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Theorem dvivthlem1 25985

Description: Lemma for dvivth 25987. (Contributed by Mario Carneiro, 24-Feb-2015.)

**Hypotheses**
Ref	Expression
dvivth.1	⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
dvivth.2	⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
dvivth.3	⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
dvivth.4	⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
dvivth.5	⊢ (𝜑 → 𝑀 < 𝑁)
dvivth.6	⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
dvivth.7	⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))

**Assertion**
Ref	Expression
dvivthlem1	⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)

Distinct variable groups: 𝑥,𝑦,𝐴 𝑥,𝐵,𝑦 𝑥,𝐹,𝑦 𝑥,𝐺 𝑥,𝑀,𝑦 𝑥,𝐶,𝑦 𝑥,𝑁,𝑦 𝜑,𝑥,𝑦 Allowed substitution hint: 𝐺(𝑦)

Proof of Theorem dvivthlem1 Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ioossre 13420	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvivth.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 3974	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4		dvivth.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
5	1, 4	sselid 3974	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
6		dvivth.5	. . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁)
7	3, 5, 6	ltled 11394	. . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
8		dvivth.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
9		cncff 24857	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	10	ffvelcdmda 7093	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℝ)
12		dvfre 25927	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13	10, 1, 12	sylancl 584	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
14		dvivth.4	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
15	4, 14	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
16	13, 15	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
17	2, 14	eleqtrrd 2828	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ dom (ℝ D 𝐹))
18	13, 17	ffvelcdmd 7094	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
19		iccssre 13441	. . . . . . . . . . . 12 ⊢ ((((ℝ D 𝐹)‘𝑁) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑀) ∈ ℝ) → (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)) ⊆ ℝ)
20	16, 18, 19	syl2anc 582	. . . . . . . . . . 11 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)) ⊆ ℝ)
21		dvivth.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)))
22	20, 21	sseldd 3977	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
23	22	adantr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
24	1	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
25	24	sselda 3976	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
26	23, 25	remulcld 11276	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℝ)
27	11, 26	resubcld 11674	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((𝐹‘𝑦) − (𝐶 · 𝑦)) ∈ ℝ)
28		dvivth.7	. . . . . . 7 ⊢ 𝐺 = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))
29	27, 28	fmptd 7123	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
30		iccssioo2 13432	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐴(,)𝐵) ∧ 𝑁 ∈ (𝐴(,)𝐵)) → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
31	2, 4, 30	syl2anc 582	. . . . . 6 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
32	29, 31	fssresd 6764	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ)
33		ax-resscn 11197	. . . . . 6 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
35		fss 6739	. . . . . . . . 9 ⊢ ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
36	29, 33, 35	sylancl 584	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℂ)
37	28	oveq2i 7430	. . . . . . . . . . 11 ⊢ (ℝ D 𝐺) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦))))
38		reelprrecn 11232	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
40	11	recnd 11274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℂ)
41	14	feq2d 6709	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
42	13, 41	mpbid 231	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
43	42	ffvelcdmda 7093	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
44	10	feqmptd 6966	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)))
45	44	oveq2d 7435	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))))
46	42	feqmptd 6966	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
47	45, 46	eqtr3d 2767	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
48	26	recnd 11274	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℂ)
49		remulcl 11225	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
50	22, 49	sylan 578	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
51	50	recnd 11274	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℂ)
52	22	adantr 479	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℝ)
53	34	sselda 3976	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
54		1cnd 11241	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
55	39	dvmptid 25933	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
56	22	recnd 11274	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℂ)
57	39, 53, 54, 55, 56	dvmptcmul 25940	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ (𝐶 · 1)))
58	56	mulridd 11263	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 1) = 𝐶)
59	58	mpteq2dv 5251	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐶 · 1)) = (𝑦 ∈ ℝ ↦ 𝐶))
60	57, 59	eqtrd 2765	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ 𝐶))
61		eqid 2725	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
62	61	tgioo2 24763	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
63		iooretop 24726	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
64	63	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
65	39, 51, 52, 60, 24, 62, 61, 64	dvmptres 25939	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐶 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 𝐶))
66	39, 40, 43, 47, 48, 23, 65	dvmptsub 25943	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
67	37, 66	eqtrid 2777	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
68	67	dmeqd 5908	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐺) = dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
69		dmmptg 6248	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)(((ℝ D 𝐹)‘𝑦) − 𝐶) ∈ V → dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵))
70		ovex 7452	. . . . . . . . . . 11 ⊢ (((ℝ D 𝐹)‘𝑦) − 𝐶) ∈ V
71	70	a1i 11	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴(,)𝐵) → (((ℝ D 𝐹)‘𝑦) − 𝐶) ∈ V)
72	69, 71	mprg 3056	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵)
73	68, 72	eqtrdi 2781	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
74		dvcn 25895	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ 𝐺:(𝐴(,)𝐵)⟶ℂ ∧ (𝐴(,)𝐵) ⊆ ℝ) ∧ dom (ℝ D 𝐺) = (𝐴(,)𝐵)) → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
