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Theorem dvivth 25988

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 25432 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 24859	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
8	7	ffvelcdmda 7040	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
9	8	renegcld 11578	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → -(𝐹‘𝑤) ∈ ℝ)
10	9	fmpttd 7071	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ)
11		ax-resscn 11097	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
12		ssid 3958	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
13		cncfss 24865	. . . . . . . . . . . . . . 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 3933	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
16		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))
17	16	negfcncf 24890	. . . . . . . . . . . . 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 24864	. . . . . . . . . . . 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 11132	. . . . . . . . . . . . 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 7040	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
27	26	recnd 11174	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℂ)
28		fvexd 6859	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑤) ∈ V)
29	25	feqmptd 6912	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤)))
30	29	oveq2d 7386	. . . . . . . . . . . . 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 13337	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴(,)𝐵) ⊆ ℝ
32		dvfre 25928	. . . . . . . . . . . . . . . . 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 6656	. . . . . . . . . . . . . . . 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 6912	. . . . . . . . . . . . 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 25943	. . . . . . . . . . 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 5864	. . . . . . . . . 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 6210	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑤) ∈ V → dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵))
43		negex 11392	. . . . . . . . . . . 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 7041	. . . . . . . . . . . . 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 7041	. . . . . . . . . . . . 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 13359	. . . . . . . . . . . . . . 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 3936	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℝ)
58		iccneg 13402	. . . . . . . . . . . 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 6846	. . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑁))
63	62	negeqd 11388	. . . . . . . . . . . . . 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 11392	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑁) ∈ V
66	63, 64, 65	fvmpt 6951	. . . . . . . . . . . . 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 6846	. . . . . . . . . . . 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 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑀 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑀))
71	70	negeqd 11388	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑀 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑀))
72		negex 11392	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑀) ∈ V
73	71, 64, 72	fvmpt 6951	. . . . . . . . . . . . 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 7388	. . . . . . . . . 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 25987	. . . . . . . 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 5897	. . . . . . . 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 11392	. . . . . . . 8 ⊢ -𝑥 ∈ V
83	64	elrnmpt 5917	. . . . . . . 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 11174	. . . . . . . . . 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 25936	. . . . . . . . 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 11502	. . . . . . . 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 6673	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
95		fvelrnb 6904	. . . . . 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 3941	. 2 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
100		fveq2 6844	. . . . 5 ⊢ (𝑀 = 𝑁 → ((ℝ D 𝐹)‘𝑀) = ((ℝ D 𝐹)‘𝑁))
101	100	oveq1d 7385	. . . 4 ⊢ (𝑀 = 𝑁 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)))
102	52	rexrd 11196	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ^*)
103		iccid 13320	. . . . 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 6673	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) Fn dom (ℝ D 𝐹))
107		fnfvelrn 7036	. . . . . 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 4767	. . . 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 3970	. 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 25987	. . . 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 3941	. 2 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
122	31, 1	sselid 3933	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
123	31, 3	sselid 3933	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
124	122, 123	lttri4d 11288	. 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 3442 ⊆ wss 3903 {csn 4582 {cpr 4584 class class class wbr 5100 ↦ cmpt 5181 dom cdm 5634 ran crn 5635 Fn wfn 6497 ⟶wf 6498 ‘cfv 6502 (class class class)co 7370 ℂcc 11038 ℝcr 11039 · cmul 11045 ℝ^*cxr 11179 < clt 11180 − cmin 11378 -cneg 11379 (,)cioo 13275 [,]cicc 13278 –cn→ccncf 24842 D cdv 25837

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 5226 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117 ax-pre-sup 11118 ax-addf 11119

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 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-se 5588 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-isom 6511 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-of 7634 df-om 7821 df-1st 7945 df-2nd 7946 df-supp 8115 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-1o 8409 df-2o 8410 df-er 8647 df-map 8779 df-pm 8780 df-ixp 8850 df-en 8898 df-dom 8899 df-sdom 8900 df-fin 8901 df-fsupp 9279 df-fi 9328 df-sup 9359 df-inf 9360 df-oi 9429 df-card 9865 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-div 11809 df-nn 12160 df-2 12222 df-3 12223 df-4 12224 df-5 12225 df-6 12226 df-7 12227 df-8 12228 df-9 12229 df-n0 12416 df-z 12503 df-dec 12622 df-uz 12766 df-q 12876 df-rp 12920 df-xneg 13040 df-xadd 13041 df-xmul 13042 df-ioo 13279 df-ico 13281 df-icc 13282 df-fz 13438 df-fzo 13585 df-seq 13939 df-exp 13999 df-hash 14268 df-cj 15036 df-re 15037 df-im 15038 df-sqrt 15172 df-abs 15173 df-struct 17088 df-sets 17105 df-slot 17123 df-ndx 17135 df-base 17151 df-ress 17172 df-plusg 17204 df-mulr 17205 df-starv 17206 df-sca 17207 df-vsca 17208 df-ip 17209 df-tset 17210 df-ple 17211 df-ds 17213 df-unif 17214 df-hom 17215 df-cco 17216 df-rest 17356 df-topn 17357 df-0g 17375 df-gsum 17376 df-topgen 17377 df-pt 17378 df-prds 17381 df-xrs 17437 df-qtop 17442 df-imas 17443 df-xps 17445 df-mre 17519 df-mrc 17520 df-acs 17522 df-mgm 18579 df-sgrp 18658 df-mnd 18674 df-submnd 18723 df-mulg 19015 df-cntz 19263 df-cmn 19728 df-psmet 21318 df-xmet 21319 df-met 21320 df-bl 21321 df-mopn 21322 df-fbas 21323 df-fg 21324 df-cnfld 21327 df-top 22855 df-topon 22872 df-topsp 22894 df-bases 22907 df-cld 22980 df-ntr 22981 df-cls 22982 df-nei 23059 df-lp 23097 df-perf 23098 df-cn 23188 df-cnp 23189 df-haus 23276 df-cmp 23348 df-tx 23523 df-hmeo 23716 df-fil 23807 df-fm 23899 df-flim 23900 df-flf 23901 df-xms 24281 df-ms 24282 df-tms 24283 df-cncf 24844 df-limc 25840 df-dv 25841

This theorem is referenced by: dvne0 25989

W3C validator