dvfsumle - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dvfsumle		Structured version Visualization version GIF version

Theorem dvfsumle 25185

Description: Compare a finite sum to an integral (the integral here is given as a function with a known derivative). (Contributed by Mario Carneiro, 14-May-2016.)

**Hypotheses**
Ref	Expression
dvfsumle.m	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
dvfsumle.a	⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
dvfsumle.v	⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
dvfsumle.b	⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
dvfsumle.c	⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
dvfsumle.d	⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
dvfsumle.x	⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
dvfsumle.l	⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)

**Assertion**
Ref	Expression
dvfsumle	⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Distinct variable groups: 𝐴,𝑘 𝑥,𝑘,𝑀 𝑘,𝑁,𝑥 𝜑,𝑘,𝑥 𝑥,𝑋 𝑥,𝐶 𝑥,𝐷 𝑥,𝑉 Allowed substitution hints: 𝐴(𝑥) 𝐵(𝑥,𝑘) 𝐶(𝑘) 𝐷(𝑘) 𝑉(𝑘) 𝑋(𝑘)

Proof of Theorem dvfsumle Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13694	. . . 4 ⊢ (𝑀..^𝑁) ∈ Fin
2	1	a1i 11	. . 3 ⊢ (𝜑 → (𝑀..^𝑁) ∈ Fin)
3		dvfsumle.x	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℝ)
4		dvfsumle.m	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
5		eluzel2 12587	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑀 ∈ ℤ)
6	4, 5	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ ℤ)
7		eluzelz 12592	. . . . . . . . . . 11 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → 𝑁 ∈ ℤ)
8	4, 7	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ ℤ)
9		fzval2 13242	. . . . . . . . . 10 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
10	6, 8, 9	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑀...𝑁) = ((𝑀[,]𝑁) ∩ ℤ))
11		inss1 4162	. . . . . . . . 9 ⊢ ((𝑀[,]𝑁) ∩ ℤ) ⊆ (𝑀[,]𝑁)
12	10, 11	eqsstrdi 3975	. . . . . . . 8 ⊢ (𝜑 → (𝑀...𝑁) ⊆ (𝑀[,]𝑁))
13	12	sselda 3921	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → 𝑦 ∈ (𝑀[,]𝑁))
14		dvfsumle.a	. . . . . . . . . 10 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
15		cncff 24056	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
17		eqid 2738	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) = (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴)
18	17	fmpt 6984	. . . . . . . . 9 ⊢ (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ↔ (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
19	16, 18	sylibr 233	. . . . . . . 8 ⊢ (𝜑 → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
20		nfcsb1v 3857	. . . . . . . . . 10 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴
21	20	nfel1 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ
22		csbeq1a 3846	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝐴 = ⦋𝑦 / 𝑥⦌𝐴)
23	22	eleq1d 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
24	21, 23	rspc 3549	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀[,]𝑁) → (∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ))
25	19, 24	mpan9 507	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
26	13, 25	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
27	26	ralrimiva 3103	. . . . 5 ⊢ (𝜑 → ∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
28		fzofzp1 13484	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → (𝑘 + 1) ∈ (𝑀...𝑁))
29		csbeq1 3835	. . . . . . 7 ⊢ (𝑦 = (𝑘 + 1) → ⦋𝑦 / 𝑥⦌𝐴 = ⦋(𝑘 + 1) / 𝑥⦌𝐴)
30	29	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = (𝑘 + 1) → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ))
31	30	rspccva 3560	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ (𝑘 + 1) ∈ (𝑀...𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
32	27, 28, 31	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋(𝑘 + 1) / 𝑥⦌𝐴 ∈ ℝ)
33		elfzofz 13403	. . . . 5 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ (𝑀...𝑁))
34		csbeq1 3835	. . . . . . 7 ⊢ (𝑦 = 𝑘 → ⦋𝑦 / 𝑥⦌𝐴 = ⦋𝑘 / 𝑥⦌𝐴)
35	34	eleq1d 2823	. . . . . 6 ⊢ (𝑦 = 𝑘 → (⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ↔ ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ))
36	35	rspccva 3560	. . . . 5 ⊢ ((∀𝑦 ∈ (𝑀...𝑁)⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ ∧ 𝑘 ∈ (𝑀...𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
37	27, 33, 36	syl2an 596	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ⦋𝑘 / 𝑥⦌𝐴 ∈ ℝ)
38	32, 37	resubcld 11403	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) ∈ ℝ)
39		elfzoelz 13387	. . . . . . . . . 10 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑘 ∈ ℤ)
40	39	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℤ)
41	40	zred 12426	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ)
42	41	recnd 11003	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℂ)
43		ax-1cn 10929	. . . . . . 7 ⊢ 1 ∈ ℂ
