intlewftc - Mathbox for metakunt

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Theorem intlewftc 40914

Description: Inequality inference by invoking fundamental theorem of calculus. (Contributed by metakunt, 22-Jul-2024.)

**Hypotheses**
Ref	Expression
intlewftc.1	⊢ (𝜑 → 𝐴 ∈ ℝ)
intlewftc.2	⊢ (𝜑 → 𝐵 ∈ ℝ)
intlewftc.3	⊢ (𝜑 → 𝐴 ≤ 𝐵)
intlewftc.4	⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
intlewftc.5	⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))
intlewftc.6	⊢ (𝜑 → 𝐷 = (ℝ D 𝐹))
intlewftc.7	⊢ (𝜑 → 𝐸 = (ℝ D 𝐺))
intlewftc.8	⊢ (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ))
intlewftc.9	⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ))
intlewftc.10	⊢ (𝜑 → 𝐷 ∈ 𝐿¹)
intlewftc.11	⊢ (𝜑 → 𝐸 ∈ 𝐿¹)
intlewftc.12	⊢ (𝜑 → 𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃))
intlewftc.13	⊢ (𝜑 → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))
intlewftc.14	⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ≤ 𝑄)
intlewftc.15	⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐺‘𝐴))

**Assertion**
Ref	Expression
intlewftc	⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐺‘𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝜑,𝑥 Allowed substitution hints: 𝐷(𝑥) 𝑃(𝑥) 𝑄(𝑥) 𝐸(𝑥) 𝐹(𝑥) 𝐺(𝑥)

