Theorem sxbrsigalem0 34308

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 4920	. . 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 4133	. . . 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 2736	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) = (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
4	3	rnmptss 7118	. . . . . . . 8 ⊢ (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5		pnfxr 11294	. . . . . . . . . . 11 ⊢ +∞ ∈ ℝ*
6		icossre 13450	. . . . . . . . . . 11 ⊢ ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
7	5, 6	mpan2 691	. . . . . . . . . 10 ⊢ (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8		xpss1 5678	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑒[,)+∞) ∈ V
11		reex 11225	. . . . . . . . . . 11 ⊢ ℝ ∈ V
12	10, 11	xpex 7752	. . . . . . . . . 10 ⊢ ((𝑒[,)+∞) × ℝ) ∈ V
13	12	elpw 4584	. . . . . . . . 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 3058	. . . . . . 7 ⊢ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
16	15	sseli 3959	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
17	16	elpwid 4589	. . . . 5 ⊢ (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18		eqid 2736	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
19	18	rnmptss 7118	. . . . . . . 8 ⊢ (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20		icossre 13450	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
21	5, 20	mpan2 691	. . . . . . . . . 10 ⊢ (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22		xpss2 5679	. . . . . . . . . 10 ⊢ ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
23	21, 22	syl 17	. . . . . . . . 9 ⊢ (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24		ovex 7443	. . . . . . . . . . 11 ⊢ (𝑓[,)+∞) ∈ V
25	11, 24	xpex 7752	. . . . . . . . . 10 ⊢ (ℝ × (𝑓[,)+∞)) ∈ V
26	25	elpw 4584	. . . . . . . . 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 3058	. . . . . . 7 ⊢ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
29	28	sseli 3959	. . . . . 6 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
30	29	elpwid 4589	. . . . 5 ⊢ (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
31	17, 30	jaoi 857	. . . 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 3059	. 2 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34		rexr 11286	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ℝ*)
35	5	a1i 11	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → +∞ ∈ ℝ*)
36		ltpnf 13141	. . . . . . . . . . 11 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) < +∞)
37		lbico1 13422	. . . . . . . . . . 11 ⊢ (((1st ‘𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st ‘𝑧) < +∞) → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
38	34, 35, 36, 37	syl3anc 1373	. . . . . . . . . 10 ⊢ ((1st ‘𝑧) ∈ ℝ → (1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞))
39	38	anim1i 615	. . . . . . . . 9 ⊢ (((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ) → ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ))
40	39	anim2i 617	. . . . . . . 8 ⊢ ((𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ((1st ‘𝑧)[,)+∞) ∧ (2nd ‘𝑧) ∈ ℝ)))
41		elxp7 8028	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st ‘𝑧) ∈ ℝ ∧ (2nd ‘𝑧) ∈ ℝ)))
42		elxp7 8028	. . . . . . . 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 8025	. . . . . . . 8 ⊢ (𝑧 ∈ (ℝ × ℝ) → (1st ‘𝑧) ∈ ℝ)
45		oveq1 7417	. . . . . . . . . 10 ⊢ (𝑒 = (1st ‘𝑧) → (𝑒[,)+∞) = ((1st ‘𝑧)[,)+∞))
46	45	xpeq1d 5688	. . . . . . . . 9 ⊢ (𝑒 = (1st ‘𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st ‘𝑧)[,)+∞) × ℝ))
47		ovex 7443	. . . . . . . . . 10 ⊢ ((1st ‘𝑧)[,)+∞) ∈ V
48	47, 11	xpex 7752	. . . . . . . . 9 ⊢ (((1st ‘𝑧)[,)+∞) × ℝ) ∈ V
49	46, 3, 48	fvmpt 6991	. . . . . . . 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 2838	. . . . . 6 ⊢ (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st ‘𝑧)))
52		elfvunirn 6913	. . . . . 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 3967	. . . 4 ⊢ (ℝ × ℝ) ⊆ ∪ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
55		ssun3 4160	. . . 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 4911	. . 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 4013	. 2 ⊢ (ℝ × ℝ) ⊆ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
59	33, 58	eqssi 3980	1 ⊢ ∪ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)

Syntax hints:   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  Vcvv 3464   ∪ cun 3929   ⊆ wss 3931  𝒫 cpw 4580  ∪ cuni 4888   class class class wbr 5124   ↦ cmpt 5206   × cxp 5657  ran crn 5660  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  ℝcr 11133  +∞cpnf 11271  ℝ^*cxr 11273   < clt 11274  [,)cico 13369

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 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-pre-lttri 11208  ax-pre-lttrn 11209

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 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-po 5566  df-so 5567  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-ico 13373

This theorem is referenced by:  sxbrsigalem3  34309  sxbrsigalem2  34323