arearect - Mathbox for Jon Pennant

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Theorem arearect 43176

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 13489	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ⊆ ℝ)
5	2, 3, 4	mp2an 691	. . . . . 6 ⊢ (𝐴[,]𝐵) ⊆ ℝ
6		arearect.3	. . . . . . 7 ⊢ 𝐶 ∈ ℝ
7		arearect.4	. . . . . . 7 ⊢ 𝐷 ∈ ℝ
8		iccssre 13489	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ⊆ ℝ)
9	6, 7, 8	mp2an 691	. . . . . 6 ⊢ (𝐶[,]𝐷) ⊆ ℝ
10		xpss12 5715	. . . . . 6 ⊢ (((𝐴[,]𝐵) ⊆ ℝ ∧ (𝐶[,]𝐷) ⊆ ℝ) → ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ))
11	5, 9, 10	mp2an 691	. . . . 5 ⊢ ((𝐴[,]𝐵) × (𝐶[,]𝐷)) ⊆ (ℝ × ℝ)
12	1, 11	eqsstri 4043	. . . 4 ⊢ 𝑆 ⊆ (ℝ × ℝ)
13		iftrue 4554	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (𝐷 − 𝐶))
14	1	imaeq1i 6086	. . . . . . . . . . . . . . 15 ⊢ (𝑆 “ {𝑥}) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
15		iftrue 4554	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (𝐶[,]𝐷))
16		xpimasn 6216	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = (𝐶[,]𝐷))
17	15, 16	eqtr4d 2783	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
18		iffalse 4557	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = ∅)
19		disjsn 4736	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴[,]𝐵))
20		xpima1 6214	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
21	19, 20	sylbir 235	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = ∅)
22	18, 21	eqtr4d 2783	. . . . . . . . . . . . . . . 16 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
23	17, 22	pm2.61i 182	. . . . . . . . . . . . . . 15 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
24	14, 23	eqtr4i 2771	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ {𝑥}) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)
25	24	fveq2i 6923	. . . . . . . . . . . . 13 ⊢ (vol‘(𝑆 “ {𝑥})) = (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅))
26	15	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘(𝐶[,]𝐷)))
27	25, 26	eqtrid 2792	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘(𝐶[,]𝐷)))
28		iccmbl 25620	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ∈ dom vol)
29	6, 7, 28	mp2an 691	. . . . . . . . . . . . . 14 ⊢ (𝐶[,]𝐷) ∈ dom vol
30		mblvol 25584	. . . . . . . . . . . . . 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 25577	. . . . . . . . . . . . . 14 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ ∧ 𝐶 ≤ 𝐷) → (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶))
34	6, 7, 32, 33	mp3an 1461	. . . . . . . . . . . . 13 ⊢ (vol*‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
35	31, 34	eqtri 2768	. . . . . . . . . . . 12 ⊢ (vol‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
36	27, 35	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (𝐷 − 𝐶))
37	13, 36	eqtr4d 2783	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
38		iffalse 4557	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = 0)
39	18	fveq2d 6924	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘∅))
40	25, 39	eqtrid 2792	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘∅))
41		0mbl 25593	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
42		mblvol 25584	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
44		ovol0 25547	. . . . . . . . . . . . 13 ⊢ (vol*‘∅) = 0
45	43, 44	eqtri 2768	. . . . . . . . . . . 12 ⊢ (vol‘∅) = 0
46	40, 45	eqtrdi 2796	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = 0)
47	38, 46	eqtr4d 2783	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
48	37, 47	pm2.61i 182	. . . . . . . . 9 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥}))
49	48	eqcomi 2749	. . . . . . . 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 11598	. . . . . . . 8 ⊢ (𝐷 − 𝐶) ∈ ℝ
52		0re 11292	. . . . . . . 8 ⊢ 0 ∈ ℝ
53	51, 52	ifcli 4595	. . . . . . 7 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) ∈ ℝ
54	50, 53	eqeltrdi 2852	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
55		volf 25583	. . . . . . . 8 ⊢ vol:dom vol⟶(0[,]+∞)
56		ffun 6750	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → Fun vol)
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ Fun vol
58	29, 41	ifcli 4595	. . . . . . . 8 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) ∈ dom vol
59	24, 58	eqeltri 2840	. . . . . . 7 ⊢ (𝑆 “ {𝑥}) ∈ dom vol
60		fvimacnv 7086	. . . . . . 7 ⊢ ((Fun vol ∧ (𝑆 “ {𝑥}) ∈ dom vol) → ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)))
61	57, 59, 60	mp2an 691	. . . . . 6 ⊢ ((vol‘(𝑆 “ {𝑥})) ∈ ℝ ↔ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
62	54, 61	sylib 218	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
63	62	rgen 3069	. . . 4 ⊢ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)
64	5	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝐴[,]𝐵) ⊆ ℝ)
65		rembl 25594	. . . . . . 7 ⊢ ℝ ∈ dom vol
66	65	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → ℝ ∈ dom vol)
67	36, 51	eqeltrdi 2852	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
68	67	adantl 481	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
69		eldifn 4155	. . . . . . . 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 481	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (ℝ ∖ (𝐴[,]𝐵))) → (vol‘(𝑆 “ {𝑥})) = 0)
72	36	mpteq2ia 5269	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶))
73	51	recni 11304	. . . . . . . . . 10 ⊢ (𝐷 − 𝐶) ∈ ℂ
74		ax-resscn 11241	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
75	5, 74	sstri 4018	. . . . . . . . . 10 ⊢ (𝐴[,]𝐵) ⊆ ℂ
76		ssid 4031	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
77		cncfmptc 24957	. . . . . . . . . 10 ⊢ (((𝐷 − 𝐶) ∈ ℂ ∧ (𝐴[,]𝐵) ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ))
78	73, 75, 76, 77	mp3an 1461	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ ((𝐴[,]𝐵)–cn→ℂ)
79		cniccibl 25896	. . . . . . . . 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 1461	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶)) ∈ 𝐿¹
81	72, 80	eqeltri 2840	. . . . . . 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 25861	. . . . 5 ⊢ (0 ∈ ℝ → (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
84	52, 83	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
85		dmarea 27018	. . . 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 1341	. . 3 ⊢ 𝑆 ∈ dom area
87		areaval 27025	. . 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 25833	. . . 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 3073	. . 3 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
91		iccmbl 25620	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
92	2, 3, 91	mp2an 691	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ dom vol
93		mblvol 25584	. . . . . . . 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 25577	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐴 ≤ 𝐵) → (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴))
97	2, 3, 95, 96	mp3an 1461	. . . . . . 7 ⊢ (vol*‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
98	94, 97	eqtri 2768	. . . . . 6 ⊢ (vol‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
99	3, 2	resubcli 11598	. . . . . 6 ⊢ (𝐵 − 𝐴) ∈ ℝ
100	98, 99	eqeltri 2840	. . . . 5 ⊢ (vol‘(𝐴[,]𝐵)) ∈ ℝ
101		itgconst 25874	. . . . 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 1461	. . . 4 ⊢ ∫(𝐴[,]𝐵)(𝐷 − 𝐶) d𝑥 = ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵)))
103		itgss2 25868	. . . . 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 7459	. . . 4 ⊢ ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵))) = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
106	102, 104, 105	3eqtr3i 2776	. . 3 ⊢ ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
107	90, 106	eqtri 2768	. 2 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
108	99	recni 11304	. . 3 ⊢ (𝐵 − 𝐴) ∈ ℂ
109	73, 108	mulcomi 11298	. 2 ⊢ ((𝐷 − 𝐶) · (𝐵 − 𝐴)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))
110	88, 107, 109	3eqtri 2772	1 ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∖ cdif 3973 ∩ cin 3975 ⊆ wss 3976 ∅c0 4352 ifcif 4548 {csn 4648 class class class wbr 5166 ↦ cmpt 5249 × cxp 5698 ^◡ccnv 5699 dom cdm 5700 “ cima 5703 Fun wfun 6567 ⟶wf 6569 ‘cfv 6573 (class class class)co 7448 ℂcc 11182 ℝcr 11183 0cc0 11184 · cmul 11189 +∞cpnf 11321 ≤ cle 11325 − cmin 11520 [,]cicc 13410 –cn→ccncf 24921 vol*covol 25516 volcvol 25517 𝐿¹cibl 25671 ∫citg 25672 areacarea 27016

