  df-ixp 8176  df-en 8223  df-dom 8224  df-sdom 8225  df-fin 8226  df-fsupp 8545  df-sup 8617  df-oi 8684  df-card 9078  df-pnf 10393  df-mnf 10394  df-xr 10395  df-ltxr 10396  df-le 10397  df-sub 10587  df-neg 10588  df-nn 11351  df-2 11414  df-3 11415  df-4 11416  df-5 11417  df-6 11418  df-7 11419  df-8 11420  df-9 11421  df-n0 11619  df-z 11705  df-dec 11822  df-uz 11969  df-fz 12620  df-fzo 12761  df-seq 13096  df-hash 13411  df-struct 16224  df-ndx 16225  df-slot 16226  df-base 16228  df-sets 16229  df-ress 16230  df-plusg 16318  df-mulr 16319  df-sca 16321  df-vsca 16322  df-ip 16323  df-tset 16324  df-ple 16325  df-ds 16327  df-hom 16329  df-cco 16330  df-0g 16455  df-gsum 16456  df-prds 16461  df-pws 16463  df-mre 16599  df-mrc 16600  df-acs 16602  df-mgm 17595  df-sgrp 17637  df-mnd 17648  df-mhm 17688  df-submnd 17689  df-grp 17779  df-minusg 17780  df-sbg 17781  df-mulg 17895  df-subg 17942  df-ghm 18009  df-cntz 18100  df-cmn 18548  df-abl 18549  df-mgp 18844  df-ur 18856  df-srg 18860  df-ring 18903  df-cring 18904  df-rnghom 19071  df-subrg 19134  df-lmod 19221  df-lss 19289  df-lsp 19331  df-assa 19673  df-asp 19674  df-ascl 19675  df-psr 19717  df-mvr 19718  df-mpl 19719  df-opsr 19721  df-evls 19866  df-psr1 19910  df-ply1 19912  df-evls1 20040

This theorem is referenced by:  evls1scasrng  20063