Theorem sxbrsigalem0 34236

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 4963	. . 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 4176	. . . 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 2740	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
4	3	rnmptss 7157	. . . . . . . 8 ⊢ (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5		pnfxr 11344	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ*
6		icossre 13488	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
7	5, 6	mpan2 690	. . . . . . . . . 10 ⊢ (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8		xpss1 5719	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑒[,)+∞) ∈ V
11		reex 11275	. . . . . . . . . . 11 ⊢ ℝ ∈ V
12	10, 11	xpex 7788	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
13	12	elpw 4626	. . . . . . . . 9 ⊢ (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
14	9, 13	sylibr 234	. . . . . . . 8 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
15	4, 14	mprg 3073	. . . . . . 7 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
16	15	sseli 4004	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
17	16	elpwid 4631	. . . . 5 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18		eqid 2740	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
19	18	rnmptss 7157	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20		icossre 13488	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
21	5, 20	mpan2 690	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22		xpss2 5720	. . . . . . . . . 10 ⊢ ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24		ovex 7481	. . . . . . . . . . 11 ⊢ (𝑓[,)+∞) ∈ V
25	11, 24	xpex 7788	. . . . . . . . . 10 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
26	25	elpw 4626	. . . . . . . . 9 ⊢ ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
27	23, 26	sylibr 234	. . . . . . . 8 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
28	19, 27	mprg 3073	. . . . . . 7 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
29	28	sseli 4004	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
30	29	elpwid 4631	. . . . 5 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
31	17, 30	jaoi 856	. . . 4 ⊢ ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
32	2, 31	sylbi 217	. . 3 ⊢ (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
33	1, 32	mprgbir 3074	. 2 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34		rexr 11336	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ℝ*)
35	5	a1i 11	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → +∞ ∈ ℝ*)
36		ltpnf 13183	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) < +∞)
37		lbico1 13461	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st ‘𝑧) < +∞) → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
38	34, 35, 36, 37	syl3anc 1371	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
39	38	anim1i 614	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ))
40	39	anim2i 616	. . . . . . . 8 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
41		elxp7 8065	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)))
42		elxp7 8065	. . . . . . . 8 ⊢ (𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
43	40, 41, 42	3imtr4i 292	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st ‘𝑧)[,)+∞) × ℝ))
44		xp1st 8062	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
45		oveq1 7455	. . . . . . . . . 10 ⊢ (𝑒 = (1st ‘𝑧) → (𝑒[,)+∞) = ((1st ‘𝑧)[,)+∞))
46	45	xpeq1d 5729	. . . . . . . . 9 ⊢ (𝑒 = (1st ‘𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st ‘𝑧)[,)+∞) × ℝ))
47		ovex 7481	. . . . . . . . . 10 ⊢ ((1st ‘𝑧)[,)+∞) ∈ V
48	47, 11	xpex 7788	. . . . . . . . 9 ⊢ (((1st ‘𝑧)[,)+∞) × ℝ) ∈ V
49	46, 3, 48	fvmpt 7029	. . . . . . . 8 ⊢ ((1st ‘𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
50	44, 49	syl 17	. . . . . . 7 ⊢ (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) = (((1st ‘𝑧)[,)+∞) × ℝ))
51	43, 50	eleqtrrd 2847	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)))
52		elfvunirn 6952	. . . . . 6 ⊢ (𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
53	51, 52	syl 17	. . . . 5 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
54	53	ssriv 4012	. . . 4 ⊢ (ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
55		ssun3 4203	. . . 4 ⊢ ((ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
56	54, 55	ax-mp 5	. . 3 ⊢ (ℝ × ℝ) ⊆ (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
57		uniun 4954	. . 3 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
58	56, 57	sseqtrri 4046	. 2 ⊢ (ℝ × ℝ) ⊆ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
59	33, 58	eqssi 4025	1 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Syntax hints:   ∧ wa 395   ∨ wo 846   = wceq 1537   ∈ wcel 2108  Vcvv 3488   ∪ cun 3974   ⊆ wss 3976  𝒫 cpw 4622  ∪ cuni 4931   class class class wbr 5166   ↦ cmpt 5249   × cxp 5698  ran crn 5701  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  ℝcr 11183  +∞cpnf 11321  ℝ^*cxr 11323   < clt 11324  [,)cico 13409

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-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-pre-lttri 11258  ax-pre-lttrn 11259

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-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-po 5607  df-so 5608  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-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-ico 13413

This theorem is referenced by:  sxbrsigalem3  34237  sxbrsigalem2  34251