arearect - Mathbox for Jon Pennant

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Theorem arearect 40690

Description: The area of a rectangle whose sides are parallel to the coordinate axes in (ℝ × ℝ) is its width multiplied by its height. (Contributed by Jon Pennant, 19-Mar-2019.)

**Hypotheses**
Ref	Expression
arearect.1	⊢ 𝐴 ∈ ℝ
arearect.2	⊢ 𝐵 ∈ ℝ
arearect.3	⊢ 𝐶 ∈ ℝ
arearect.4	⊢ 𝐷 ∈ ℝ
arearect.5	⊢ 𝐴 ≤ 𝐵
arearect.6	⊢ 𝐶 ≤ 𝐷
arearect.7	⊢ 𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))

**Assertion**
Ref	Expression
arearect	⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))

Proof of Theorem arearect Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		arearect.7	. . . . 5 ⊢ 𝑆 = ((𝐴[,]𝐵) × (𝐶[,]𝐷))
2		arearect.1	. . . . . . 7 ⊢ 𝐴 ∈ ℝ
3		arearect.2	. . . . . . 7 ⊢ 𝐵 ∈ ℝ
4		iccssre 12982	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
5	2, 3, 4	mp2an 692	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ℝ
6		arearect.3	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
7		arearect.4	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
8		iccssre 12982	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ⊆ ℝ)
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ (𝐶[,]𝐷) ⊆ ℝ
10		xpss12 5551	. . . . . 6 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ) → ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ))
11	5, 9, 10	mp2an 692	. . . . 5 ⊢ ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ)
12	1, 11	eqsstri 3921	. . . 4 ⊢ 𝑆 ⊆ (ℝ × ℝ)
13		iftrue 4431	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (𝐷 − 𝐶))
14	1	imaeq1i 5911	. . . . . . . . . . . . . . 15 ⊢ (𝑆 “ {𝑥}) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
15		iftrue 4431	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (𝐶[,]𝐷))
16		xpimasn 6028	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = (𝐶[,]𝐷))
17	15, 16	eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
18		iffalse 4434	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = ∅)
19		disjsn 4613	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴[,]𝐵))
20		xpima1 6026	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
21	19, 20	sylbir 238	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
22	18, 21	eqtr4d 2774	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
23	17, 22	pm2.61i 185	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
24	14, 23	eqtr4i 2762	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ {𝑥}) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)
25	24	fveq2i 6698	. . . . . . . . . . . . 13 ⊢ (vol‘(𝑆 “ {𝑥})) = (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅))
26	15	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘(𝐶[,]𝐷)))
27	25, 26	syl5eq 2783	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘(𝐶[,]𝐷)))
28		iccmbl 24417	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ∈ dom vol)
29	6, 7, 28	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (𝐶[,]𝐷) ∈ dom vol
30		mblvol 24381	. . . . . . . . . . . . . 14 ⊢ ((𝐶[,]𝐷) ∈ dom vol → (vol‘(𝐶[,]𝐷)) = (vol*‘(𝐶[,]𝐷)))
31	29, 30	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘(𝐶[,]𝐷)) = (vol*‘(𝐶[,]𝐷))
32		arearect.6	. . . . . . . . . . . . . 14 ⊢ 𝐶 ≤ 𝐷
33		ovolicc 24374	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ∧ 𝐶 ≤ 𝐷) → (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶))
34	6, 7, 32, 33	mp3an 1463	. . . . . . . . . . . . 13 ⊢ (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
35	31, 34	eqtri 2759	. . . . . . . . . . . 12 ⊢ (vol‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
36	27, 35	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (𝐷 − 𝐶))
37	13, 36	eqtr4d 2774	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
38		iffalse 4434	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = 0)
39	18	fveq2d 6699	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘∅))
40	25, 39	syl5eq 2783	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘∅))
41		0mbl 24390	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
42		mblvol 24381	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
44		ovol0 24344	. . . . . . . . . . . . 13 ⊢ (vol*‘∅) = 0
45	43, 44	eqtri 2759	. . . . . . . . . . . 12 ⊢ (vol‘∅) = 0
46	40, 45	eqtrdi 2787	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = 0)
47	38, 46	eqtr4d 2774	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
48	37, 47	pm2.61i 185	. . . . . . . . 9 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥}))
49	48	eqcomi 2745	. . . . . . . 8 ⊢ (vol‘(𝑆 “ {𝑥})) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0)
50	49	a1i 11	. . . . . . 7 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0))
51	7, 6	resubcli 11105	. . . . . . . 8 ⊢ (𝐷 − 𝐶) ∈ ℝ
52		0re 10800	. . . . . . . 8 ⊢ 0 ∈ ℝ
53	51, 52	ifcli 4472	. . . . . . 7 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) ∈ ℝ
54	50, 53	eqeltrdi 2839	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
55		volf 24380	. . . . . . . 8 ⊢ vol:dom vol⟶(0[,]+∞)
56		ffun 6526	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → Fun vol)
