arearect - Mathbox for Jon Pennant

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Theorem arearect 43198

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 13332	. . . . . . 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 13332	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ⊆ ℝ)
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ (𝐶[,]𝐷) ⊆ ℝ
10		xpss12 5634	. . . . . 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 3982	. . . 4 ⊢ 𝑆 ⊆ (ℝ × ℝ)
13		iftrue 4482	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (𝐷 − 𝐶))
14	1	imaeq1i 6008	. . . . . . . . . . . . . . 15 ⊢ (𝑆 “ {𝑥}) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
15		iftrue 4482	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (𝐶[,]𝐷))
16		xpimasn 6134	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}) = (𝐶[,]𝐷))
17	15, 16	eqtr4d 2767	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥}))
18		iffalse 4485	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = ∅)
19		disjsn 4663	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴[,]𝐵))
20		xpima1 6132	. . . . . . . . . . . . . . . . . 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 2767	. . . . . . . . . . . . . . . 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 2755	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ {𝑥}) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)
25	24	fveq2i 6825	. . . . . . . . . . . . 13 ⊢ (vol‘(𝑆 “ {𝑥})) = (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅))
26	15	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘(𝐶[,]𝐷)))
27	25, 26	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘(𝐶[,]𝐷)))
28		iccmbl 25465	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ∈ dom vol)
29	6, 7, 28	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (𝐶[,]𝐷) ∈ dom vol
30		mblvol 25429	. . . . . . . . . . . . . 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 25422	. . . . . . . . . . . . . 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 2752	. . . . . . . . . . . 12 ⊢ (vol‘(𝐶[,]𝐷)) = (𝐷 − 𝐶)
36	27, 35	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (𝐷 − 𝐶))
37	13, 36	eqtr4d 2767	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (vol‘(𝑆 “ {𝑥})))
38		iffalse 4485	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = 0)
39	18	fveq2d 6826	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘∅))
40	25, 39	eqtrid 2776	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘∅))
41		0mbl 25438	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
42		mblvol 25429	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
44		ovol0 25392	. . . . . . . . . . . . 13 ⊢ (vol*‘∅) = 0
45	43, 44	eqtri 2752	. . . . . . . . . . . 12 ⊢ (vol‘∅) = 0
46	40, 45	eqtrdi 2780	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = 0)
47	38, 46	eqtr4d 2767	. . . . . . . . . 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 2738	. . . . . . . 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 11426	. . . . . . . 8 ⊢ (𝐷 − 𝐶) ∈ ℝ
52		0re 11117	. . . . . . . 8 ⊢ 0 ∈ ℝ
53	51, 52	ifcli 4524	. . . . . . 7 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) ∈ ℝ
54	50, 53	eqeltrdi 2836	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
55		volf 25428	. . . . . . . 8 ⊢ vol:dom vol⟶(0[,]+∞)
56		ffun 6655	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → Fun vol)
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ Fun vol
58	29, 41	ifcli 4524	. . . . . . . 8 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) ∈ dom vol
59	24, 58	eqeltri 2824	. . . . . . 7 ⊢ (𝑆 “ {𝑥}) ∈ dom vol
60		fvimacnv 6987	. . . . . . 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 218	. . . . 5 ⊢ (𝑥 ∈ ℝ → (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ))
63	62	rgen 3046	. . . 4 ⊢ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)
64	5	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝐴[,]𝐵) ⊆ ℝ)
65		rembl 25439	. . . . . . 7 ⊢ ℝ ∈ dom vol
66	65	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → ℝ ∈ dom vol)
67	36, 51	eqeltrdi 2836	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
68	67	adantl 481	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
69		eldifn 4083	. . . . . . . 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 5187	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶))
73	51	recni 11129	. . . . . . . . . 10 ⊢ (𝐷 − 𝐶) ∈ ℂ
74		ax-resscn 11066	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
75	5, 74	sstri 3945	. . . . . . . . . 10 ⊢ (𝐴[,]𝐵) ⊆ ℂ
76		ssid 3958	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
77		cncfmptc 24803	. . . . . . . . . 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 25740	. . . . . . . . 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 2824	. . . . . . 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 25705	. . . . 5 ⊢ (0 ∈ ℝ → (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
84	52, 83	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
85		dmarea 26865	. . . 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 1342	. . 3 ⊢ 𝑆 ∈ dom area
87		areaval 26872	. . 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 25677	. . . 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 3050	. . 3 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
91		iccmbl 25465	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
92	2, 3, 91	mp2an 692	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ dom vol
93		mblvol 25429	. . . . . . . 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 25422	. . . . . . . 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 2752	. . . . . 6 ⊢ (vol‘(𝐴[,]𝐵)) = (𝐵 − 𝐴)
99	3, 2	resubcli 11426	. . . . . 6 ⊢ (𝐵 − 𝐴) ∈ ℝ
100	98, 99	eqeltri 2824	. . . . 5 ⊢ (vol‘(𝐴[,]𝐵)) ∈ ℝ
101		itgconst 25718	. . . . 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 25712	. . . . 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 7360	. . . 4 ⊢ ((𝐷 − 𝐶) · (vol‘(𝐴[,]𝐵))) = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
106	102, 104, 105	3eqtr3i 2760	. . 3 ⊢ ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
107	90, 106	eqtri 2752	. 2 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ((𝐷 − 𝐶) · (𝐵 − 𝐴))
108	99	recni 11129	. . 3 ⊢ (𝐵 − 𝐴) ∈ ℂ
109	73, 108	mulcomi 11123	. 2 ⊢ ((𝐷 − 𝐶) · (𝐵 − 𝐴)) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))
110	88, 107, 109	3eqtri 2756	1 ⊢ (area‘𝑆) = ((𝐵 − 𝐴) · (𝐷 − 𝐶))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 ↔ wb 206 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∖ cdif 3900 ∩ cin 3902 ⊆ wss 3903 ∅c0 4284 ifcif 4476 {csn 4577 class class class wbr 5092 ↦ cmpt 5173 × cxp 5617 ^◡ccnv 5618 dom cdm 5619 “ cima 5622 Fun wfun 6476 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 · cmul 11014 +∞cpnf 11146 ≤ cle 11150 − cmin 11347 [,]cicc 13251 –cn→ccncf 24767 vol*covol 25361 volcvol 25362 𝐿¹cibl 25516 ∫citg 25517 areacarea 26863

