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Theorem dvivthlem1 26236

Description: Lemma for dvivth 26238. (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 13461	. . . . 5 ⊢ (𝐴(,)𝐵) ⊆ ℝ
2		dvivth.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
3	1, 2	sselid 3929	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℝ)
4		dvivth.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
5	1, 4	sselid 3929	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℝ)
6		dvivth.5	. . . . 5 ⊢ (𝜑 → 𝑀 < 𝑁)
7	3, 5, 6	ltled 11383	. . . 4 ⊢ (𝜑 → 𝑀 ≤ 𝑁)
8		dvivth.3	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
9		cncff 25122	. . . . . . . . . 10 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
10	8, 9	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
11	10	ffvelcdmda 7078	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℝ)
12		dvfre 26179	. . . . . . . . . . . . . 14 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
13	10, 1, 12	sylancl 598	. . . . . . . . . . . . 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 2863	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
16	13, 15	ffvelcdmd 7079	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
17	2, 14	eleqtrrd 2863	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑀 ∈ dom (ℝ D 𝐹))
18	13, 17	ffvelcdmd 7079	. . . . . . . . . . . 12 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
19		iccssre 13483	. . . . . . . . . . . 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 596	. . . . . . . . . . 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 3932	. . . . . . . . . 10 ⊢ (𝜑 → 𝐶 ∈ ℝ)
23	22	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝐶 ∈ ℝ)
24	1	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴(,)𝐵) ⊆ ℝ)
25	24	sselda 3931	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → 𝑦 ∈ ℝ)
26	23, 25	remulcld 11264	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℝ)
27	11, 26	resubcld 11667	. . . . . . 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 7108	. . . . . 6 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℝ)
30		iccssioo2 13473	. . . . . . 7 ⊢ ((𝑀 ∈ (𝐴(,)𝐵) ∧ 𝑁 ∈ (𝐴(,)𝐵)) → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
31	2, 4, 30	syl2anc 596	. . . . . 6 ⊢ (𝜑 → (𝑀[,]𝑁) ⊆ (𝐴(,)𝐵))
32	29, 31	fssresd 6743	. . . . 5 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ)
33		ax-resscn 11182	. . . . . 6 ⊢ ℝ ⊆ ℂ
34	33	a1i 11	. . . . . . . 8 ⊢ (𝜑 → ℝ ⊆ ℂ)
35		fss 6720	. . . . . . . . 9 ⊢ ((𝐺:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴(,)𝐵)⟶ℂ)
36	29, 33, 35	sylancl 598	. . . . . . . 8 ⊢ (𝜑 → 𝐺:(𝐴(,)𝐵)⟶ℂ)
37	28	oveq2i 7425	. . . . . . . . . . 11 ⊢ (ℝ D 𝐺) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦))))
38		reelprrecn 11217	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
39	38	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
40	11	recnd 11262	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑦) ∈ ℂ)
41	14	feq2d 6687	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
42	13, 41	mpbid 235	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
43	42	ffvelcdmda 7078	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑦) ∈ ℝ)
44	10	feqmptd 6947	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 = (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦)))
45	44	oveq2d 7430	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))))
46	42	feqmptd 6947	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D 𝐹) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
47	45, 46	eqtr3d 2797	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑦)))
48	26	recnd 11262	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴(,)𝐵)) → (𝐶 · 𝑦) ∈ ℂ)
49		remulcl 11210	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
50	22, 49	sylan 592	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℝ)
51	50	recnd 11262	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝐶 · 𝑦) ∈ ℂ)
52	22	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝐶 ∈ ℝ)
53	34	sselda 3931	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℂ)
54		1cnd 11227	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → 1 ∈ ℂ)
55	39	dvmptid 26185	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ 𝑦)) = (𝑦 ∈ ℝ ↦ 1))
56	22	recnd 11262	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐶 ∈ ℂ)