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ Fun vol
58	29, 41	ifcli 4472	. . . . . . . 8 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) ∈ dom vol
59	24, 58	eqeltri 2827	. . . . . . 7 ⊢ (𝑆 “ {𝑥}) ∈ dom vol
60		fvimacnv 6851	. . . . . . 7 ⊢ ((Fun vol ∧ (𝑆 “ {𝑥}) ∈ dom vol) → ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)))
61	57, 59, 60	mp2an 692	. . . . . 6 ⊢ ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
62	54, 61	sylib 221	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
63	62	rgen 3061	. . . 4 ⊢ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)
64	5	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝐴[,]𝐵) ⊆ ℝ)
65		rembl 24391	. . . . . . 7 ⊢ ℝ ∈ dom vol
66	65	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → ℝ ∈ dom vol)
67	36, 51	eqeltrdi 2839	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
68	67	adantl 485	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
69		eldifn 4028	. . . . . . . 8 ⊢ (𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵)) → ¬ 𝑥 ∈ (𝐴[,]𝐵))
70	69, 46	syl 17	. . . . . . 7 ⊢ (𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) = 0)
71	70	adantl 485	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))) → (vol‘(𝑆 “ {𝑥})) = 0)
72	36	mpteq2ia 5131	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶))
73	51	recni 10812	. . . . . . . . . 10 ⊢ (𝐷 − 𝐶) ∈ ℂ
74		ax-resscn 10751	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
75	5, 74	sstri 3896	. . . . . . . . . 10 ⊢ (𝐴[,]𝐵) ⊆ ℂ
76		ssid 3909	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
77		cncfmptc 23763	. . . . . . . . . 10 ⊢ (((𝐷 − 𝐶) ∈ ℂ ∧ (𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
78	73, 75, 76, 77	mp3an 1463	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ)
79		cniccibl 24692	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ)) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ 𝐿¹)
80	2, 3, 78, 79	mp3an 1463	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ 𝐿¹
81	72, 80	eqeltri 2827	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
82	81	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
83	64, 66, 68, 71, 82	iblss2 24657	. . . . 5 ⊢ (0 ∈ ℝ → (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
84	52, 83	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
85		dmarea 25794	. . . 4 ⊢ (𝑆 ∈ dom area ↔ (𝑆 ⊆ (ℝ × ℝ) ∧ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ) ∧ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹))
86	12, 63, 84, 85	mpbir3an 1343	. . 3 ⊢ 𝑆 ∈ dom area
87		areaval 25801	. . 3 ⊢ (𝑆 ∈ dom area → (area‘𝑆) = ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥)
88	86, 87	ax-mp 5	. 2 ⊢ (area‘𝑆) = ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥
89		itgeq2 24629	. . . 4 ⊢ (∀𝑥 ∈ ℝ (vol‘(𝑆 “ {𝑥})) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) → ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥)
90	89, 50	mprg 3065	. . 3 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
91		iccmbl 24417	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
92	2, 3, 91	mp2an 692	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ dom vol
93		mblvol 24381	. . . . . . . 8 ⊢ ((𝐴[,]𝐵) ∈ dom vol → (vol‘(𝐴[,]𝐵)) = (vol*‘(𝐴[,]𝐵)))
94	92, 93	ax-mp 5	. . . . . . 7 ⊢ (vol‘(𝐴[,]𝐵)) = (vol*‘(𝐴[,]𝐵))
95		arearect.5	. . . . . . . 8 ⊢ 𝐴 ≤ 𝐵
96		ovolicc 24374	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴))
97	2, 3, 95, 96	mp3an 1463	. . . . . . 7 ⊢ (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
98	94, 97	eqtri 2759	. . . . . 6 ⊢ (vol‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
99	3, 2	resubcli 11105	. . . . . 6 ⊢ (𝐵 − 𝐴) ∈ ℝ
100	98, 99	eqeltri 2827	. . . . 5 ⊢ (vol‘(𝐴[,]𝐵)) ∈ ℝ
101		itgconst 24670	. . . . 5 ⊢ (((𝐴[,]𝐵) ∈ dom vol ∧ (vol‘(𝐴[,]𝐵)) ∈ ℝ ∧ (𝐷 − 𝐶) ∈ ℂ) → ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵))))
102	92, 100, 73, 101	mp3an 1463	. . . 4 ⊢ ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵)))
103		itgss2 24664	. . . . 5 ⊢ ((𝐴[,]𝐵) ⊆ ℝ → ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥)
104	5, 103	ax-mp 5	. . . 4 ⊢ ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
105	98	oveq2i 7202	. . . 4 ⊢ ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵))) = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
106	102, 104, 105	3eqtr3i 2767	. . 3 ⊢ ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
107	90, 106	eqtri 2759	. 2 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
108	99	recni 10812	. . 3 ⊢ (𝐵 − 𝐴) ∈ ℂ
109	73, 108	mulcomi 10806	. 2 ⊢ ((𝐷 − 𝐶) · (𝐵 − 𝐴)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))
110	88, 107, 109	3eqtri 2763	1 ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∖ cdif 3850 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ifcif 4425 {csn 4527 class class class wbr 5039 ↦ cmpt 5120 × cxp 5534 ^◡ccnv 5535 dom cdm 5536 “ cima 5539 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 ℝcr 10693 0cc0 10694 · cmul 10699 +∞cpnf 10829 ≤ cle 10833 − cmin 11027 [,]cicc 12903 –cn→ccncf 23727 vol*covol 24313 volcvol 24314 𝐿¹cibl 24468 ∫citg 24469 areacarea 25792

