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Theorem dvivth 25936

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 25380 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 24806	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐹:(𝐴(,)𝐵)⟶ℝ)
7	5, 6	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹:(𝐴(,)𝐵)⟶ℝ)
8	7	ffvelcdmda 7088	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
9	8	renegcld 11665	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ (𝐴(,)𝐵)) → -(𝐹‘𝑤) ∈ ℝ)
10	9	fmpttd 7119	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)):(𝐴(,)𝐵)⟶ℝ)
11		ax-resscn 11189	. . . . . . . . . . . 12 ⊢ ℝ ⊆ ℂ
12		ssid 4000	. . . . . . . . . . . . . . 15 ⊢ ℂ ⊆ ℂ
13		cncfss 24812	. . . . . . . . . . . . . . 15 ⊢ ((ℝ ⊆ ℂ ∧ ℂ ⊆ ℂ) → ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ))
14	11, 12, 13	mp2an 691	. . . . . . . . . . . . . 14 ⊢ ((𝐴(,)𝐵)–cn→ℝ) ⊆ ((𝐴(,)𝐵)–cn→ℂ)
15	14, 5	sselid 3976	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹 ∈ ((𝐴(,)𝐵)–cn→ℂ))
16		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -(𝐹‘𝑤))
17	16	negfcncf 24837	. . . . . . . . . . . . 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 24811	. . . . . . . . . . . 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 586	. . . . . . . . . . 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 11224	. . . . . . . . . . . . 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 7088	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℝ)
27	26	recnd 11266	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → (𝐹‘𝑤) ∈ ℂ)
28		fvexd 6906	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) ∧ 𝑤 ∈ (𝐴(,)𝐵)) → ((ℝ D 𝐹)‘𝑤) ∈ V)
29	25	feqmptd 6961	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝐹 = (𝑤 ∈ (𝐴(,)𝐵) ↦ (𝐹‘𝑤)))
30	29	oveq2d 7430	. . . . . . . . . . . . 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 13411	. . . . . . . . . . . . . . . . 17 ⊢ (𝐴(,)𝐵) ⊆ ℝ
32		dvfre 25876	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹:(𝐴(,)𝐵)⟶ℝ ∧ (𝐴(,)𝐵) ⊆ ℝ) → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
33	7, 31, 32	sylancl 585	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → (ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ)
34		dvivth.4	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → dom (ℝ D 𝐹) = (𝐴(,)𝐵))
35	34	feq2d 6702	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℝ D 𝐹):dom (ℝ D 𝐹)⟶ℝ ↔ (ℝ D 𝐹):(𝐴(,)𝐵)⟶ℝ))
36	33, 35	mpbid 231	. . . . . . . . . . . . . . 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 6961	. . . . . . . . . . . . 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 2770	. . . . . . . . . . . 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 25891	. . . . . . . . . . 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 5902	. . . . . . . . . 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 6240	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ (𝐴(,)𝐵)-((ℝ D 𝐹)‘𝑤) ∈ V → dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵))
43		negex 11482	. . . . . . . . . . . 12 ⊢ -((ℝ D 𝐹)‘𝑤) ∈ V
44	43	a1i 11	. . . . . . . . . . 11 ⊢ (𝑤 ∈ (𝐴(,)𝐵) → -((ℝ D 𝐹)‘𝑤) ∈ V)
45	42, 44	mprg 3063	. . . . . . . . . 10 ⊢ dom (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝐴(,)𝐵)
46	41, 45	eqtrdi 2784	. . . . . . . . 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 770	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑀 < 𝑁)
48		simprr 772	. . . . . . . . . . 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 7089	. . . . . . . . . . . . 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 2832	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ dom (ℝ D 𝐹))
52	33, 51	ffvelcdmd 7089	. . . . . . . . . . . . 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 13432	. . . . . . . . . . . . . . 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 583	. . . . . . . . . . . . . 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 3979	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ ℝ)
58		iccneg 13475	. . . . . . . . . . . 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 1369	. . . . . . . . . . 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 231	. . . . . . . . . 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 6893	. . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑁 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑁))
63	62	negeqd 11478	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑁 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑁))