57	39, 53, 54, 55, 56	dvmptcmul 26192	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ (𝐶 · 1)))
58	56	mulridd 11251	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝐶 · 1) = 𝐶)
59	58	mpteq2dv 5199	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑦 ∈ ℝ ↦ (𝐶 · 1)) = (𝑦 ∈ ℝ ↦ 𝐶))
60	57, 59	eqtrd 2795	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (𝐶 · 𝑦))) = (𝑦 ∈ ℝ ↦ 𝐶))
61		tgioo4 25032	. . . . . . . . . . . . 13 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
62		eqid 2760	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
63		iooretop 24992	. . . . . . . . . . . . . 14 ⊢ (𝐴(,)𝐵) ∈ (topGen‘ran (,))
64	63	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝐵) ∈ (topGen‘ran (,)))
65	39, 51, 52, 60, 24, 61, 62, 64	dvmptres 26191	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ (𝐶 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ 𝐶))
66	39, 40, 43, 47, 48, 23, 65	dvmptsub 26195	. . . . . . . . . . 11 ⊢ (𝜑 → (ℝ D (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝐶 · 𝑦)))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
67	37, 66	eqtrid 2807	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
68	67	dmeqd 5889	. . . . . . . . 9 ⊢ (𝜑 → dom (ℝ D 𝐺) = dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)))
69		dmmptg 6238	. . . . . . . . . 10 ⊢ (∀𝑦 ∈ (𝐴(,)𝐵)(((ℝ D 𝐹)‘𝑦) − 𝐶) ∈ V → dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵))
70		ovex 7447	. . . . . . . . . . 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 3082	. . . . . . . . 9 ⊢ dom (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝐴(,)𝐵)
73	68, 72	eqtrdi 2811	. . . . . . . 8 ⊢ (𝜑 → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
74		dvcn 26149	. . . . . . . 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 1400	. . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ))
76		rescncf 25126	. . . . . . 7 ⊢ ((𝑀[,]𝑁) ⊆ (𝐴(,)𝐵) → (𝐺 ∈ ((𝐴(,)𝐵)–cn→ℂ) → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℂ)))
77	31, 75, 76	sylc 66	. . . . . 6 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℂ))
78		cncfcdm 25127	. . . . . 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 599	. . . . 5 ⊢ (𝜑 → ((𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ) ↔ (𝐺 ↾ (𝑀[,]𝑁)):(𝑀[,]𝑁)⟶ℝ))
80	32, 79	mpbird 260	. . . 4 ⊢ (𝜑 → (𝐺 ↾ (𝑀[,]𝑁)) ∈ ((𝑀[,]𝑁)–cn→ℝ))
81	3, 5, 7, 80	evthicc 25688	. . 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 500	. 2 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥))
83		fvres 6898	. . . . . . . 8 ⊢ (𝑧 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) = (𝐺‘𝑧))
84		fvres 6898	. . . . . . . 8 ⊢ (𝑥 ∈ (𝑀[,]𝑁) → ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) = (𝐺‘𝑥))
85	83, 84	breqan12rd 5120	. . . . . . 7 ⊢ ((𝑥 ∈ (𝑀[,]𝑁) ∧ 𝑧 ∈ (𝑀[,]𝑁)) → (((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) ↔ (𝐺‘𝑧) ≤ (𝐺‘𝑥)))
86	85	ralbidva 3183	. . . . . 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 487	. . . . 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 13487	. . . . . 6 ⊢ (𝑀(,)𝑁) ⊆ (𝑀[,]𝑁)
89		ssralv 4000	. . . . . 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 256	. . . 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 3931	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → 𝑥 ∈ (𝐴(,)𝐵))
93	42	ffvelcdmda 7078	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
94	92, 93	syldan 603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
95	94	recnd 11262	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
96	95	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℂ)
97	56	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℂ)
98	67	fveq1d 6881	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
99	98	adantr 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥))
100		fveq2 6879	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((ℝ D 𝐹)‘𝑦) = ((ℝ D 𝐹)‘𝑥))
101	100	oveq1d 7429	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℝ D 𝐹)‘𝑦) − 𝐶) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
102		eqid 2760	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶)) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))
103		ovex 7447	. . . . . . . . . . . 12 ⊢ (((ℝ D 𝐹)‘𝑥) − 𝐶) ∈ V
104	101, 102, 103	fvmpt 6987	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴(,)𝐵) → ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