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-inf2 9234 ax-cc 10014 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-pre-sup 10772 ax-addf 10773 ax-mulf 10774

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-disj 5005 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-se 5495 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-of 7447 df-ofr 7448 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-2o 8181 df-oadd 8184 df-omul 8185 df-er 8369 df-map 8488 df-pm 8489 df-ixp 8557 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-fsupp 8964 df-fi 9005 df-sup 9036 df-inf 9037 df-oi 9104 df-dju 9482 df-card 9520 df-acn 9523 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-q 12510 df-rp 12552 df-xneg 12669 df-xadd 12670 df-xmul 12671 df-ioo 12904 df-ioc 12905 df-ico 12906 df-icc 12907 df-fz 13061 df-fzo 13204 df-fl 13332 df-mod 13408 df-seq 13540 df-exp 13601 df-hash 13862 df-cj 14627 df-re 14628 df-im 14629 df-sqrt 14763 df-abs 14764 df-limsup 14997 df-clim 15014 df-rlim 15015 df-sum 15215 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-hom 16773 df-cco 16774 df-rest 16881 df-topn 16882 df-0g 16900 df-gsum 16901 df-topgen 16902 df-pt 16903 df-prds 16906 df-xrs 16961 df-qtop 16966 df-imas 16967 df-xps 16969 df-mre 17043 df-mrc 17044 df-acs 17046 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-submnd 18173 df-mulg 18443 df-cntz 18665 df-cmn 19126 df-psmet 20309 df-xmet 20310 df-met 20311 df-bl 20312 df-mopn 20313 df-cnfld 20318 df-top 21745 df-topon 21762 df-topsp 21784 df-bases 21797 df-cn 22078 df-cnp 22079 df-cmp 22238 df-tx 22413 df-hmeo 22606 df-xms 23172 df-ms 23173 df-tms 23174 df-cncf 23729 df-ovol 24315 df-vol 24316 df-mbf 24470 df-itg1 24471 df-itg2 24472 df-ibl 24473 df-itg 24474 df-0p 24521 df-area 25793

This theorem is referenced by: (None)

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