44		pncan2 11228	. . . . . . 7 ⊢ ((𝑘 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑘 + 1) − 𝑘) = 1)
45	42, 43, 44	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑘 + 1) − 𝑘) = 1)
46	45	oveq2d 7291	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = (𝑋 · 1))
47	3	recnd 11003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ∈ ℂ)
48		peano2re 11148	. . . . . . . 8 ⊢ (𝑘 ∈ ℝ → (𝑘 + 1) ∈ ℝ)
49	41, 48	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ)
50	49	recnd 11003	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℂ)
51	47, 50, 42	subdid 11431	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · ((𝑘 + 1) − 𝑘)) = ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)))
52	47	mulid1d 10992	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑋 · 1) = 𝑋)
53	46, 51, 52	3eqtr3d 2786	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) = 𝑋)
54		eqid 2738	. . . . . 6 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
55	54	mulcn 24030	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
56	6	zred 12426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℝ)
57	56	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ)
58	8	zred 12426	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℝ)
59	58	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ)
60		elfzole1 13395	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (𝑀..^𝑁) → 𝑀 ≤ 𝑘)
61	60	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ≤ 𝑘)
62	28	adantl 482	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑀...𝑁))
63		elfzle2 13260	. . . . . . . . . . 11 ⊢ ((𝑘 + 1) ∈ (𝑀...𝑁) → (𝑘 + 1) ≤ 𝑁)
64	62, 63	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ≤ 𝑁)
65		iccss 13147	. . . . . . . . . 10 ⊢ (((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) ∧ (𝑀 ≤ 𝑘 ∧ (𝑘 + 1) ≤ 𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
66	57, 59, 61, 64, 65	syl22anc 836	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁))
67		iccssre 13161	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀[,]𝑁) ⊆ ℝ)
68	56, 58, 67	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ ℝ)
69	68	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℝ)
70	66, 69	sstrd 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℝ)
71		ax-resscn 10928	. . . . . . . 8 ⊢ ℝ ⊆ ℂ
72	70, 71	sstrdi 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘[,](𝑘 + 1)) ⊆ ℂ)
73	71	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ⊆ ℂ)
74		cncfmptc 24075	. . . . . . 7 ⊢ ((𝑋 ∈ ℝ ∧ (𝑘[,](𝑘 + 1)) ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
75	3, 72, 73, 74	syl3anc 1370	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑋) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
76		cncfmptid 24076	. . . . . . 7 ⊢ (((𝑘[,](𝑘 + 1)) ⊆ ℝ ∧ ℝ ⊆ ℂ) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
77	70, 71, 76	sylancl 586	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝑦) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
78		remulcl 10956	. . . . . 6 ⊢ ((𝑋 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑋 · 𝑦) ∈ ℝ)
79	54, 55, 75, 77, 71, 78	cncfmpt2ss 24079	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ (𝑋 · 𝑦)) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
80		reelprrecn 10963	. . . . . . . 8 ⊢ ℝ ∈ {ℝ, ℂ}
81	80	a1i 11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ℝ ∈ {ℝ, ℂ})
82	57	rexrd 11025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑀 ∈ ℝ^*)
83		iooss1 13114	. . . . . . . . . . 11 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑀 ≤ 𝑘) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
84	82, 61, 83	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)(𝑘 + 1)))
85	59	rexrd 11025	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑁 ∈ ℝ^*)
86		iooss2 13115	. . . . . . . . . . 11 ⊢ ((𝑁 ∈ ℝ^* ∧ (𝑘 + 1) ≤ 𝑁) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
87	85, 64, 86	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
88	84, 87	sstrd 3931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ (𝑀(,)𝑁))
89		ioossicc 13165	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
90	69, 71	sstrdi 3933	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀[,]𝑁) ⊆ ℂ)
91	89, 90	sstrid 3932	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑀(,)𝑁) ⊆ ℂ)
92	88, 91	sstrd 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℂ)
93	92	sselda 3921	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑦 ∈ ℂ)
94		1cnd 10970	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 1 ∈ ℂ)
95	73	sselda 3921	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
96		1cnd 10970	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
97	81	dvmptid 25121	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
98		ioossre 13140	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ⊆ ℝ
99	98	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ⊆ ℝ)
100	54	tgioo2 23966	. . . . . . . 8 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