105	92, 104	syl 18	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((𝑦 ∈ (𝐴(,)𝐵) ↦ (((ℝ D 𝐹)‘𝑦) − 𝐶))‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
106	99, 105	eqtrd 2795	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → ((ℝ D 𝐺)‘𝑥) = (((ℝ D 𝐹)‘𝑥) − 𝐶))
107	106	adantr 486	. . . . . . . 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 739	. . . . . . . . 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 783	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝑁))
111	88, 31	sstrid 3942	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
112	111	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑁) ⊆ (𝐴(,)𝐵))
113	92	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
114	73	ad2antrr 739	. . . . . . . . . 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 2863	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
116		simprr 785	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))
117		fveq2 6879	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
118	117	breq1d 5113	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ (𝐺‘𝑤) ≤ (𝐺‘𝑥)))
119	118	cbvralvw 3240	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) ↔ ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
120	116, 119	sylib 221	. . . . . . . . 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 26216	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) = 0)
122	107, 121	eqtr3d 2797	. . . . . . 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 11603	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 ∈ (𝑀(,)𝑁) ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
124	123	exp32 426	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
125		vex 3454	. . . . . . 7 ⊢ 𝑥 ∈ V
126	125	elpr 4609	. . . . . 6 ⊢ (𝑥 ∈ {𝑀, 𝑁} ↔ (𝑥 = 𝑀 ∨ 𝑥 = 𝑁))
127	106	adantr 486	. . . . . . . . . . 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 739	. . . . . . . . . . . 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 783	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑀)
131		eliooord 13459	. . . . . . . . . . . . . . . . 17 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
132	2, 131	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑀 ∧ 𝑀 < 𝐵))
133	132	simpld 500	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 < 𝑀)
134		ne0i 4287	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → (𝐴(,)𝐵) ≠ ∅)
135		ndmioo 13426	. . . . . . . . . . . . . . . . . . 19 ⊢ (¬ (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝐴(,)𝐵) = ∅)
136	135	necon1ai 2982	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐴(,)𝐵) ≠ ∅ → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
137	2, 134, 136	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
138	137	simpld 500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝐴 ∈ ℝ^*)
139	5	rexrd 11284	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑁 ∈ ℝ^*)
140		elioo2 13440	. . . . . . . . . . . . . . . 16 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^*) → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
141	138, 139, 140	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑀 ∈ (𝐴(,)𝑁) ↔ (𝑀 ∈ ℝ ∧ 𝐴 < 𝑀 ∧ 𝑀 < 𝑁)))
142	3, 133, 6, 141	mpbir3and 1361	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝑁))
143	142	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑀 ∈ (𝐴(,)𝑁))
144	130, 143	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝑁))
145	137	simprd 501	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐵 ∈ ℝ^*)
146		eliooord 13459	. . . . . . . . . . . . . . . . 17 ⊢ (𝑁 ∈ (𝐴(,)𝐵) → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
147	4, 146	syl 18	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (𝐴 < 𝑁 ∧ 𝑁 < 𝐵))
148	147	simprd 501	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 < 𝐵)
149	139, 145, 148	xrltled 13202	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ≤ 𝐵)
150		iooss2 13435	. . . . . . . . . . . . . 14 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝑁 ≤ 𝐵) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
151	145, 149, 150	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
152	151	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝐴(,)𝑁) ⊆ (𝐴(,)𝐵))
153	92	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