64		eqid 2728	. . . . . . . . . . . . . 14 ⊢ (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)) = (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤))
65		negex 11482	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑁) ∈ V
66	63, 64, 65	fvmpt 6999	. . . . . . . . . . . . 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 2768	. . . . . . . . . . 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 6893	. . . . . . . . . . . 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 6891	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑀 → ((ℝ D 𝐹)‘𝑤) = ((ℝ D 𝐹)‘𝑀))
71	70	negeqd 11478	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑀 → -((ℝ D 𝐹)‘𝑤) = -((ℝ D 𝐹)‘𝑀))
72		negex 11482	. . . . . . . . . . . . . 14 ⊢ -((ℝ D 𝐹)‘𝑀) ∈ V
73	71, 64, 72	fvmpt 6999	. . . . . . . . . . . . 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 2768	. . . . . . . . . . 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 7432	. . . . . . . . . 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 2832	. . . . . . . . 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 2728	. . . . . . . . 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 25935	. . . . . . . 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 5934	. . . . . . . 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 2831	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → -𝑥 ∈ ran (𝑤 ∈ (𝐴(,)𝐵) ↦ -((ℝ D 𝐹)‘𝑤)))
82		negex 11482	. . . . . . . 8 ⊢ -𝑥 ∈ V
83	64	elrnmpt 5952	. . . . . . . 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 217	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)-𝑥 = -((ℝ D 𝐹)‘𝑤))
86	57	recnd 11266	. . . . . . . . . 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 25884	. . . . . . . . 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 11591	. . . . . . . 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 2735	. . . . . . . 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 3172	. . . . . 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 231	. . . . 5 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → ∃𝑤 ∈ (𝐴(,)𝐵)((ℝ D 𝐹)‘𝑤) = 𝑥)
94	37	ffnd 6717	. . . . . 6 ⊢ ((𝜑 ∧ (𝑀 < 𝑁 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → (ℝ D 𝐹) Fn (𝐴(,)𝐵))
95		fvelrnb 6953	. . . . . 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 3984	. 2 ⊢ ((𝜑 ∧ 𝑀 < 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
100		fveq2 6891	. . . . 5 ⊢ (𝑀 = 𝑁 → ((ℝ D 𝐹)‘𝑀) = ((ℝ D 𝐹)‘𝑁))
101	100	oveq1d 7429	. . . 4 ⊢ (𝑀 = 𝑁 → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = (((ℝ D 𝐹)‘𝑁)[,]((ℝ D 𝐹)‘𝑁)))
102	52	rexrd 11288	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ℝ^*)
103		iccid 13395	. . . . 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 2790	. . 3 ⊢ ((𝜑 ∧ 𝑀 = 𝑁) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) = {((ℝ D 𝐹)‘𝑁)})
106	33	ffnd 6717	. . . . . 6 ⊢ (𝜑 → (ℝ D 𝐹) Fn dom (ℝ D 𝐹))
107		fnfvelrn 7084	. . . . . 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 583	. . . . 5 ⊢ (𝜑 → ((ℝ D 𝐹)‘𝑁) ∈ ran (ℝ D 𝐹))
109	108	snssd 4808	. . . 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 4016	. 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 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑁 < 𝑀)
117		simprr 772	. . . . 5 ⊢ ((𝜑 ∧ (𝑁 < 𝑀 ∧ 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))) → 𝑥 ∈ (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)))
118		eqid 2728	. . . . 5 ⊢ (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦))) = (𝑦 ∈ (𝐴(,)𝐵) ↦ ((𝐹‘𝑦) − (𝑥 · 𝑦)))
119	112, 113, 114, 115, 116, 117, 118	dvivthlem2 25935	. . . 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 3984	. 2 ⊢ ((𝜑 ∧ 𝑁 < 𝑀) → (((ℝ D 𝐹)‘𝑀)[,]((ℝ D 𝐹)‘𝑁)) ⊆ ran (ℝ D 𝐹))
122	31, 1	sselid 3976	. . 3 ⊢ (𝜑 → 𝑀 ∈ ℝ)
123	31, 3	sselid 3976	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℝ)
124	122, 123	lttri4d 11379	. 2 ⊢ (𝜑 → (𝑀 < 𝑁 ∨ 𝑀 = 𝑁 ∨ 𝑁 < 𝑀))
125	99, 111, 121, 124	mpjao3dan 1429	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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∃wrex 3066 Vcvv 3470 ⊆ wss 3945 {csn 4624 {cpr 4626 class class class wbr 5142 ↦ cmpt 5225 dom cdm 5672 ran crn 5673 Fn wfn 6537 ⟶wf 6538 ‘cfv 6542 (class class class)co 7414 ℂcc 11130 ℝcr 11131 · cmul 11137 ℝ^*cxr 11271 < clt 11272 − cmin 11468 -cneg 11469 (,)cioo 13350 [,]cicc 13353 –cn→ccncf 24789 D cdv 25785

