arearect - Mathbox for Jon Pennant

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Theorem arearect 43478

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 13347	. . . . . . 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 13347	. . . . . . 7 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ⊆ ℝ)
9	6, 7, 8	mp2an 692	. . . . . 6 ⊢ (𝐶[,]𝐷) ⊆ ℝ
10		xpss12 5639	. . . . . 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 3980	. . . 4 ⊢ 𝑆 ⊆ (ℝ × ℝ)
13		iftrue 4485	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = (𝐷 − 𝐶))
14	1	imaeq1i 6016	. . . . . . . . . . . . . . 15 ⊢ (𝑆 “ {𝑥}) = (((𝐴[,]𝐵) × (𝐶[,]𝐷)) “ {𝑥})
15		iftrue 4485	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = (𝐶[,]𝐷))
16		xpimasn 6143	. . . . . . . . . . . . . . . . 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 4488	. . . . . . . . . . . . . . . . 17 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) = ∅)
19		disjsn 4668	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐴[,]𝐵) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ (𝐴[,]𝐵))
20		xpima1 6141	. . . . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . . 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 2762	. . . . . . . . . . . . . 14 ⊢ (𝑆 “ {𝑥}) = if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)
25	24	fveq2i 6837	. . . . . . . . . . . . 13 ⊢ (vol‘(𝑆 “ {𝑥})) = (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅))
26	15	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘(𝐶[,]𝐷)))
27	25, 26	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘(𝐶[,]𝐷)))
28		iccmbl 25525	. . . . . . . . . . . . . . 15 ⊢ ((𝐶 ∈ ℝ ∧ 𝐷 ∈ ℝ) → (𝐶[,]𝐷) ∈ dom vol)
29	6, 7, 28	mp2an 692	. . . . . . . . . . . . . 14 ⊢ (𝐶[,]𝐷) ∈ dom vol
30		mblvol 25489	. . . . . . . . . . . . . 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 25482	. . . . . . . . . . . . . 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 4488	. . . . . . . . . . 11 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) = 0)
39	18	fveq2d 6838	. . . . . . . . . . . . 13 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅)) = (vol‘∅))
40	25, 39	eqtrid 2783	. . . . . . . . . . . 12 ⊢ (¬ 𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) = (vol‘∅))
41		0mbl 25498	. . . . . . . . . . . . . 14 ⊢ ∅ ∈ dom vol
42		mblvol 25489	. . . . . . . . . . . . . 14 ⊢ (∅ ∈ dom vol → (vol‘∅) = (vol*‘∅))
43	41, 42	ax-mp 5	. . . . . . . . . . . . 13 ⊢ (vol‘∅) = (vol*‘∅)
44		ovol0 25452	. . . . . . . . . . . . 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 182	. . . . . . . . 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 11445	. . . . . . . 8 ⊢ (𝐷 − 𝐶) ∈ ℝ
52		0re 11136	. . . . . . . 8 ⊢ 0 ∈ ℝ
53	51, 52	ifcli 4527	. . . . . . 7 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) ∈ ℝ
54	50, 53	eqeltrdi 2844	. . . . . 6 ⊢ (𝑥 ∈ ℝ → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
55		volf 25488	. . . . . . . 8 ⊢ vol:dom vol⟶(0[,]+∞)
56		ffun 6665	. . . . . . . 8 ⊢ (vol:dom vol⟶(0[,]+∞) → Fun vol)
57	55, 56	ax-mp 5	. . . . . . 7 ⊢ Fun vol
58	29, 41	ifcli 4527	. . . . . . . 8 ⊢ if(𝑥 ∈ (𝐴[,]𝐵), (𝐶[,]𝐷), ∅) ∈ dom vol
59	24, 58	eqeltri 2832	. . . . . . 7 ⊢ (𝑆 “ {𝑥}) ∈ dom vol
60		fvimacnv 6998	. . . . . . 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 3053	. . . 4 ⊢ ∀𝑥 ∈ ℝ (𝑆 “ {𝑥}) ∈ (^◡vol “ ℝ)
64	5	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → (𝐴[,]𝐵) ⊆ ℝ)
65		rembl 25499	. . . . . . 7 ⊢ ℝ ∈ dom vol
66	65	a1i 11	. . . . . 6 ⊢ (0 ∈ ℝ → ℝ ∈ dom vol)
67	36, 51	eqeltrdi 2844	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴[,]𝐵) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
68	67	adantl 481	. . . . . 6 ⊢ ((0 ∈ ℝ ∧ 𝑥 ∈ (𝐴[,]𝐵)) → (vol‘(𝑆 “ {𝑥})) ∈ ℝ)
69		eldifn 4084	. . . . . . . 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 5193	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴[,]𝐵) ↦ (vol‘(𝑆 “ {𝑥}))) = (𝑥 ∈ (𝐴[,]𝐵) ↦ (𝐷 − 𝐶))
73	51	recni 11148	. . . . . . . . . 10 ⊢ (𝐷 − 𝐶) ∈ ℂ
74		ax-resscn 11085	. . . . . . . . . . 11 ⊢ ℝ ⊆ ℂ
75	5, 74	sstri 3943	. . . . . . . . . 10 ⊢ (𝐴[,]𝐵) ⊆ ℂ
76		ssid 3956	. . . . . . . . . 10 ⊢ ℂ ⊆ ℂ
77		cncfmptc 24863	. . . . . . . . . 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 25800	. . . . . . . . 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 2832	. . . . . . 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 25765	. . . . 5 ⊢ (0 ∈ ℝ → (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹)
84	52, 83	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ℝ ↦ (vol‘(𝑆 “ {𝑥}))) ∈ 𝐿¹
85		dmarea 26925	. . . 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 26932	. . 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 25737	. . . 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 3057	. . 3 ⊢ ∫ℝ(vol‘(𝑆 “ {𝑥})) d𝑥 = ∫ℝif(𝑥 ∈ (𝐴[,]𝐵), (𝐷 − 𝐶), 0) d𝑥
91		iccmbl 25525	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴[,]𝐵) ∈ dom vol)
92	2, 3, 91	mp2an 692	. . . . 5 ⊢ (𝐴[,]𝐵) ∈ dom vol
93		mblvol 25489	. . . . . . . 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 25482	. . . . . . . 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 11445	. . . . . 6 ⊢ (𝐵 − 𝐴) ∈ ℝ
100	98, 99	eqeltri 2832	. . . . 5 ⊢ (vol‘(𝐴[,]𝐵)) ∈ ℝ
101		itgconst 25778	. . . . 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 25772	. . . . 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 7369	. . . 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 11148	. . 3 ⊢ (𝐵 − 𝐴) ∈ ℂ
109	73, 108	mulcomi 11142	. 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 206 = wceq 1541 ∈ wcel 2113 ∀wral 3051 ∖ cdif 3898 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 ifcif 4479 {csn 4580 class class class wbr 5098 ↦ cmpt 5179 × cxp 5622 ^◡ccnv 5623 dom cdm 5624 “ cima 5627 Fun wfun 6486 ⟶wf 6488 ‘cfv 6492 (class class class)co 7358 ℂcc 11026 ℝcr 11027 0cc0 11028 · cmul 11033 +∞cpnf 11165 ≤ cle 11169 − cmin 11366 [,]cicc 13266 –cn→ccncf 24827 vol*covol 25421 volcvol 25422 𝐿¹cibl 25576 ∫citg 25577 areacarea 26923

