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Theorem dvivthlem1 26067

Description: Lemma for dvivth 26069. (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 13468	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvivth.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 4006	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4		dvivth.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
5	1, 4	sselid 4006	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
6		dvivth.5	. . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁)
7	3, 5, 6	ltled 11438	. . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
8		dvivth.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
9		cncff 24938	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	8, 9	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	10	ffvelcdmda 7118	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℝ)
12		dvfre 26009	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13	10, 1, 12	sylancl 585	. . . . . . . . . . . . 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 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
16	13, 15	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
17	2, 14	eleqtrrd 2847	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ dom (ℝ D 𝐹))
18	13, 17	ffvelcdmd 7119	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
19		iccssre 13489	. . . . . . . . . . . 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 583	. . . . . . . . . . 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 4009	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
24	1	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
25	24	sselda 4008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
26	23, 25	remulcld 11320	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℝ)
27	11, 26	resubcld 11718	. . . . . . 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 7148	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
30		iccssioo2 13480	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐴(,)𝐵) ∧ 𝑁 ∈ (𝐴(,)𝐵)) → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
31	2, 4, 30	syl2anc 583	. . . . . 6 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
32	29, 31	fssresd 6788	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ)
33		ax-resscn 11241	. . . . . 6 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
35		fss 6763	. . . . . . . . 9 ⊢ ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
36	29, 33, 35	sylancl 585	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℂ)
37	28	oveq2i 7459	. . . . . . . . . . 11 ⊢ (ℝ D 𝐺) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦))))
38		reelprrecn 11276	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
40	11	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℂ)
41	14	feq2d 6733	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
42	13, 41	mpbid 232	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
43	42	ffvelcdmda 7118	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
44	10	feqmptd 6990	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)))
45	44	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))))
46	42	feqmptd 6990	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
47	45, 46	eqtr3d 2782	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
48	26	recnd 11318	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℂ)
49		remulcl 11269	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
50	22, 49	sylan 579	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
51	50	recnd 11318	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℂ)
52	22	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℝ)
53	34	sselda 4008	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
54		1cnd 11285	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
55	39	dvmptid 26015	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
56	22	recnd 11318	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℂ)
57	39, 53, 54, 55, 56	dvmptcmul 26022	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ (𝐶 · 1)))
58	56	mulridd 11307	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 1) = 𝐶)
59	58	mpteq2dv 5268	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐶 · 1)) = (𝑦 ∈ ℝ ↦ 𝐶))
60	57, 59	eqtrd 2780	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ 𝐶))
61		eqid 2740	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
62	61	tgioo2 24844	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
63		iooretop 24807	. . . . . . . . . . . . . 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 26021	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐶 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 𝐶))
66	39, 40, 43, 47, 48, 23, 65	dvmptsub 26025	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
67	37, 66	eqtrid 2792	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
68	67	dmeqd 5930	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐺) = dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
69		dmmptg 6273	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)(((ℝ D 𝐹)‘𝑦) − 𝐶) ∈ V → dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵))
70		ovex 7481	. . . . . . . . . . 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 3073	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵)
73	68, 72	eqtrdi 2796	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
74		dvcn 25977	. . . . . . . 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 1373	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
76		rescncf 24942	. . . . . . 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 24943	. . . . . 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 586	. . . . 5 ⊢ (𝜑 → ((𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ) ↔ (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ))
80	32, 79	mpbird 257	. . . 4 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
81	3, 5, 7, 80	evthicc 25513	. . 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 494	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥))
83		fvres 6939	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) = (𝐺‘𝑧))
84		fvres 6939	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) = (𝐺‘𝑥))
85	83, 84	breqan12rd 5183	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ∧ 𝑧 ∈ (𝑀[,]𝑁)) → (((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ↔ (𝐺‘𝑧) ≤ (𝐺‘𝑥)))
86	85	ralbidva 3182	. . . . . 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 481	. . . . 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 13493	. . . . . 6 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
89		ssralv 4077	. . . . . 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 253	. . . 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 4008	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵))
93	42	ffvelcdmda 7118	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
94	92, 93	syldan 590	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
95	94	recnd 11318	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
96	95	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
97	56	ad2antrr 725	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℂ)
98	67	fveq1d 6922	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
99	98	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
100		fveq2 6920	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
101	100	oveq1d 7463	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℝ D 𝐹)‘𝑦) − 𝐶) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
102		eqid 2740	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))
103		ovex 7481	. . . . . . . . . . . 12 ⊢ (((ℝ D 𝐹)‘𝑥) − 𝐶) ∈ V
104	101, 102, 103	fvmpt 7029	. . . . . . . . . . 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 2780	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
107	106	adantr 480	. . . . . . . 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 725	. . . . . . . . 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 770	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝑁))
111	88, 31	sstrid 4020	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
112	111	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
113	92	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
114	73	ad2antrr 725	. . . . . . . . . 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 2847	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
116		simprr 772	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
117		fveq2 6920	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
118	117	breq1d 5176	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ (𝐺‘𝑤) ≤ (𝐺‘𝑥)))
119	118	cbvralvw 3243	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
120	116, 119	sylib 218	. . . . . . . . 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 26046	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = 0)
122	107, 121	eqtr3d 2782	. . . . . . 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 11655	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
124	123	exp32 420	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