75	34, 36, 24, 73, 74	syl31anc 1370	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
76		rescncf 24861	. . . . . . 7 ⊢ ((𝑀[,]𝑁) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ) → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℂ)))
77	31, 75, 76	sylc 65	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℂ))
78		cncfcdm 24862	. . . . . 6 ⊢ ((ℝ ⊆ ℂ ∧ (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℂ)) → ((𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ) ↔ (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ))
79	33, 77, 78	sylancr 585	. . . . 5 ⊢ (𝜑 → ((𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ) ↔ (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ))
80	32, 79	mpbird 256	. . . 4 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
81	3, 5, 7, 80	evthicc 25432	. . 3 ⊢ (𝜑 → (∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ∧ ∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑧)))
82	81	simpld 493	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥))
83		fvres 6915	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) = (𝐺‘𝑧))
84		fvres 6915	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) = (𝐺‘𝑥))
85	83, 84	breqan12rd 5166	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ∧ 𝑧 ∈ (𝑀[,]𝑁)) → (((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ↔ (𝐺‘𝑧) ≤ (𝐺‘𝑥)))
86	85	ralbidva 3165	. . . . . 6 ⊢ (𝑥 ∈ (𝑀[,]𝑁) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ↔ ∀𝑧 ∈ (𝑀[,]𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥)))
87	86	adantl 480	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ↔ ∀𝑧 ∈ (𝑀[,]𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥)))
88		ioossicc 13445	. . . . . 6 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
89		ssralv 4045	. . . . . 6 ⊢ ((𝑀(,)𝑁) ⊆ (𝑀[,]𝑁) → (∀𝑧 ∈ (𝑀[,]𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥)))
90	88, 89	ax-mp 5	. . . . 5 ⊢ (∀𝑧 ∈ (𝑀[,]𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
91	87, 90	biimtrdi 252	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥)))
92	31	sselda 3976	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵))
93	42	ffvelcdmda 7093	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
94	92, 93	syldan 589	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
95	94	recnd 11274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
96	95	adantr 479	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
97	56	ad2antrr 724	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℂ)
98	67	fveq1d 6898	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
99	98	adantr 479	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
100		fveq2 6896	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
101	100	oveq1d 7434	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℝ D 𝐹)‘𝑦) − 𝐶) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
102		eqid 2725	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))
103		ovex 7452	. . . . . . . . . . . 12 ⊢ (((ℝ D 𝐹)‘𝑥) − 𝐶) ∈ V
104	101, 102, 103	fvmpt 7004	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
105	92, 104	syl 17	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
106	99, 105	eqtrd 2765	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
107	106	adantr 479	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
108	29	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
109	1	a1i 11	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝐵) ⊆ ℝ)
110		simprl 769	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝑁))
111	88, 31	sstrid 3988	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
112	111	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
113	92	adantr 479	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
114	73	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
115	113, 114	eleqtrrd 2828	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
116		simprr 771	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
117		fveq2 6896	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
118	117	breq1d 5159	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ (𝐺‘𝑤) ≤ (𝐺‘𝑥)))
119	118	cbvralvw 3224	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
120	116, 119	sylib 217	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
121	108, 109, 110, 112, 115, 120	dvferm 25964	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = 0)
122	107, 121	eqtr3d 2767	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) − 𝐶) = 0)
123	96, 97, 122	subeq0d 11611	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
124	123	exp32 419	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
125		vex 3465	. . . . . . 7 ⊢ 𝑥 ∈ V
126	125	elpr 4654	. . . . . 6 ⊢ (𝑥 ∈ {𝑀, 𝑁} ↔ (𝑥 = 𝑀 ∨ 𝑥 = 𝑁))
127	106	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
128	29	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
129	1	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝐵) ⊆ ℝ)
130		simprl 769	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑀)
131		eliooord 13418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
132	2, 131	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
133	132	simpld 493	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < 𝑀)
134		ne0i 4334	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
135		ndmioo 13386	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
136	135	necon1ai 2957	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
137	2, 134, 136	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
138	137	simpld 493	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
139	5	rexrd 11296	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
140		elioo2 13400	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^*) → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
141	138, 139, 140	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
142	3, 133, 6, 141	mpbir3and 1339	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝑁))
143	142	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑀 ∈ (𝐴(,)𝑁))
144	130, 143	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝑁))
145	137	simprd 494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
146		eliooord 13418	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
147	4, 146	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
148	147	simprd 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 < 𝐵)
149	139, 145, 148	xrltled 13164	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ≤ 𝐵)
150		iooss2 13395	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑁 ≤ 𝐵) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
151	145, 149, 150	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
152	151	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
153	92	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
154	73	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
155	153, 154	eleqtrrd 2828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