Proof of Theorem intlewftc Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		intlewftc.4	. . . . . 6 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ))
2		cncff 24400	. . . . . 6 ⊢ (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐹:(𝐴[,]𝐵)⟶ℝ)
3	1, 2	syl 17	. . . . 5 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℝ)
4		intlewftc.2	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ)
5		intlewftc.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐵)
6	4	leidd 11776	. . . . . . 7 ⊢ (𝜑 → 𝐵 ≤ 𝐵)
7	4, 5, 6	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵))
8		intlewftc.1	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ ℝ)
9		elicc2 13385	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
10	8, 4, 9	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐵 ∈ (𝐴[,]𝐵) ↔ (𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵 ∧ 𝐵 ≤ 𝐵)))
11	7, 10	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ (𝐴[,]𝐵))
12	3, 11	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℝ)
13	8	leidd 11776	. . . . . . 7 ⊢ (𝜑 → 𝐴 ≤ 𝐴)
14	8, 13, 5	3jca 1128	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵))
15		elicc2 13385	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
16	8, 4, 15	syl2anc 584	. . . . . 6 ⊢ (𝜑 → (𝐴 ∈ (𝐴[,]𝐵) ↔ (𝐴 ∈ ℝ ∧ 𝐴 ≤ 𝐴 ∧ 𝐴 ≤ 𝐵)))
17	14, 16	mpbird 256	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ (𝐴[,]𝐵))
18	3, 17	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℝ)
19	12, 18	resubcld 11638	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) − (𝐹‘𝐴)) ∈ ℝ)
20		intlewftc.5	. . . . . 6 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ))
21		cncff 24400	. . . . . 6 ⊢ (𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ) → 𝐺:(𝐴[,]𝐵)⟶ℝ)
22	20, 21	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℝ)
23	22, 11	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℝ)
24	22, 17	ffvelcdmd 7084	. . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℝ)
25	23, 24	resubcld 11638	. . 3 ⊢ (𝜑 → ((𝐺‘𝐵) − (𝐺‘𝐴)) ∈ ℝ)
26		intlewftc.10	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝐿¹)
27		intlewftc.12	. . . . . . 7 ⊢ (𝜑 → 𝐷 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃))
28	27	eleq1d 2818	. . . . . 6 ⊢ (𝜑 → (𝐷 ∈ 𝐿¹ ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃) ∈ 𝐿¹))
29	26, 28	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃) ∈ 𝐿¹)
30		intlewftc.11	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝐿¹)
31		intlewftc.13	. . . . . . 7 ⊢ (𝜑 → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))
32	31	eleq1d 2818	. . . . . 6 ⊢ (𝜑 → (𝐸 ∈ 𝐿¹ ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄) ∈ 𝐿¹))
33	30, 32	mpbid 231	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄) ∈ 𝐿¹)
34		intlewftc.8	. . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ))
35		cncff 24400	. . . . . . . 8 ⊢ (𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐷:(𝐴(,)𝐵)⟶ℝ)
36	34, 35	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐷:(𝐴(,)𝐵)⟶ℝ)
37	27	feq1d 6699	. . . . . . 7 ⊢ (𝜑 → (𝐷:(𝐴(,)𝐵)⟶ℝ ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃):(𝐴(,)𝐵)⟶ℝ))
38	36, 37	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃):(𝐴(,)𝐵)⟶ℝ)
39	38	fvmptelcdm 7109	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ∈ ℝ)
40		intlewftc.9	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ))
41		cncff 24400	. . . . . . . 8 ⊢ (𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ) → 𝐸:(𝐴(,)𝐵)⟶ℝ)
42	40, 41	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝐸:(𝐴(,)𝐵)⟶ℝ)
43	31	feq1d 6699	. . . . . . 7 ⊢ (𝜑 → (𝐸:(𝐴(,)𝐵)⟶ℝ ↔ (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄):(𝐴(,)𝐵)⟶ℝ))
44	42, 43	mpbid 231	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄):(𝐴(,)𝐵)⟶ℝ)
45	44	fvmptelcdm 7109	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑄 ∈ ℝ)
46		intlewftc.14	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴(,)𝐵)) → 𝑃 ≤ 𝑄)
47	29, 33, 39, 45, 46	itgle 25318	. . . 4 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑃 d𝑥 ≤ ∫(𝐴(,)𝐵)𝑄 d𝑥)