125		vex 3492	. . . . . . 7 ⊢ 𝑥 ∈ V
126	125	elpr 4672	. . . . . 6 ⊢ (𝑥 ∈ {𝑀, 𝑁} ↔ (𝑥 = 𝑀 ∨ 𝑥 = 𝑁))
127	106	adantr 480	. . . . . . . . . . 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 725	. . . . . . . . . . . 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 770	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑀)
131		eliooord 13466	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
132	2, 131	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
133	132	simpld 494	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < 𝑀)
134		ne0i 4364	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
135		ndmioo 13434	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
136	135	necon1ai 2974	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
137	2, 134, 136	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
138	137	simpld 494	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
139	5	rexrd 11340	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
140		elioo2 13448	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^*) → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
141	138, 139, 140	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
142	3, 133, 6, 141	mpbir3and 1342	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝑁))
143	142	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑀 ∈ (𝐴(,)𝑁))
144	130, 143	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝑁))
145	137	simprd 495	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
146		eliooord 13466	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
147	4, 146	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
148	147	simprd 495	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 < 𝐵)
149	139, 145, 148	xrltled 13212	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ≤ 𝐵)
150		iooss2 13443	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑁 ≤ 𝐵) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
151	145, 149, 150	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
152	151	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
153	92	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
154	73	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
155	153, 154	eleqtrrd 2847	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
156		simprr 772	. . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
158	130	oveq1d 7463	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑥(,)𝑁) = (𝑀(,)𝑁))
159	157, 158	raleqtrrdv 3338	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑥(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
160	128, 129, 144, 152, 155, 159	dvferm1 26043	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) ≤ 0)
161	127, 160	eqbrtrrd 5190	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) − 𝐶) ≤ 0)
162	94	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
163	22	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
164	162, 163	suble0d 11881	. . . . . . . . . 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))
165	161, 164	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
166		elicc2 13472	. . . . . . . . . . . . . 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))))
167	16, 18, 166	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐶 ∈ (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑀)) ↔ (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))))
168	21, 167	mpbid 232	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀)))
169	168	simp3d 1144	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
170	169	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
171	130	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D 𝐹)‘𝑀))
172	170, 171	breqtrrd 5194	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
173	162, 163	letri3d 11432	. . . . . . . . 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))))
174	165, 172, 173	mpbir2and 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
175	174	exp32 420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑀 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
176		simprl 770	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑁)
177	176	fveq2d 6924	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = ((ℝ D 𝐹)‘𝑁))
178	168	simp2d 1143	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
179	178	ad2antrr 725	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
180	177, 179	eqbrtrd 5188	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
181	29	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐺:(𝐴(,)𝐵)⟶ℝ)
182	1	a1i 11	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝐵) ⊆ ℝ)
183	3	rexrd 11340	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
184		elioo2 13448	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
185	183, 145, 184	syl2anc 583	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
186	5, 6, 148, 185	mpbir3and 1342	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (𝑀(,)𝐵))
187	186	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑁 ∈ (𝑀(,)𝐵))
188	176, 187	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝐵))
189	138, 183, 133	xrltled 13212	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑀)
190		iooss1 13442	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑀) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
191	138, 189, 190	syl2anc 583	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
192	191	ad2antrr 725	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
193	92	adantr 480	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
194	73	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
195	193, 194	eleqtrrd 2847	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
196		simprr 772	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
197	196, 119	sylib 218	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
198	176	oveq2d 7464	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑥) = (𝑀(,)𝑁))
199	197, 198	raleqtrrdv 3338	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑥)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
200	181, 182, 188, 192, 195, 199	dvferm2 26045	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ ((ℝ D 𝐺)‘𝑥))
201	106	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
202	200, 201	breqtrd 5192	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ (((ℝ D 𝐹)‘𝑥) − 𝐶))
203	94	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
204	22	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
205	203, 204	subge0d 11880	. . . . . . . . . 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)))
206	202, 205	mpbid 232	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
207	203, 204	letri3d 11432	. . . . . . . . 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))))
208	180, 206, 207	mpbir2and 712	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
209	208	exp32 420	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑁 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
210	175, 209	jaod 858	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((𝑥 = 𝑀 ∨ 𝑥 = 𝑁) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
211	126, 210	biimtrid 242	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ {𝑀, 𝑁} → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
212		elun 4176	. . . . . . 7 ⊢ (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
213		prunioo 13541	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
214	183, 139, 7, 213	syl3anc 1371	. . . . . . . 8 ⊢ (𝜑 → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
215	214	eleq2d 2830	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
216	212, 215	bitr3id 285	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
217	216	biimpar 477	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
218	124, 211, 217	mpjaod 859	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
219	91, 218	syld 47	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
220	219	reximdva 3174	. 2 ⊢ (𝜑 → (∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶))
221	82, 220	mpd 15	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 846 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 Vcvv 3488 ∪ cun 3974 ⊆ wss 3976 ∅c0 4352 {cpr 4650 class class class wbr 5166 ↦ cmpt 5249 dom cdm 5700 ran crn 5701 ↾ cres 5702 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 1c1 11185 · cmul 11189 ℝ^*cxr 11323 < clt 11324 ≤ cle 11325 − cmin 11520 (,)cioo 13407 [,]cicc 13410 TopOpenctopn 17481 topGenctg 17497 ℂ_fldccnfld 21387 –cn→ccncf 24921 D cdv 25918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-fbas 21384 df-fg 21385 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cld 23048 df-ntr 23049 df-cls 23050 df-nei 23127 df-lp 23165 df-perf 23166 df-cn 23256 df-cnp 23257 df-haus 23344 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-fil 23875 df-fm 23967 df-flim 23968 df-flf 23969 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-limc 25921 df-dv 25922

This theorem is referenced by: dvivthlem2 26068

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