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Theorem dvivth 24607

Description: Darboux' theorem, or the intermediate value theorem for derivatives. A differentiable function's derivative satisfies the intermediate value property, even though it may not be continuous (so that ivthicc 24059 does not directly apply). (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))

**Assertion**
Ref	Expression
dvivth	⊢ (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))

Proof of Theorem dvivth Dummy variables 𝑥 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dvivth.1	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ∈ (𝐴(,)𝐵))
2	1	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 ∈ (𝐴(,)𝐵))
3		dvivth.2	. . . . . . . . . 10 ⊢ (𝜑 → 𝑁 ∈ (𝐴(,)𝐵))
4	3	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 ∈ (𝐴(,)𝐵))
5		dvivth.3	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
6		cncff 23501	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
8	7	ffvelrnda 6851	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
9	8	renegcld 11067	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → -(𝐹‘𝑤) ∈ ℝ)
10	9	fmpttd 6879	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ)
11		ax-resscn 10594	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
12		ssid 3989	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
13		cncfss 23507	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
14	11, 12, 13	mp2an 690	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ)
15	14, 5	sseldi 3965	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
16		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))
17	16	negfcncf 23527	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ) → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
18	15, 17	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℂ))
19		cncffvrn 23506	. . . . . . . . . . . 12 ⊢ ((ℝ ⊆ ℂ ∧ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℂ)) → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ))
20	11, 18, 19	sylancr 589	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ))
21	10, 20	mpbird 259	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
22	21	adantr 483	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
23		reelprrecn 10629	. . . . . . . . . . . . 13 ⊢ ℝ ∈ {ℝ, ℂ}
24	23	a1i 11	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ℝ ∈ {ℝ, ℂ})
25	7	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
26	25	ffvelrnda 6851	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
27	26	recnd 10669	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℂ)
28		fvexd 6685	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑤) ∈ V)
29	25	feqmptd 6733	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤)))
30	29	oveq2d 7172	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) = (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤))))
31		ioossre 12799	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴(,)𝐵) ⊆ ℝ
32		dvfre 24548	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	7, 31, 32	sylancl 588	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34		dvivth.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
35	34	feq2d 6500	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
36	33, 35	mpbid 234	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
37	36	adantr 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
38	37	feqmptd 6733	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑤)))
39	30, 38	eqtr3d 2858	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ ((ℝ D 𝐹)‘𝑤)))
40	24, 27, 28, 39	dvmptneg 24563	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
41	40	dmeqd 5774	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → dom (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))) = dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
42		dmmptg 6096	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑤) ∈ V → dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵))
43		negex 10884	. . . . . . . . . . . 12 ⊢ -((ℝ D 𝐹)‘𝑤) ∈ V
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑤) ∈ V)
45	42, 44	mprg 3152	. . . . . . . . . 10 ⊢ dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵)
46	41, 45	syl6eq 2872	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → dom (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))) = (𝐴(,)𝐵))
47		simprl 769	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 < 𝑁)
48		simprr 771	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))
49	36, 1	ffvelrnd 6852	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
50	49	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
51	3, 34	eleqtrrd 2916	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
52	33, 51	ffvelrnd 6852	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
53	52	adantr 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
54		iccssre 12819	. . . . . . . . . . . . . . 15 ⊢ ((((ℝ D 𝐹)‘𝑀) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ∈ ℝ) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ℝ)
55	49, 52, 54	syl2anc 586	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ℝ)
56	55	adantr 483	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ℝ)
57	56, 48	sseldd 3968	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℝ)
58		iccneg 12859	. . . . . . . . . . . 12 ⊢ ((((ℝ D 𝐹)‘𝑀) ∈ ℝ ∧ ((ℝ D 𝐹)‘𝑁) ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ↔ -𝑥 ∈ (-((ℝ D 𝐹)‘𝑁)[,]-((ℝ D 𝐹)‘𝑀))))
59	50, 53, 57, 58	syl3anc 1367	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ↔ -𝑥 ∈ (-((ℝ D 𝐹)‘𝑁)[,]-((ℝ D 𝐹)‘𝑀))))
60	48, 59	mpbid 234	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ (-((ℝ D 𝐹)‘𝑁)[,]-((ℝ D 𝐹)‘𝑀)))
61	40	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑁) = ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑁))
62		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑁))
63	62	negeqd 10880	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑁))
64		eqid 2821	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))
65		negex 10884	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑁) ∈ V
66	63, 64, 65	fvmpt 6768	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ (𝐴(,)𝐵) → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑁) = -((ℝ D 𝐹)‘𝑁))
67	4, 66	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑁) = -((ℝ D 𝐹)‘𝑁))