154	73	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
155	153, 154	eleqtrrd 2863	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
156		simprr 785	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
158	130	oveq1d 7429	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑥(,)𝑁) = (𝑀(,)𝑁))
159	157, 158	raleqtrrdv 3323	. . . . . . . . . . . 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 26213	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐺)‘𝑥) ≤ 0)
161	127, 160	eqbrtrrd 5129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (((ℝ D 𝐹)‘𝑥) − 𝐶) ≤ 0)
162	94	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
163	22	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
164	162, 163	suble0d 11830	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
166		elicc2 13465	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 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 235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐶 ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ≤ 𝐶 ∧ 𝐶 ≤ ((ℝ D 𝐹)‘𝑀)))
169	168	simp3d 1162	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
170	169	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑀))
171	130	fveq2d 6883	. . . . . . . . . 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 5133	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
173	162, 163	letri3d 11377	. . . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑀 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
175	174	exp32 426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑀 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
176		simprl 783	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 = 𝑁)
177	176	fveq2d 6883	. . . . . . . . . 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 1161	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
179	178	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑁) ≤ 𝐶)
180	177, 179	eqbrtrd 5127	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ≤ 𝐶)
181	29	ad2antrr 739	. . . . . . . . . . . 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 11284	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑀 ∈ ℝ^*)
184		elioo2 13440	. . . . . . . . . . . . . . . 16 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
185	183, 145, 184	syl2anc 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑁 ∈ (𝑀(,)𝐵) ↔ (𝑁 ∈ ℝ ∧ 𝑀 < 𝑁 ∧ 𝑁 < 𝐵)))
186	5, 6, 148, 185	mpbir3and 1361	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (𝑀(,)𝐵))
187	186	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑁 ∈ (𝑀(,)𝐵))
188	176, 187	eqeltrd 2860	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝑀(,)𝐵))
189	138, 183, 133	xrltled 13202	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ≤ 𝑀)
190		iooss1 13434	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ≤ 𝑀) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
191	138, 189, 190	syl2anc 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
192	191	ad2antrr 739	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝐵) ⊆ (𝐴(,)𝐵))
193	92	adantr 486	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ (𝐴(,)𝐵))
194	73	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → dom (ℝ D 𝐺) = (𝐴(,)𝐵))
195	193, 194	eleqtrrd 2863	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝑥 ∈ dom (ℝ D 𝐺))
196		simprr 785	. . . . . . . . . . . . . 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 221	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ∀𝑤 ∈ (𝑀(,)𝑁)(𝐺‘𝑤) ≤ (𝐺‘𝑥))
198	176	oveq2d 7430	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → (𝑀(,)𝑥) = (𝑀(,)𝑁))
199	197, 198	raleqtrrdv 3323	. . . . . . . . . . . 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 26215	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ ((ℝ D 𝐺)‘𝑥))
201	106	adantr 486	. . . . . . . . . . 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 5131	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 0 ≤ (((ℝ D 𝐹)‘𝑥) − 𝐶))
203	94	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) ∈ ℝ)
204	22	ad2antrr 739	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ∈ ℝ)
205	203, 204	subge0d 11829	. . . . . . . . . 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 235	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → 𝐶 ≤ ((ℝ D 𝐹)‘𝑥))