101		iooretop 23929	. . . . . . . . 9 ⊢ (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,))
102	101	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘(,)(𝑘 + 1)) ∈ (topGen‘ran (,)))
103	81, 95, 96, 97, 99, 100, 54, 102	dvmptres 25127	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑦)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 1))
104	81, 93, 94, 103, 47	dvmptcmul 25128	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)))
105	52	mpteq2dv 5176	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 1)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
106	104, 105	eqtrd 2778	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ (𝑋 · 𝑦))) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ 𝑋))
107		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑦𝐴
108	107, 20, 22	cbvmpt 5185	. . . . . 6 ⊢ (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) = (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)
109	66	resmptd 5948	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) = (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴))
110	14	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ))
111		rescncf 24060	. . . . . . . 8 ⊢ ((𝑘[,](𝑘 + 1)) ⊆ (𝑀[,]𝑁) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ∈ ((𝑀[,]𝑁)–cn→ℝ) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ)))
112	66, 110, 111	sylc 65	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴) ↾ (𝑘[,](𝑘 + 1))) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
113	109, 112	eqeltrrd 2840	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑘[,](𝑘 + 1)) ↦ 𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
114	108, 113	eqeltrrid 2844	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑦 ∈ (𝑘[,](𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴) ∈ ((𝑘[,](𝑘 + 1))–cn→ℝ))
115	16	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀[,]𝑁) ↦ 𝐴):(𝑀[,]𝑁)⟶ℝ)
116	115, 18	sylibr 233	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ)
117	89	sseli 3917	. . . . . . . 8 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → 𝑦 ∈ (𝑀[,]𝑁))
118	24	impcom 408	. . . . . . . 8 ⊢ ((∀𝑥 ∈ (𝑀[,]𝑁)𝐴 ∈ ℝ ∧ 𝑦 ∈ (𝑀[,]𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
119	116, 117, 118	syl2an 596	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℝ)
120	119	recnd 11003	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
121	89	sseli 3917	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑀(,)𝑁) → 𝑥 ∈ (𝑀[,]𝑁))
122	16	fvmptelrn 6987	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
123	122	adantlr 712	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝐴 ∈ ℝ)
124	121, 123	sylan2 593	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐴 ∈ ℝ)
125	124	fmpttd 6989	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ)
126		ioossre 13140	. . . . . . . . . 10 ⊢ (𝑀(,)𝑁) ⊆ ℝ
127		dvfre 25115	. . . . . . . . . 10 ⊢ (((𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴):(𝑀(,)𝑁)⟶ℝ ∧ (𝑀(,)𝑁) ⊆ ℝ) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
128	125, 126, 127	sylancl 586	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ)
129		dvfsumle.b	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
130	129	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
131	130	dmeqd 5814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵))
132		dvfsumle.v	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
133	132	adantlr 712	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑀(,)𝑁)) → 𝐵 ∈ 𝑉)
134	133	ralrimiva 3103	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉)
135		dmmptg 6145	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ 𝑉 → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
136	134, 135	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑀(,)𝑁))
137	131, 136	eqtrd 2778	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (𝑀(,)𝑁))
138	130, 137	feq12d 6588	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)):dom (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴))⟶ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ))
139	128, 138	mpbid 231	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
140		eqid 2738	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵)
141	140	fmpt 6984	. . . . . . . 8 ⊢ (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ ↔ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵):(𝑀(,)𝑁)⟶ℝ)
142	139, 141	sylibr 233	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ)
143		nfcsb1v 3857	. . . . . . . . 9 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
144	143	nfel1 2923	. . . . . . . 8 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ
145		csbeq1a 3846	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
146	145	eleq1d 2823	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐵 ∈ ℝ ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
147	144, 146	rspc 3549	. . . . . . 7 ⊢ (𝑦 ∈ (𝑀(,)𝑁) → (∀𝑥 ∈ (𝑀(,)𝑁)𝐵 ∈ ℝ → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ))
