arearect - Mathbox for Jon Pennant

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Theorem arearect 43660

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 13373	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
5	2, 3, 4	mp2an 698	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ℝ
6		arearect.3	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
7		arearect.4	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
8		iccssre 13373	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ⊆ ℝ)
9	6, 7, 8	mp2an 698	. . . . . 6 ⊢ (𝐶[,]𝐷) ⊆ ℝ
10		xpss12 5633	. . . . . 6 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ) → ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ))
11	5, 9, 10	mp2an 698	. . . . 5 ⊢ ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ)
12	1, 11	eqsstri 3961	. . . 4 ⊢ 𝑆 ⊆ (ℝ × ℝ)
13		iftrue 4460	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (𝐷 − 𝐶))
14	1	imaeq1i 6009	. . . . . . . . . . . . . . 15 ⊢ (𝑆 “ {𝑥}) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
15		iftrue 4460	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (𝐶[,]𝐷))
16		xpimasn 6136	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = (𝐶[,]𝐷))
17	15, 16	eqtr4d 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
18		iffalse 4463	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = ∅)
19		disjsn 4643	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴[,]𝐵))
20		xpima1 6134	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
21	19, 20	sylbir 236	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
22	18, 21	eqtr4d 2777	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
23	17, 22	pm2.61i 183	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
24	14, 23	eqtr4i 2765	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ {𝑥}) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)
25	24	fveq2i 6830	. . . . . . . . . . . . 13 ⊢ (vol‘(𝑆 “ {𝑥})) = (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅))
26	15	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘(𝐶[,]𝐷)))
27	25, 26	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘(𝐶[,]𝐷)))
28		iccmbl 25551	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ∈ dom vol)
29	6, 7, 28	mp2an 698	. . . . . . . . . . . . . 14 ⊢ (𝐶[,]𝐷) ∈ dom vol
30		mblvol 25515	. . . . . . . . . . . . . 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 25508	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ∧ 𝐶 ≤ 𝐷) → (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶))
34	6, 7, 32, 33	mp3an 1469	. . . . . . . . . . . . 13 ⊢ (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
35	31, 34	eqtri 2762	. . . . . . . . . . . 12 ⊢ (vol‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
36	27, 35	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (𝐷 − 𝐶))
37	13, 36	eqtr4d 2777	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
38		iffalse 4463	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = 0)
39	18	fveq2d 6831	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘∅))
40	25, 39	eqtrid 2786	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘∅))
41		0mbl 25524	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
42		mblvol 25515	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
44		ovol0 25478	. . . . . . . . . . . . 13 ⊢ (vol*‘∅) = 0
45	43, 44	eqtri 2762	. . . . . . . . . . . 12 ⊢ (vol‘∅) = 0
46	40, 45	eqtrdi 2790	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = 0)
47	38, 46	eqtr4d 2777	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
48	37, 47	pm2.61i 183	. . . . . . . . 9 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥}))
49	48	eqcomi 2748	. . . . . . . 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 11447	. . . . . . . 8 ⊢ (𝐷 − 𝐶) ∈ ℝ
52		0re 11137	. . . . . . . 8 ⊢ 0 ∈ ℝ
53	51, 52	ifcli 4502	. . . . . . 7 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) ∈ ℝ
54	50, 53	eqeltrdi 2847	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
55		volf 25514	. . . . . . . 8 ⊢ vol:dom vol⟶(0[,]+∞)
56		ffun 6658	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → Fun vol)
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ Fun vol
58	29, 41	ifcli 4502	. . . . . . . 8 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) ∈ dom vol
59	24, 58	eqeltri 2835	. . . . . . 7 ⊢ (𝑆 “ {𝑥}) ∈ dom vol
60		fvimacnv 6994	. . . . . . 7 ⊢ ((Fun vol ∧ (𝑆 “ {𝑥}) ∈ dom vol) → ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)))
61	57, 59, 60	mp2an 698	. . . . . 6 ⊢ ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
62	54, 61	sylib 219	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
63	62	rgen 3055	. . . 4 ⊢ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)
64	5	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝐴[,]𝐵) ⊆ ℝ)
65		rembl 25525	. . . . . . 7 ⊢ ℝ ∈ dom vol
66	65	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → ℝ ∈ dom vol)
67	36, 51	eqeltrdi 2847	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
68	67	adantl 482	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
69		eldifn 4062	. . . . . . . 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 482	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))) → (vol‘(𝑆 “ {𝑥})) = 0)
72	36	mpteq2ia 5167	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶))
73	51	recni 11150	. . . . . . . . . 10 ⊢ (𝐷 − 𝐶) ∈ ℂ
74		ax-resscn 11086	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
75	5, 74	sstri 3924	. . . . . . . . . 10 ⊢ (𝐴[,]𝐵) ⊆ ℂ
76		ssid 3937	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
77		cncfmptc 24897	. . . . . . . . . 10 ⊢ (((𝐷 − 𝐶) ∈ ℂ ∧ (𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
78	73, 75, 76, 77	mp3an 1469	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ)
79		cniccibl 25826	. . . . . . . . 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 1469	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ 𝐿¹
81	72, 80	eqeltri 2835	. . . . . . 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 25791	. . . . 5 ⊢ (0 ∈ ℝ → (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
84	52, 83	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
85		dmarea 26939	. . . 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 1348	. . 3 ⊢ 𝑆 ∈ dom area
87		areaval 26946	. . 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 25763	. . . 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 3059	. . 3 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
91		iccmbl 25551	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
92	2, 3, 91	mp2an 698	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ dom vol
93		mblvol 25515	. . . . . . . 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 25508	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴))
97	2, 3, 95, 96	mp3an 1469	. . . . . . 7 ⊢ (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
98	94, 97	eqtri 2762	. . . . . 6 ⊢ (vol‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
99	3, 2	resubcli 11447	. . . . . 6 ⊢ (𝐵 − 𝐴) ∈ ℝ
100	98, 99	eqeltri 2835	. . . . 5 ⊢ (vol‘(𝐴[,]𝐵)) ∈ ℝ
101		itgconst 25804	. . . . 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 1469	. . . 4 ⊢ ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵)))
103		itgss2 25798	. . . . 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 7367	. . . 4 ⊢ ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵))) = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
106	102, 104, 105	3eqtr3i 2770	. . 3 ⊢ ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
107	90, 106	eqtri 2762	. 2 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
108	99	recni 11150	. . 3 ⊢ (𝐵 − 𝐴) ∈ ℂ
109	73, 108	mulcomi 11144	. 2 ⊢ ((𝐷 − 𝐶) · (𝐵 − 𝐴)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))
110	88, 107, 109	3eqtri 2766	1 ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 207 = wceq 1547 ∈ wcel 2119 ∀wral 3053 ∖ cdif 3880 ∩ cin 3882 ⊆ wss 3883 ∅c0 4261 ifcif 4454 {csn 4555 class class class wbr 5072 ↦ cmpt 5153 × cxp 5616 ^◡ccnv 5617 dom cdm 5618 “ cima 5621 Fun wfun 6479 ⟶wf 6481 ‘cfv 6485 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 · cmul 11034 +∞cpnf 11167 ≤ cle 11171 − cmin 11368 [,]cicc 13292 –cn→ccncf 24861 vol*covol 25447 volcvol 25448 𝐿¹cibl 25602 ∫citg 25603 areacarea 26937