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9552 ax-cc 10347 ax-cnex 11084 ax-resscn 11085 ax-1cn 11086 ax-icn 11087 ax-addcl 11088 ax-addrcl 11089 ax-mulcl 11090 ax-mulrcl 11091 ax-mulcom 11092 ax-addass 11093 ax-mulass 11094 ax-distr 11095 ax-i2m1 11096 ax-1ne0 11097 ax-1rid 11098 ax-rnegex 11099 ax-rrecex 11100 ax-cnre 11101 ax-pre-lttri 11102 ax-pre-lttrn 11103 ax-pre-ltadd 11104 ax-pre-mulgt0 11105 ax-pre-sup 11106 ax-addf 11107

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-disj 5066 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-ofr 7623 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-oadd 8401 df-omul 8402 df-er 8635 df-map 8767 df-pm 8768 df-ixp 8838 df-en 8886 df-dom 8887 df-sdom 8888 df-fin 8889 df-fsupp 9267 df-fi 9316 df-sup 9347 df-inf 9348 df-oi 9417 df-dju 9815 df-card 9853 df-acn 9856 df-pnf 11170 df-mnf 11171 df-xr 11172 df-ltxr 11173 df-le 11174 df-sub 11368 df-neg 11369 df-div 11797 df-nn 12148 df-2 12210 df-3 12211 df-4 12212 df-5 12213 df-6 12214 df-7 12215 df-8 12216 df-9 12217 df-n0 12404 df-z 12491 df-dec 12610 df-uz 12754 df-q 12864 df-rp 12908 df-xneg 13028 df-xadd 13029 df-xmul 13030 df-ioo 13267 df-ioc 13268 df-ico 13269 df-icc 13270 df-fz 13426 df-fzo 13573 df-fl 13714 df-mod 13792 df-seq 13927 df-exp 13987 df-hash 14256 df-cj 15024 df-re 15025 df-im 15026 df-sqrt 15160 df-abs 15161 df-limsup 15396 df-clim 15413 df-rlim 15414 df-sum 15612 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17139 df-ress 17160 df-plusg 17192 df-mulr 17193 df-starv 17194 df-sca 17195 df-vsca 17196 df-ip 17197 df-tset 17198 df-ple 17199 df-ds 17201 df-unif 17202 df-hom 17203 df-cco 17204 df-rest 17344 df-topn 17345 df-0g 17363 df-gsum 17364 df-topgen 17365 df-pt 17366 df-prds 17369 df-xrs 17425 df-qtop 17430 df-imas 17431 df-xps 17433 df-mre 17507 df-mrc 17508 df-acs 17510 df-mgm 18567 df-sgrp 18646 df-mnd 18662 df-submnd 18711 df-mulg 19000 df-cntz 19248 df-cmn 19713 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-cnfld 21312 df-top 22840 df-topon 22857 df-topsp 22879 df-bases 22892 df-cn 23173 df-cnp 23174 df-cmp 23333 df-tx 23508 df-hmeo 23701 df-xms 24266 df-ms 24267 df-tms 24268 df-cncf 24829 df-ovol 25423 df-vol 25424 df-mbf 25578 df-itg1 25579 df-itg2 25580 df-ibl 25581 df-itg 25582 df-0p 25629 df-area 26924

This theorem is referenced by: (None)

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