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Theorem dvivth 25956

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 25403 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 480	. . . . . . . . 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 480	. . . . . . . . 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 24838	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
8	7	ffvelcdmda 7028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
9	8	renegcld 11565	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → -(𝐹‘𝑤) ∈ ℝ)
10	9	fmpttd 7059	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ)
11		ax-resscn 11084	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
12		ssid 3945	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
13		cncfss 24844	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
14	11, 12, 13	mp2an 693	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ)
15	14, 5	sselid 3920	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
16		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))
17	16	negfcncf 24868	. . . . . . . . . . . . 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		cncfcdm 24843	. . . . . . . . . . . 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 588	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ) ↔ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ))
21	10, 20	mpbird 257	. . . . . . . . . 10 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
22	21	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) ∈ ((𝐴(,)𝐵)–cn→ℝ))
23		reelprrecn 11119	. . . . . . . . . . . . 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 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
26	25	ffvelcdmda 7028	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
27	26	recnd 11161	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℂ)
28		fvexd 6847	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑤) ∈ V)
29	25	feqmptd 6900	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤)))
30	29	oveq2d 7374	. . . . . . . . . . . . 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 13324	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴(,)𝐵) ⊆ ℝ
32		dvfre 25896	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	7, 31, 32	sylancl 587	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34		dvivth.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
35	34	feq2d 6644	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
36	33, 35	mpbid 232	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
37	36	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ)
38	37	feqmptd 6900	. . . . . . . . . . . . 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 2774	. . . . . . . . . . . 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 25911	. . . . . . . . . . 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 5852	. . . . . . . . . 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 6198	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑤) ∈ V → dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵))
43		negex 11379	. . . . . . . . . . . 12 ⊢ -((ℝ D 𝐹)‘𝑤) ∈ V
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑤) ∈ V)
45	42, 44	mprg 3058	. . . . . . . . . 10 ⊢ dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵)
46	41, 45	eqtrdi 2788	. . . . . . . . 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 771	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 < 𝑁)
48		simprr 773	. . . . . . . . . . 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	ffvelcdmd 7029	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
50	49	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D 𝐹)‘𝑀) ∈ ℝ)
51	3, 34	eleqtrrd 2840	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
52	33, 51	ffvelcdmd 7029	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
53	52	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ((ℝ D 𝐹)‘𝑁) ∈ ℝ)
54		iccssre 13346	. . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ℝ)
56	55	adantr 480	. . . . . . . . . . . . 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 3923	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℝ)
58		iccneg 13389	. . . . . . . . . . . 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 1374	. . . . . . . . . . 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 232	. . . . . . . . . 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 6834	. . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑁))
63	62	negeqd 11375	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑁))
64		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))
65		negex 11379	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑁) ∈ V
66	63, 64, 65	fvmpt 6939	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 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 6834	. . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑀 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑀))
71	70	negeqd 11375	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑀 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑀))
72		negex 11379	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑀) ∈ V
73	71, 64, 72	fvmpt 6939	. . . . . . . . . . . . 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 2772	. . . . . . . . . . 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 7376	. . . . . . . . . 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 2840	. . . . . . . . 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 2737	. . . . . . . . 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 25955	. . . . . . . 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 5885	. . . . . . . 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 2839	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
82		negex 11379	. . . . . . . 8 ⊢ -𝑥 ∈ V
83	64	elrnmpt 5905	. . . . . . . 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 218	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤))
86	57	recnd 11161	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℂ)
87	86	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → 𝑥 ∈ ℂ)
88	24, 27, 28, 39	dvmptcl 25904	. . . . . . . . 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 11489	. . . . . . . 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 2744	. . . . . . . 8 ⊢ (𝑥 = ((ℝ D 𝐹)‘𝑤) ↔ ((ℝ D 𝐹)‘𝑤) = 𝑥)
91	89, 90	bitrdi 287	. . . . . . 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 3160	. . . . . 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 232	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥)
94	37	ffnd 6661	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
95		fvelrnb 6892	. . . . . 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 257	. . . 4 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ran (ℝ D 𝐹))
98	97	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) → 𝑥 ∈ ran (ℝ D 𝐹)))
99	98	ssrdv 3928	. 2 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
100		fveq2 6832	. . . . 5 ⊢ (𝑀 = 𝑁 → ((ℝ D 𝐹)‘𝑀) = ((ℝ D 𝐹)‘𝑁))
101	100	oveq1d 7373	. . . 4 ⊢ (𝑀 = 𝑁 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)))
102	52	rexrd 11183	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ^*)
103		iccid 13307	. . . . 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 2794	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = {((ℝ D 𝐹)‘𝑁)})
106	33	ffnd 6661	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) Fn dom (ℝ D 𝐹))
107		fnfvelrn 7024	. . . . . 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 585	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ran (ℝ D 𝐹))
109	108	snssd 4753	. . . 4 ⊢ (𝜑 → {((ℝ D 𝐹)‘𝑁)} ⊆ ran (ℝ D 𝐹))
110	109	adantr 480	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → {((ℝ D 𝐹)‘𝑁)} ⊆ ran (ℝ D 𝐹))
111	105, 110	eqsstrd 3957	. 2 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
112	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 ∈ (𝐴(,)𝐵))
113	1	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 ∈ (𝐴(,)𝐵))
114	5	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ))
115	34	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
116		simprl 771	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 < 𝑀)
117		simprr 773	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))
118		eqid 2737	. . . . 5 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦)))
119	112, 113, 114, 115, 116, 117, 118	dvivthlem2 25955	. . . 4 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ran (ℝ D 𝐹))
120	119	expr 456	. . 3 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) → 𝑥 ∈ ran (ℝ D 𝐹)))
121	120	ssrdv 3928	. 2 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
122	31, 1	sselid 3920	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
123	31, 3	sselid 3920	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
124	122, 123	lttri4d 11275	. 2 ⊢ (𝜑 → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
125	99, 111, 121, 124	mpjao3dan 1435	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 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 ∃wrex 3062 Vcvv 3430 ⊆ wss 3890 {csn 4568 {cpr 4570 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5622 ran crn 5623 Fn wfn 6485 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ℂcc 11025 ℝcr 11026 · cmul 11032 ℝ^*cxr 11166 < clt 11167 − cmin 11365 -cneg 11366 (,)cioo 13262 [,]cicc 13265 –cn→ccncf 24821 D cdv 25808

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-z 12490 df-dec 12609 df-uz 12753 df-q 12863 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-ioo 13266 df-ico 13268 df-icc 13269 df-fz 13425 df-fzo 13572 df-seq 13926 df-exp 13986 df-hash 14255 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-rest 17343 df-topn 17344 df-0g 17362 df-gsum 17363 df-topgen 17364 df-pt 17365 df-prds 17368 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-cmp 23330 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-xms 24263 df-ms 24264 df-tms 24265 df-cncf 24823 df-limc 25811 df-dv 25812

This theorem is referenced by: dvne0 25957

W3C validator