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cc 10329 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087 ax-addf 11088

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-iin 4944 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-of 7613 df-ofr 7614 df-om 7800 df-1st 7924 df-2nd 7925 df-supp 8094 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-2o 8389 df-oadd 8392 df-omul 8393 df-er 8625 df-map 8755 df-pm 8756 df-ixp 8825 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-fsupp 9252 df-fi 9301 df-sup 9332 df-inf 9333 df-oi 9402 df-dju 9797 df-card 9835 df-acn 9838 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-q 12850 df-rp 12894 df-xneg 13014 df-xadd 13015 df-xmul 13016 df-ioo 13252 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-fzo 13558 df-fl 13696 df-mod 13774 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-limsup 15378 df-clim 15395 df-rlim 15396 df-sum 15594 df-struct 17058 df-sets 17075 df-slot 17093 df-ndx 17105 df-base 17121 df-ress 17142 df-plusg 17174 df-mulr 17175 df-starv 17176 df-sca 17177 df-vsca 17178 df-ip 17179 df-tset 17180 df-ple 17181 df-ds 17183 df-unif 17184 df-hom 17185 df-cco 17186 df-rest 17326 df-topn 17327 df-0g 17345 df-gsum 17346 df-topgen 17347 df-pt 17348 df-prds 17351 df-xrs 17406 df-qtop 17411 df-imas 17412 df-xps 17414 df-mre 17488 df-mrc 17489 df-acs 17491 df-mgm 18514 df-sgrp 18593 df-mnd 18609 df-submnd 18658 df-mulg 18947 df-cntz 19196 df-cmn 19661 df-psmet 21253 df-xmet 21254 df-met 21255 df-bl 21256 df-mopn 21257 df-cnfld 21262 df-top 22779 df-topon 22796 df-topsp 22818 df-bases 22831 df-cn 23112 df-cnp 23113 df-cmp 23272 df-tx 23447 df-hmeo 23640 df-xms 24206 df-ms 24207 df-tms 24208 df-cncf 24769 df-ovol 25363 df-vol 25364 df-mbf 25518 df-itg1 25519 df-itg2 25520 df-ibl 25521 df-itg 25522 df-0p 25569 df-area 26864

This theorem is referenced by: (None)

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