156		simprr 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
157	156, 119	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
158	130	oveq1d 7434	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑥(,)𝑁) = (𝑀(,)𝑁))
159	158	raleqdv 3314	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (∀𝑤 ∈ (𝑥(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥) ↔ ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥)))
160	157, 159	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑥(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
161	128, 129, 144, 152, 155, 160	dvferm1 25961	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) ≤ 0)
162	127, 161	eqbrtrrd 5173	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) − 𝐶) ≤ 0)
163	94	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
164	22	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
165	163, 164	suble0d 11837	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((((ℝ D 𝐹)‘𝑥) − 𝐶) ≤ 0 ↔ ((ℝ D 𝐹)‘𝑥) ≤ 𝐶))
166	162, 165	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
167		elicc2 13424	. . . . . . . . . . . . . 14 ⊢ ((((ℝ D 𝐹)‘𝑁) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑀) ∈ ℝ) → (𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)) ↔ (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))))
168	16, 18, 167	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)) ↔ (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))))
169	21, 168	mpbid 231	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀)))
170	169	simp3d 1141	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
171	170	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
172	130	fveq2d 6900	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D 𝐹)‘𝑀))
173	171, 172	breqtrrd 5177	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
174	163, 164	letri3d 11388	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) = 𝐶 ↔ (((ℝ D 𝐹)‘𝑥) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))))
175	166, 173, 174	mpbir2and 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
176	175	exp32 419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑀 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
177		simprl 769	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑁)
178	177	fveq2d 6900	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D 𝐹)‘𝑁))
179	169	simp2d 1140	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
180	179	ad2antrr 724	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
181	178, 180	eqbrtrd 5171	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
182	29	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
183	1	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝐵) ⊆ ℝ)
184	3	rexrd 11296	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
185		elioo2 13400	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
186	184, 145, 185	syl2anc 582	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
187	5, 6, 148, 186	mpbir3and 1339	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (𝑀(,)𝐵))
188	187	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑁 ∈ (𝑀(,)𝐵))
189	177, 188	eqeltrd 2825	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝐵))
190	138, 184, 133	xrltled 13164	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑀)
191		iooss1 13394	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑀) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
192	138, 190, 191	syl2anc 582	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
193	192	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
194	92	adantr 479	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
195	73	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
196	194, 195	eleqtrrd 2828	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
197		simprr 771	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
198	197, 119	sylib 217	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
199	177	oveq2d 7435	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑥) = (𝑀(,)𝑁))
200	199	raleqdv 3314	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (∀𝑤 ∈ (𝑀(,)𝑥)(𝐺‘𝑤) ≤ (𝐺‘𝑥) ↔ ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥)))
201	198, 200	mpbird 256	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑥)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
202	182, 183, 189, 193, 196, 201	dvferm2 25963	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ ((ℝ D 𝐺)‘𝑥))
203	106	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
204	202, 203	breqtrd 5175	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ (((ℝ D 𝐹)‘𝑥) − 𝐶))
205	94	adantr 479	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
206	22	ad2antrr 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
207	205, 206	subge0d 11836	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (0 ≤ (((ℝ D 𝐹)‘𝑥) − 𝐶) ↔ 𝐶 ≤ ((ℝ D 𝐹)‘𝑥)))
208	204, 207	mpbid 231	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
209	205, 206	letri3d 11388	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) = 𝐶 ↔ (((ℝ D 𝐹)‘𝑥) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))))
210	181, 208, 209	mpbir2and 711	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
211	210	exp32 419	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑁 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
212	176, 211	jaod 857	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((𝑥 = 𝑀 ∨ 𝑥 = 𝑁) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
213	126, 212	biimtrid 241	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ {𝑀, 𝑁} → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
214		elun 4145	. . . . . . 7 ⊢ (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
215		prunioo 13493	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
216	184, 139, 7, 215	syl3anc 1368	. . . . . . . 8 ⊢ (𝜑 → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
217	216	eleq2d 2811	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
218	214, 217	bitr3id 284	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
219	218	biimpar 476	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
220	124, 213, 219	mpjaod 858	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
221	91, 220	syld 47	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
222	221	reximdva 3157	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶))
223	82, 222	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ wo 845 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∃wrex 3059 Vcvv 3461 ∪ cun 3942 ⊆ wss 3944 ∅c0 4322 {cpr 4632 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5678 ran crn 5679 ↾ cres 5680 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 ℝcr 11139 0cc0 11140 1c1 11141 · cmul 11145 ℝ^*cxr 11279 < clt 11280 ≤ cle 11281 − cmin 11476 (,)cioo 13359 [,]cicc 13362 TopOpenctopn 17406 topGenctg 17422 ℂ_fldccnfld 21296 –cn→ccncf 24840 D cdv 25836