48	39	itgmpt 25291	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑃 d𝑥 = ∫(𝐴(,)𝐵)((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡) d𝑡)
49	27	fveq1d 6890	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷‘𝑡) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡))
50	49	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐷‘𝑡) = ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡))
51	50	eqcomd 2738	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡) = (𝐷‘𝑡))
52	51	itgeq2dv 25290	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐷‘𝑡) d𝑡)
53		intlewftc.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷 = (ℝ D 𝐹))
54	53	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐷 = (ℝ D 𝐹))
55	54	fveq1d 6890	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐷‘𝑡) = ((ℝ D 𝐹)‘𝑡))
56	55	itgeq2dv 25290	. . . . . . . 8 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐷‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡)
57		ax-resscn 11163	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
58	57	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ⊆ ℂ)
59		fss 6731	. . . . . . . . . . . 12 ⊢ ((𝐷:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐷:(𝐴(,)𝐵)⟶ℂ)
60	36, 58, 59	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐷:(𝐴(,)𝐵)⟶ℂ)
61		ssidd 4004	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
62		cncfcdm 24405	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℝ)) → (𝐷 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ 𝐷:(𝐴(,)𝐵)⟶ℂ))
63	61, 34, 62	syl2anc 584	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐷 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ 𝐷:(𝐴(,)𝐵)⟶ℂ))
64	60, 63	mpbird 256	. . . . . . . . . 10 ⊢ (𝜑 → 𝐷 ∈ ((𝐴(,)𝐵)–cn→ℂ))
65	53	eleq1d 2818	. . . . . . . . . 10 ⊢ (𝜑 → (𝐷 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ)))
66	64, 65	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ ((𝐴(,)𝐵)–cn→ℂ))
67	53, 26	eqeltrrd 2834	. . . . . . . . 9 ⊢ (𝜑 → (ℝ D 𝐹) ∈ 𝐿¹)
68		fss 6731	. . . . . . . . . . 11 ⊢ ((𝐹:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:(𝐴[,]𝐵)⟶ℂ)
69	3, 58, 68	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(𝐴[,]𝐵)⟶ℂ)
70		cncfcdm 24405	. . . . . . . . . . 11 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℝ)) → (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ 𝐹:(𝐴[,]𝐵)⟶ℂ))
71	61, 1, 70	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜑 → (𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ 𝐹:(𝐴[,]𝐵)⟶ℂ))
72	69, 71	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ ((𝐴[,]𝐵)–cn→ℂ))
73	8, 4, 5, 66, 67, 72	ftc2 25552	. . . . . . . 8 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐹)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
74	56, 73	eqtrd 2772	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐷‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
75	52, 74	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑃)‘𝑡) d𝑡 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
76	48, 75	eqtrd 2772	. . . . 5 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑃 d𝑥 = ((𝐹‘𝐵) − (𝐹‘𝐴)))
77	45	itgmpt 25291	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑄 d𝑥 = ∫(𝐴(,)𝐵)((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄)‘𝑡) d𝑡)
78	31	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐸 = (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄))
79	78	eqcomd 2738	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄) = 𝐸)
80	79	fveq1d 6890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → ((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄)‘𝑡) = (𝐸‘𝑡))
81	80	itgeq2dv 25290	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((𝑥 ∈ (𝐴(,)𝐵) ↦ 𝑄)‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)(𝐸‘𝑡) d𝑡)
82	77, 81	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑄 d𝑥 = ∫(𝐴(,)𝐵)(𝐸‘𝑡) d𝑡)
83		intlewftc.7	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 = (ℝ D 𝐺))
84	83	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → 𝐸 = (ℝ D 𝐺))
85	84	fveq1d 6890	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝐴(,)𝐵)) → (𝐸‘𝑡) = ((ℝ D 𝐺)‘𝑡))