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189 ax-1cn 11190 ax-icn 11191 ax-addcl 11192 ax-addrcl 11193 ax-mulcl 11194 ax-mulrcl 11195 ax-mulcom 11196 ax-addass 11197 ax-mulass 11198 ax-distr 11199 ax-i2m1 11200 ax-1ne0 11201 ax-1rid 11202 ax-rnegex 11203 ax-rrecex 11204 ax-cnre 11205 ax-pre-lttri 11206 ax-pre-lttrn 11207 ax-pre-ltadd 11208 ax-pre-mulgt0 11209 ax-pre-sup 11210 ax-addf 11211

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7679 df-om 7865 df-1st 7987 df-2nd 7988 df-supp 8160 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-2o 8481 df-er 8718 df-map 8840 df-pm 8841 df-ixp 8910 df-en 8958 df-dom 8959 df-sdom 8960 df-fin 8961 df-fsupp 9380 df-fi 9428 df-sup 9459 df-inf 9460 df-oi 9527 df-card 9956 df-pnf 11274 df-mnf 11275 df-xr 11276 df-ltxr 11277 df-le 11278 df-sub 11470 df-neg 11471 df-div 11896 df-nn 12237 df-2 12299 df-3 12300 df-4 12301 df-5 12302 df-6 12303 df-7 12304 df-8 12305 df-9 12306 df-n0 12497 df-z 12583 df-dec 12702 df-uz 12847 df-q 12957 df-rp 13001 df-xneg 13118 df-xadd 13119 df-xmul 13120 df-ioo 13354 df-ico 13356 df-icc 13357 df-fz 13511 df-fzo 13654 df-seq 13993 df-exp 14053 df-hash 14316 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17109 df-sets 17126 df-slot 17144 df-ndx 17156 df-base 17174 df-ress 17203 df-plusg 17239 df-mulr 17240 df-starv 17241 df-sca 17242 df-vsca 17243 df-ip 17244 df-tset 17245 df-ple 17246 df-ds 17248 df-unif 17249 df-hom 17250 df-cco 17251 df-rest 17397 df-topn 17398 df-0g 17416 df-gsum 17417 df-topgen 17418 df-pt 17419 df-prds 17422 df-xrs 17477 df-qtop 17482 df-imas 17483 df-xps 17485 df-mre 17559 df-mrc 17560 df-acs 17562 df-mgm 18593 df-sgrp 18672 df-mnd 18688 df-submnd 18734 df-mulg 19017 df-cntz 19261 df-cmn 19730 df-psmet 21264 df-xmet 21265 df-met 21266 df-bl 21267 df-mopn 21268 df-fbas 21269 df-fg 21270 df-cnfld 21273 df-top 22789 df-topon 22806 df-topsp 22828 df-bases 22842 df-cld 22916 df-ntr 22917 df-cls 22918 df-nei 22995 df-lp 23033 df-perf 23034 df-cn 23124 df-cnp 23125 df-haus 23212 df-cmp 23284 df-tx 23459 df-hmeo 23652 df-fil 23743 df-fm 23835 df-flim 23836 df-flf 23837 df-xms 24219 df-ms 24220 df-tms 24221 df-cncf 24791 df-limc 25788 df-dv 25789

This theorem is referenced by: dvne0 25937

W3C validator