148	142, 147	mpan9 507	. . . . . 6 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑀(,)𝑁)) → ⦋𝑦 / 𝑥⦌𝐵 ∈ ℝ)
149	107, 20, 22	cbvmpt 5185	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)
150	149	oveq2i 7286	. . . . . . 7 ⊢ (ℝ D (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐴)) = (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴))
151		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑦𝐵
152	151, 143, 145	cbvmpt 5185	. . . . . . 7 ⊢ (𝑥 ∈ (𝑀(,)𝑁) ↦ 𝐵) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵)
153	130, 150, 152	3eqtr3g 2801	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑀(,)𝑁) ↦ ⦋𝑦 / 𝑥⦌𝐵))
154	81, 120, 148, 153, 88, 100, 54, 102	dvmptres 25127	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (ℝ D (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐴)) = (𝑦 ∈ (𝑘(,)(𝑘 + 1)) ↦ ⦋𝑦 / 𝑥⦌𝐵))
155		dvfsumle.l	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑘 ∈ (𝑀..^𝑁) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1)))) → 𝑋 ≤ 𝐵)
156	155	anassrs 468	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑥 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ 𝐵)
157	156	ralrimiva 3103	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵)
158		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑋
159		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥 ≤
160	158, 159, 143	nfbr 5121	. . . . . . 7 ⊢ Ⅎ𝑥 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵
161	145	breq2d 5086	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑋 ≤ 𝐵 ↔ 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
162	160, 161	rspc 3549	. . . . . 6 ⊢ (𝑦 ∈ (𝑘(,)(𝑘 + 1)) → (∀𝑥 ∈ (𝑘(,)(𝑘 + 1))𝑋 ≤ 𝐵 → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵))
163	157, 162	mpan9 507	. . . . 5 ⊢ (((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) ∧ 𝑦 ∈ (𝑘(,)(𝑘 + 1))) → 𝑋 ≤ ⦋𝑦 / 𝑥⦌𝐵)
164	41	rexrd 11025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ ℝ^*)
165	49	rexrd 11025	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ ℝ^*)
166	41	lep1d 11906	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ≤ (𝑘 + 1))
167		lbicc2 13196	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
168	164, 165, 166, 167	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑘 ∈ (𝑘[,](𝑘 + 1)))
169		ubicc2 13197	. . . . . 6 ⊢ ((𝑘 ∈ ℝ^* ∧ (𝑘 + 1) ∈ ℝ^* ∧ 𝑘 ≤ (𝑘 + 1)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
170	164, 165, 166, 169	syl3anc 1370	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → (𝑘 + 1) ∈ (𝑘[,](𝑘 + 1)))
171		oveq2 7283	. . . . 5 ⊢ (𝑦 = 𝑘 → (𝑋 · 𝑦) = (𝑋 · 𝑘))
172		oveq2 7283	. . . . 5 ⊢ (𝑦 = (𝑘 + 1) → (𝑋 · 𝑦) = (𝑋 · (𝑘 + 1)))
173	41, 49, 79, 106, 114, 154, 163, 168, 170, 166, 171, 34, 172, 29	dvle 25171	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → ((𝑋 · (𝑘 + 1)) − (𝑋 · 𝑘)) ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
174	53, 173	eqbrtrrd 5098	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀..^𝑁)) → 𝑋 ≤ (⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
175	2, 3, 38, 174	fsumle 15511	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴))
176		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
177	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑀 → 𝑦 ∈ V)
178		eqeq2 2750	. . . . . 6 ⊢ (𝑦 = 𝑀 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑀))
179	178	biimpa 477	. . . . 5 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑀)
180		dvfsumle.c	. . . . 5 ⊢ (𝑥 = 𝑀 → 𝐴 = 𝐶)
181	179, 180	syl 17	. . . 4 ⊢ ((𝑦 = 𝑀 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐶)
182	177, 181	csbied 3870	. . 3 ⊢ (𝑦 = 𝑀 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐶)
183	176	a1i 11	. . . 4 ⊢ (𝑦 = 𝑁 → 𝑦 ∈ V)
184		eqeq2 2750	. . . . . 6 ⊢ (𝑦 = 𝑁 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑁))
185	184	biimpa 477	. . . . 5 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑁)
186		dvfsumle.d	. . . . 5 ⊢ (𝑥 = 𝑁 → 𝐴 = 𝐷)
187	185, 186	syl 17	. . . 4 ⊢ ((𝑦 = 𝑁 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐷)
188	183, 187	csbied 3870	. . 3 ⊢ (𝑦 = 𝑁 → ⦋𝑦 / 𝑥⦌𝐴 = 𝐷)
189	26	recnd 11003	. . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝑀...𝑁)) → ⦋𝑦 / 𝑥⦌𝐴 ∈ ℂ)
190	34, 29, 182, 188, 4, 189	telfsumo2 15515	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)(⦋(𝑘 + 1) / 𝑥⦌𝐴 − ⦋𝑘 / 𝑥⦌𝐴) = (𝐷 − 𝐶))
191	175, 190	breqtrd 5100	1 ⊢ (𝜑 → Σ𝑘 ∈ (𝑀..^𝑁)𝑋 ≤ (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 ⦋csb 3832 ∩ cin 3886 ⊆ wss 3887 {cpr 4563 class class class wbr 5074 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 ↾ cres 5591 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 ℝcr 10870 1c1 10872 + caddc 10874 · cmul 10876 ℝ^*cxr 11008 ≤ cle 11010 − cmin 11205 ℤcz 12319 ℤ_≥cuz 12582 (,)cioo 13079 [,]cicc 13082 ...cfz 13239 ..^cfzo 13382 Σcsu 15397 TopOpenctopn 17132 topGenctg 17148 ℂ_fldccnfld 20597 –cn→ccncf 24039 D cdv 25027