86	85	itgeq2dv 25290	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐸‘𝑡) d𝑡 = ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑡) d𝑡)
87		fss 6731	. . . . . . . . . . . . 13 ⊢ ((𝐸:(𝐴(,)𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐸:(𝐴(,)𝐵)⟶ℂ)
88	42, 58, 87	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸:(𝐴(,)𝐵)⟶ℂ)
89		cncfcdm 24405	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℝ)) → (𝐸 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ 𝐸:(𝐴(,)𝐵)⟶ℂ))
90	61, 40, 89	syl2anc 584	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ 𝐸:(𝐴(,)𝐵)⟶ℂ))
91	88, 90	mpbird 256	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℂ))
92	83	eleq1d 2818	. . . . . . . . . . 11 ⊢ (𝜑 → (𝐸 ∈ ((𝐴(,)𝐵)–cn→ℂ) ↔ (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ)))
93	91, 92	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
94	93, 92	mpbird 256	. . . . . . . . 9 ⊢ (𝜑 → 𝐸 ∈ ((𝐴(,)𝐵)–cn→ℂ))
95	94, 92	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐺) ∈ ((𝐴(,)𝐵)–cn→ℂ))
96	83, 30	eqeltrrd 2834	. . . . . . . 8 ⊢ (𝜑 → (ℝ D 𝐺) ∈ 𝐿¹)
97		fss 6731	. . . . . . . . . 10 ⊢ ((𝐺:(𝐴[,]𝐵)⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐺:(𝐴[,]𝐵)⟶ℂ)
98	22, 58, 97	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → 𝐺:(𝐴[,]𝐵)⟶ℂ)
99		cncfcdm 24405	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℝ)) → (𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ 𝐺:(𝐴[,]𝐵)⟶ℂ))
100	61, 20, 99	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ) ↔ 𝐺:(𝐴[,]𝐵)⟶ℂ))
101	98, 100	mpbird 256	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ ((𝐴[,]𝐵)–cn→ℂ))
102	8, 4, 5, 95, 96, 101	ftc2 25552	. . . . . . 7 ⊢ (𝜑 → ∫(𝐴(,)𝐵)((ℝ D 𝐺)‘𝑡) d𝑡 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
103	86, 102	eqtrd 2772	. . . . . 6 ⊢ (𝜑 → ∫(𝐴(,)𝐵)(𝐸‘𝑡) d𝑡 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
104	82, 103	eqtrd 2772	. . . . 5 ⊢ (𝜑 → ∫(𝐴(,)𝐵)𝑄 d𝑥 = ((𝐺‘𝐵) − (𝐺‘𝐴)))
105	76, 104	breq12d 5160	. . . 4 ⊢ (𝜑 → (∫(𝐴(,)𝐵)𝑃 d𝑥 ≤ ∫(𝐴(,)𝐵)𝑄 d𝑥 ↔ ((𝐹‘𝐵) − (𝐹‘𝐴)) ≤ ((𝐺‘𝐵) − (𝐺‘𝐴))))
106	47, 105	mpbid 231	. . 3 ⊢ (𝜑 → ((𝐹‘𝐵) − (𝐹‘𝐴)) ≤ ((𝐺‘𝐵) − (𝐺‘𝐴)))
107		intlewftc.15	. . 3 ⊢ (𝜑 → (𝐹‘𝐴) ≤ (𝐺‘𝐴))
108	19, 18, 25, 24, 106, 107	le2addd 11829	. 2 ⊢ (𝜑 → (((𝐹‘𝐵) − (𝐹‘𝐴)) + (𝐹‘𝐴)) ≤ (((𝐺‘𝐵) − (𝐺‘𝐴)) + (𝐺‘𝐴)))
109	57, 12	sselid 3979	. . . 4 ⊢ (𝜑 → (𝐹‘𝐵) ∈ ℂ)
110	57, 18	sselid 3979	. . . 4 ⊢ (𝜑 → (𝐹‘𝐴) ∈ ℂ)
111	109, 110	npcand 11571	. . 3 ⊢ (𝜑 → (((𝐹‘𝐵) − (𝐹‘𝐴)) + (𝐹‘𝐴)) = (𝐹‘𝐵))
112	57, 23	sselid 3979	. . . 4 ⊢ (𝜑 → (𝐺‘𝐵) ∈ ℂ)
113	57, 24	sselid 3979	. . . 4 ⊢ (𝜑 → (𝐺‘𝐴) ∈ ℂ)
114	112, 113	npcand 11571	. . 3 ⊢ (𝜑 → (((𝐺‘𝐵) − (𝐺‘𝐴)) + (𝐺‘𝐴)) = (𝐺‘𝐵))
115	111, 114	breq12d 5160	. 2 ⊢ (𝜑 → ((((𝐹‘𝐵) − (𝐹‘𝐴)) + (𝐹‘𝐴)) ≤ (((𝐺‘𝐵) − (𝐺‘𝐴)) + (𝐺‘𝐴)) ↔ (𝐹‘𝐵) ≤ (𝐺‘𝐵)))
116	108, 115	mpbid 231	1 ⊢ (𝜑 → (𝐹‘𝐵) ≤ (𝐺‘𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3947 class class class wbr 5147 ↦ cmpt 5230 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ℂcc 11104 ℝcr 11105 + caddc 11109 ≤ cle 11245 − cmin 11440 (,)cioo 13320 [,]cicc 13323 –cn→ccncf 24383 𝐿¹cibl 25125 ∫citg 25126 D cdv 25371

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-symdif 4241 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-disj 5113 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-ofr 7667 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-oadd 8466 df-omul 8467 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-dju 9892 df-card 9930 df-acn 9933 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-hash 14287 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-clim 15428 df-rlim 15429 df-sum 15629 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-submnd 18668 df-mulg 18945 df-cntz 19175 df-cmn 19644 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-cmp 22882 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-xms 23817 df-ms 23818 df-tms 23819 df-cncf 24385 df-ovol 24972 df-vol 24973 df-mbf 25127 df-itg1 25128 df-itg2 25129 df-ibl 25130 df-itg 25131 df-0p 25178 df-limc 25374 df-dv 25375

This theorem is referenced by: (None)

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