68	61, 67	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑁) = -((ℝ D 𝐹)‘𝑁))
69	40	fveq1d 6672	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑀) = ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑀))
70		fveq2 6670	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑀 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑀))
71	70	negeqd 10880	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑀 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑀))
72		negex 10884	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑀) ∈ V
73	71, 64, 72	fvmpt 6768	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (𝐴(,)𝐵) → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑀) = -((ℝ D 𝐹)‘𝑀))
74	2, 73	syl 17	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))‘𝑀) = -((ℝ D 𝐹)‘𝑀))
75	69, 74	eqtrd 2856	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑀) = -((ℝ D 𝐹)‘𝑀))
76	68, 75	oveq12d 7174	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑁)[,]((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑀)) = (-((ℝ D 𝐹)‘𝑁)[,]-((ℝ D 𝐹)‘𝑀)))
77	60, 76	eleqtrrd 2916	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ (((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑁)[,]((ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)))‘𝑀)))
78		eqid 2821	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ (((𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))‘𝑦) − (-𝑥 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ (((𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))‘𝑦) − (-𝑥 · 𝑦)))
79	2, 4, 22, 46, 47, 77, 78	dvivthlem2 24606	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ ran (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))))
80	40	rneqd 5808	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ran (ℝ D (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))) = ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
81	79, 80	eleqtrd 2915	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
82		negex 10884	. . . . . . . 8 ⊢ -𝑥 ∈ V
83	64	elrnmpt 5828	. . . . . . . 8 ⊢ (-𝑥 ∈ V → (-𝑥 ∈ ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) ↔ ∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤)))
84	82, 83	ax-mp 5	. . . . . . 7 ⊢ (-𝑥 ∈ ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) ↔ ∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤))
85	81, 84	sylib 220	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤))
86	57	recnd 10669	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℂ)
87	86	adantr 483	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
88	24, 27, 28, 39	dvmptcl 24556	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑤) ∈ ℂ)
89	87, 88	neg11ad 10993	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (-𝑥 = -((ℝ D 𝐹)‘𝑤) ↔ 𝑥 = ((ℝ D 𝐹)‘𝑤)))
90		eqcom 2828	. . . . . . . 8 ⊢ (𝑥 = ((ℝ D 𝐹)‘𝑤) ↔ ((ℝ D 𝐹)‘𝑤) = 𝑥)
91	89, 90	syl6bb 289	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (-𝑥 = -((ℝ D 𝐹)‘𝑤) ↔ ((ℝ D 𝐹)‘𝑤) = 𝑥))
92	91	rexbidva 3296	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤) ↔ ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥))
93	85, 92	mpbid 234	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥)
94	37	ffnd 6515	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
95		fvelrnb 6726	. . . . . 6 ⊢ ((ℝ D 𝐹) Fn (𝐴(,)𝐵) → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥))
96	94, 95	syl 17	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (𝑥 ∈ ran (ℝ D 𝐹) ↔ ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥))
97	93, 96	mpbird 259	. . . 4 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ran (ℝ D 𝐹))
98	97	expr 459	. . 3 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) → 𝑥 ∈ ran (ℝ D 𝐹)))
99	98	ssrdv 3973	. 2 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
100		fveq2 6670	. . . . 5 ⊢ (𝑀 = 𝑁 → ((ℝ D 𝐹)‘𝑀) = ((ℝ D 𝐹)‘𝑁))
101	100	oveq1d 7171	. . . 4 ⊢ (𝑀 = 𝑁 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)))
102	52	rexrd 10691	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ^*)
103		iccid 12784	. . . . 5 ⊢ (((ℝ D 𝐹)‘𝑁) ∈ ℝ^* → (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)) = {((ℝ D 𝐹)‘𝑁)})
104	102, 103	syl 17	. . . 4 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)) = {((ℝ D 𝐹)‘𝑁)})
105	101, 104	sylan9eqr 2878	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = {((ℝ D 𝐹)‘𝑁)})
106	33	ffnd 6515	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) Fn dom (ℝ D 𝐹))
107		fnfvelrn 6848	. . . . . 6 ⊢ (((ℝ D 𝐹) Fn dom (ℝ D 𝐹) ∧ 𝑁 ∈ dom (ℝ D 𝐹)) → ((ℝ D 𝐹)‘𝑁) ∈ ran (ℝ D 𝐹))
108	106, 51, 107	syl2anc 586	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ran (ℝ D 𝐹))
109	108	snssd 4742	. . . 4 ⊢ (𝜑 → {((ℝ D 𝐹)‘𝑁)} ⊆ ran (ℝ D 𝐹))
110	109	adantr 483	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → {((ℝ D 𝐹)‘𝑁)} ⊆ ran (ℝ D 𝐹))
111	105, 110	eqsstrd 4005	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
112	3	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 ∈ (𝐴(,)𝐵))
113	1	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 ∈ (𝐴(,)𝐵))
114	5	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
115	34	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
116		simprl 769	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 < 𝑀)
117		simprr 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))
118		eqid 2821	. . . . 5 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦)))
119	112, 113, 114, 115, 116, 117, 118	dvivthlem2 24606	. . . 4 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ran (ℝ D 𝐹))
120	119	expr 459	. . 3 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) → 𝑥 ∈ ran (ℝ D 𝐹)))
121	120	ssrdv 3973	. 2 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
122	31, 1	sseldi 3965	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
123	31, 3	sseldi 3965	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
124	122, 123	lttri4d 10781	. 2 ⊢ (𝜑 → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
125	99, 111, 121, 124	mpjao3dan 1427	1 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 {csn 4567 {cpr 4569 class class class wbr 5066 ↦ cmpt 5146 dom cdm 5555 ran crn 5556 Fn wfn 6350 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 · cmul 10542 ℝ^*cxr 10674 < clt 10675 − cmin 10870 -cneg 10871 (,)cioo 12739 [,]cicc 12742 –cn→ccncf 23484 D cdv 24461

