Theorem sxbrsigalem0 32238

Description: The closed half-spaces of (ℝ × ℝ) cover (ℝ × ℝ). (Contributed by Thierry Arnoux, 11-Oct-2017.)

Assertion

Ref	Expression
sxbrsigalem0	⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Distinct variable group:   𝑒,𝑓

Proof of Theorem sxbrsigalem0

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		unissb 4873	. . 3 ⊢ (∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ) ↔ ∀𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))𝑧 ⊆ (ℝ × ℝ))
2		elun 4083	. . . 4 ⊢ (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ↔ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
3		eqid 2738	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
4	3	rnmptss 6996	. . . . . . . 8 ⊢ (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5		pnfxr 11029	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ*
6		icossre 13160	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
7	5, 6	mpan2 688	. . . . . . . . . 10 ⊢ (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8		xpss1 5608	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10		ovex 7308	. . . . . . . . . . 11 ⊢ (𝑒[,)+∞) ∈ V
11		reex 10962	. . . . . . . . . . 11 ⊢ ℝ ∈ V
12	10, 11	xpex 7603	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
13	12	elpw 4537	. . . . . . . . 9 ⊢ (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
14	9, 13	sylibr 233	. . . . . . . 8 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
15	4, 14	mprg 3078	. . . . . . 7 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
16	15	sseli 3917	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
17	16	elpwid 4544	. . . . 5 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18		eqid 2738	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
19	18	rnmptss 6996	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20		icossre 13160	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
21	5, 20	mpan2 688	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22		xpss2 5609	. . . . . . . . . 10 ⊢ ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24		ovex 7308	. . . . . . . . . . 11 ⊢ (𝑓[,)+∞) ∈ V
25	11, 24	xpex 7603	. . . . . . . . . 10 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
26	25	elpw 4537	. . . . . . . . 9 ⊢ ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
27	23, 26	sylibr 233	. . . . . . . 8 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
28	19, 27	mprg 3078	. . . . . . 7 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
29	28	sseli 3917	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
30	29	elpwid 4544	. . . . 5 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
31	17, 30	jaoi 854	. . . 4 ⊢ ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
32	2, 31	sylbi 216	. . 3 ⊢ (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
33	1, 32	mprgbir 3079	. 2 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34		funmpt 6472	. . . . . 6 ⊢ Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
35		rexr 11021	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ℝ*)
36	5	a1i 11	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → +∞ ∈ ℝ*)
37		ltpnf 12856	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) < +∞)
38		lbico1 13133	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st ‘𝑧) < +∞) → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
39	35, 36, 37, 38	syl3anc 1370	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
40	39	anim1i 615	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ))
41	40	anim2i 617	. . . . . . . 8 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
42		elxp7 7866	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)))
43		elxp7 7866	. . . . . . . 8 ⊢ (𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
44	41, 42, 43	3imtr4i 292	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ))
45		xp1st 7863	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
46		oveq1 7282	. . . . . . . . . 10 ⊢ (𝑒 = (1st ‘𝑧) → (𝑒[,)+∞) = ((1st ‘𝑧)[,)+∞))
47	46	xpeq1d 5618	. . . . . . . . 9 ⊢ (𝑒 = (1st ‘𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st ‘𝑧)[,)+∞) × ℝ))
48		ovex 7308	. . . . . . . . . 10 ⊢ ((1st ‘𝑧)[,)+∞) ∈ V
49	48, 11	xpex 7603	. . . . . . . . 9 ⊢ (((1st ‘𝑧)[,)+∞) × ℝ) ∈ V
50	47, 3, 49	fvmpt 6875	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
51	45, 50	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
52	44, 51	eleqtrrd 2842	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)))
53		elunirn2 7126	. . . . . 6 ⊢ ((Fun (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∧ 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧))) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
54	34, 52, 53	sylancr 587	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
55	54	ssriv 3925	. . . 4 ⊢ (ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
56		ssun3 4108	. . . 4 ⊢ ((ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
57	55, 56	ax-mp 5	. . 3 ⊢ (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
58		uniun 4864	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
59	57, 58	sseqtrri 3958	. 2 ⊢ (ℝ × ℝ) ⊆ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
60	33, 59	eqssi 3937	1 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Syntax hints:   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  Vcvv 3432   ∪ cun 3885   ⊆ wss 3887  𝒫 cpw 4533  ∪ cuni 4839   class class class wbr 5074   ↦ cmpt 5157   × cxp 5587  ran crn 5590  Fun wfun 6427  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  ℝcr 10870  +∞cpnf 11006  ℝ^*cxr 11008   < clt 11009  [,)cico 13081

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ico 13085

This theorem is referenced by:  sxbrsigalem3  32239  sxbrsigalem2  32253