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949 ax-addf 10950 ax-mulf 10951

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-iin 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-of 7533 df-om 7713 df-1st 7831 df-2nd 7832 df-supp 7978 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-2o 8298 df-er 8498 df-map 8617 df-pm 8618 df-ixp 8686 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-fsupp 9129 df-fi 9170 df-sup 9201 df-inf 9202 df-oi 9269 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-4 12038 df-5 12039 df-6 12040 df-7 12041 df-8 12042 df-9 12043 df-n0 12234 df-z 12320 df-dec 12438 df-uz 12583 df-q 12689 df-rp 12731 df-xneg 12848 df-xadd 12849 df-xmul 12850 df-ioo 13083 df-ico 13085 df-icc 13086 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-mulr 16976 df-starv 16977 df-sca 16978 df-vsca 16979 df-ip 16980 df-tset 16981 df-ple 16982 df-ds 16984 df-unif 16985 df-hom 16986 df-cco 16987 df-rest 17133 df-topn 17134 df-0g 17152 df-gsum 17153 df-topgen 17154 df-pt 17155 df-prds 17158 df-xrs 17213 df-qtop 17218 df-imas 17219 df-xps 17221 df-mre 17295 df-mrc 17296 df-acs 17298 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-mulg 18701 df-cntz 18923 df-cmn 19388 df-psmet 20589 df-xmet 20590 df-met 20591 df-bl 20592 df-mopn 20593 df-fbas 20594 df-fg 20595 df-cnfld 20598 df-top 22043 df-topon 22060 df-topsp 22082 df-bases 22096 df-cld 22170 df-ntr 22171 df-cls 22172 df-nei 22249 df-lp 22287 df-perf 22288 df-cn 22378 df-cnp 22379 df-haus 22466 df-cmp 22538 df-tx 22713 df-hmeo 22906 df-fil 22997 df-fm 23089 df-flim 23090 df-flf 23091 df-xms 23473 df-ms 23474 df-tms 23475 df-cncf 24041 df-limc 25030 df-dv 25031

This theorem is referenced by: dvfsumge 25186

W3C validator