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-iin 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-isom 6558 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-of 7685 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-2o 8488 df-er 8725 df-map 8847 df-pm 8848 df-ixp 8917 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-fsupp 9388 df-fi 9436 df-sup 9467 df-inf 9468 df-oi 9535 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-q 12966 df-rp 13010 df-xneg 13127 df-xadd 13128 df-xmul 13129 df-ioo 13363 df-ico 13365 df-icc 13366 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-hash 14326 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-hom 17260 df-cco 17261 df-rest 17407 df-topn 17408 df-0g 17426 df-gsum 17427 df-topgen 17428 df-pt 17429 df-prds 17432 df-xrs 17487 df-qtop 17492 df-imas 17493 df-xps 17495 df-mre 17569 df-mrc 17570 df-acs 17572 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-mulg 19032 df-cntz 19280 df-cmn 19749 df-psmet 21288 df-xmet 21289 df-met 21290 df-bl 21291 df-mopn 21292 df-fbas 21293 df-fg 21294 df-cnfld 21297 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22893 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cn 23175 df-cnp 23176 df-haus 23263 df-cmp 23335 df-tx 23510 df-hmeo 23703 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24270 df-ms 24271 df-tms 24272 df-cncf 24842 df-limc 25839 df-dv 25840

This theorem is referenced by: dvivthlem2 25986

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