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-of 7409 df-om 7581 df-1st 7689 df-2nd 7690 df-supp 7831 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-2o 8103 df-oadd 8106 df-er 8289 df-map 8408 df-pm 8409 df-ixp 8462 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-fsupp 8834 df-fi 8875 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-q 12350 df-rp 12391 df-xneg 12508 df-xadd 12509 df-xmul 12510 df-ioo 12743 df-ico 12745 df-icc 12746 df-fz 12894 df-fzo 13035 df-seq 13371 df-exp 13431 df-hash 13692 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-struct 16485 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-mulr 16579 df-starv 16580 df-sca 16581 df-vsca 16582 df-ip 16583 df-tset 16584 df-ple 16585 df-ds 16587 df-unif 16588 df-hom 16589 df-cco 16590 df-rest 16696 df-topn 16697 df-0g 16715 df-gsum 16716 df-topgen 16717 df-pt 16718 df-prds 16721 df-xrs 16775 df-qtop 16780 df-imas 16781 df-xps 16783 df-mre 16857 df-mrc 16858 df-acs 16860 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-submnd 17957 df-mulg 18225 df-cntz 18447 df-cmn 18908 df-psmet 20537 df-xmet 20538 df-met 20539 df-bl 20540 df-mopn 20541 df-fbas 20542 df-fg 20543 df-cnfld 20546 df-top 21502 df-topon 21519 df-topsp 21541 df-bases 21554 df-cld 21627 df-ntr 21628 df-cls 21629 df-nei 21706 df-lp 21744 df-perf 21745 df-cn 21835 df-cnp 21836 df-haus 21923 df-cmp 21995 df-tx 22170 df-hmeo 22363 df-fil 22454 df-fm 22546 df-flim 22547 df-flf 22548 df-xms 22930 df-ms 22931 df-tms 22932 df-cncf 23486 df-limc 24464 df-dv 24465

This theorem is referenced by: dvne0 24608

W3C validator