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cc 10348 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-iin 4924 df-disj 5040 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-ofr 7621 df-om 7807 df-1st 7931 df-2nd 7932 df-supp 8101 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-2o 8396 df-oadd 8399 df-omul 8400 df-er 8633 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-dju 9816 df-card 9854 df-acn 9857 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-z 12516 df-dec 12636 df-uz 12780 df-q 12890 df-rp 12934 df-xneg 13054 df-xadd 13055 df-xmul 13056 df-ioo 13293 df-ioc 13294 df-ico 13295 df-icc 13296 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-hom 17235 df-cco 17236 df-rest 17376 df-topn 17377 df-0g 17395 df-gsum 17396 df-topgen 17397 df-pt 17398 df-prds 17401 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-submnd 18743 df-mulg 19035 df-cntz 19283 df-cmn 19748 df-psmet 21339 df-xmet 21340 df-met 21341 df-bl 21342 df-mopn 21343 df-cnfld 21348 df-top 22877 df-topon 22894 df-topsp 22916 df-bases 22929 df-cn 23210 df-cnp 23211 df-cmp 23370 df-tx 23545 df-hmeo 23738 df-xms 24303 df-ms 24304 df-tms 24305 df-cncf 24863 df-ovol 25449 df-vol 25450 df-mbf 25604 df-itg1 25605 df-itg2 25606 df-ibl 25607 df-itg 25608 df-0p 25655 df-area 26938

This theorem is referenced by: (None)

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