207	203, 204	letri3d 11377	. . . . . . . . 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 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) ∧ (𝑥 = 𝑁 ∧ ∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥))) → ((ℝ D 𝐹)‘𝑥) = 𝐶)
209	208	exp32 426	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 = 𝑁 → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
210	175, 209	jaod 873	. . . . . 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 245	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ {𝑀, 𝑁} → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶)))
212		elun 4100	. . . . . . 7 ⊢ (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
213		prunioo 13535	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℝ^* ∧ 𝑁 ∈ ℝ^* ∧ 𝑀 ≤ 𝑁) → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
214	183, 139, 7, 213	syl3anc 1398	. . . . . . . 8 ⊢ (𝜑 → ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) = (𝑀[,]𝑁))
215	214	eleq2d 2846	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ ((𝑀(,)𝑁) ∪ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
216	212, 215	bitr3id 288	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}) ↔ 𝑥 ∈ (𝑀[,]𝑁)))
217	216	biimpar 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (𝑥 ∈ (𝑀(,)𝑁) ∨ 𝑥 ∈ {𝑀, 𝑁}))
218	124, 211, 217	mpjaod 874	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀(,)𝑁)(𝐺‘𝑧) ≤ (𝐺‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
219	91, 218	syld 48	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑀[,]𝑁)) → (∀𝑧 ∈ (𝑀[,]𝑁)((𝐺 ↾ (𝑀[,]𝑁))‘𝑧) ≤ ((𝐺 ↾ (𝑀[,]𝑁))‘𝑥) → ((ℝ D 𝐹)‘𝑥) = 𝐶))
220	219	reximdva 3175	. 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 16	1 ⊢ (𝜑 → ∃𝑥 ∈ (𝑀[,]𝑁)((ℝ D 𝐹)‘𝑥) = 𝐶)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 ∀wral 3076 ∃wrex 3086 Vcvv 3450 ∪ cun 3897 ⊆ wss 3899 ∅c0 4279 {cpr 4586 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5655 ran crn 5656 ↾ cres 5657 ⟶wf 6529 ‘cfv 6533 (class class class)co 7414 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 · cmul 11130 ℝ^*cxr 11267 < clt 11268 ≤ cle 11269 − cmin 11466 (,)cioo 13399 [,]cicc 13402 TopOpenctopn 17507 topGenctg 17523 ℂ_fldccnfld 21586 –cn→ccncf 25105 D cdv 26091

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-rep 5232 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7737 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203 ax-addf 11204

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-se 5609 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6299 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6489 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-isom 6542 df-riota 7371 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8281 df-wrecs 8312 df-recs 8361 df-rdg 8400 df-1o 8456 df-2o 8457 df-er 8697 df-map 8829 df-pm 8830 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9333 df-fi 9382 df-sup 9413 df-inf 9414 df-oi 9483 df-card 9945 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13044 df-xneg 13164 df-xadd 13165 df-xmul 13166 df-ioo 13403 df-ico 13405 df-icc 13406 df-fz 13563 df-fzo 13711 df-seq 14067 df-exp 14127 df-hash 14396 df-cj 15187 df-re 15188 df-im 15189 df-sqrt 15323 df-abs 15324 df-struct 17240 df-sets 17257 df-slot 17275 df-ndx 17287 df-base 17303 df-ress 17324 df-plusg 17356 df-mulr 17357 df-starv 17358 df-sca 17359 df-vsca 17360 df-ip 17361 df-tset 17362 df-ple 17363 df-ds 17365 df-unif 17366 df-hom 17367 df-cco 17368 df-rest 17508 df-topn 17509 df-0g 17527 df-gsum 17528 df-topgen 17529 df-pt 17530 df-prds 17533 df-xrs 17589 df-qtop 17594 df-imas 17595 df-xps 17597 df-mre 17671 df-mrc 17672 df-acs 17674 df-mgm 18731 df-sgrp 18822 df-mnd 18838 df-submnd 18893 df-mulg 19192 df-cntz 19445 df-cmn 19910 df-psmet 21578 df-xmet 21579 df-met 21580 df-bl 21581 df-mopn 21582 df-fbas 21583 df-fg 21584 df-cnfld 21587 df-top 23120 df-topon 23137 df-topsp 23159 df-bases 23172 df-cld 23245 df-ntr 23246 df-cls 23247 df-nei 23324 df-lp 23362 df-perf 23363 df-cn 23453 df-cnp 23454 df-haus 23541 df-cmp 23613 df-tx 23789 df-hmeo 23982 df-fil 24073 df-fm 24165 df-flim 24166 df-flf 24167 df-xms 24547 df-ms 24548 df-tms 24549 df-cncf 25107 df-limc 26094 df-dv 26095

This theorem is used by: dvivthlem2 26237

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