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cc 10504 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262 ax-addf 11263

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-tp 4653 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-iin 5018 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-of 7714 df-ofr 7715 df-om 7904 df-1st 8030 df-2nd 8031 df-supp 8202 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-2o 8523 df-oadd 8526 df-omul 8527 df-er 8763 df-map 8886 df-pm 8887 df-ixp 8956 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-fsupp 9432 df-fi 9480 df-sup 9511 df-inf 9512 df-oi 9579 df-dju 9970 df-card 10008 df-acn 10011 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-q 13014 df-rp 13058 df-xneg 13175 df-xadd 13176 df-xmul 13177 df-ioo 13411 df-ioc 13412 df-ico 13413 df-icc 13414 df-fz 13568 df-fzo 13712 df-fl 13843 df-mod 13921 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-limsup 15517 df-clim 15534 df-rlim 15535 df-sum 15735 df-struct 17194 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-ress 17288 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17482 df-topn 17483 df-0g 17501 df-gsum 17502 df-topgen 17503 df-pt 17504 df-prds 17507 df-xrs 17562 df-qtop 17567 df-imas 17568 df-xps 17570 df-mre 17644 df-mrc 17645 df-acs 17647 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-submnd 18819 df-mulg 19108 df-cntz 19357 df-cmn 19824 df-psmet 21379 df-xmet 21380 df-met 21381 df-bl 21382 df-mopn 21383 df-cnfld 21388 df-top 22921 df-topon 22938 df-topsp 22960 df-bases 22974 df-cn 23256 df-cnp 23257 df-cmp 23416 df-tx 23591 df-hmeo 23784 df-xms 24351 df-ms 24352 df-tms 24353 df-cncf 24923 df-ovol 25518 df-vol 25519 df-mbf 25673 df-itg1 25674 df-itg2 25675 df-ibl 25676 df-itg 25677 df-0p 25724 df-area 27017

This theorem is referenced by: (None)

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