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Theorem sxbrsigalem0 32871
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.
StepHypRef Expression
1 unissb 4900 . . 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 4108 . . . 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))
43rnmptss 7070 . . . . . . . 8 (∀𝑒 ∈ ℝ ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) → ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ))
5 pnfxr 11209 . . . . . . . . . . 11 +∞ ∈ ℝ*
6 icossre 13345 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑒[,)+∞) ⊆ ℝ)
75, 6mpan2 689 . . . . . . . . . 10 (𝑒 ∈ ℝ → (𝑒[,)+∞) ⊆ ℝ)
8 xpss1 5652 . . . . . . . . . 10 ((𝑒[,)+∞) ⊆ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
97, 8syl 17 . . . . . . . . 9 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
10 ovex 7390 . . . . . . . . . . 11 (𝑒[,)+∞) ∈ V
11 reex 11142 . . . . . . . . . . 11 ℝ ∈ V
1210, 11xpex 7687 . . . . . . . . . 10 ((𝑒[,)+∞) × ℝ) ∈ V
1312elpw 4564 . . . . . . . . 9 (((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ) ↔ ((𝑒[,)+∞) × ℝ) ⊆ (ℝ × ℝ))
149, 13sylibr 233 . . . . . . . 8 (𝑒 ∈ ℝ → ((𝑒[,)+∞) × ℝ) ∈ 𝒫 (ℝ × ℝ))
154, 14mprg 3070 . . . . . . 7 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ⊆ 𝒫 (ℝ × ℝ)
1615sseli 3940 . . . . . 6 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
1716elpwid 4569 . . . . 5 (𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → 𝑧 ⊆ (ℝ × ℝ))
18 eqid 2736 . . . . . . . . 9 (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) = (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))
1918rnmptss 7070 . . . . . . . 8 (∀𝑓 ∈ ℝ (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) → ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ))
20 icossre 13345 . . . . . . . . . . 11 ((𝑓 ∈ ℝ ∧ +∞ ∈ ℝ*) → (𝑓[,)+∞) ⊆ ℝ)
215, 20mpan2 689 . . . . . . . . . 10 (𝑓 ∈ ℝ → (𝑓[,)+∞) ⊆ ℝ)
22 xpss2 5653 . . . . . . . . . 10 ((𝑓[,)+∞) ⊆ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2321, 22syl 17 . . . . . . . . 9 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
24 ovex 7390 . . . . . . . . . . 11 (𝑓[,)+∞) ∈ V
2511, 24xpex 7687 . . . . . . . . . 10 (ℝ × (𝑓[,)+∞)) ∈ V
2625elpw 4564 . . . . . . . . 9 ((ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ) ↔ (ℝ × (𝑓[,)+∞)) ⊆ (ℝ × ℝ))
2723, 26sylibr 233 . . . . . . . 8 (𝑓 ∈ ℝ → (ℝ × (𝑓[,)+∞)) ∈ 𝒫 (ℝ × ℝ))
2819, 27mprg 3070 . . . . . . 7 ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) ⊆ 𝒫 (ℝ × ℝ)
2928sseli 3940 . . . . . 6 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ∈ 𝒫 (ℝ × ℝ))
3029elpwid 4569 . . . . 5 (𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))) → 𝑧 ⊆ (ℝ × ℝ))
3117, 30jaoi 855 . . . 4 ((𝑧 ∈ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∨ 𝑧 ∈ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
322, 31sylbi 216 . . 3 (𝑧 ∈ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) → 𝑧 ⊆ (ℝ × ℝ))
331, 32mprgbir 3071 . 2 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) ⊆ (ℝ × ℝ)
34 rexr 11201 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ℝ*)
355a1i 11 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → +∞ ∈ ℝ*)
36 ltpnf 13041 . . . . . . . . . . 11 ((1st𝑧) ∈ ℝ → (1st𝑧) < +∞)
37 lbico1 13318 . . . . . . . . . . 11 (((1st𝑧) ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ (1st𝑧) < +∞) → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3834, 35, 36, 37syl3anc 1371 . . . . . . . . . 10 ((1st𝑧) ∈ ℝ → (1st𝑧) ∈ ((1st𝑧)[,)+∞))
3938anim1i 615 . . . . . . . . 9 (((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ) → ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ))
4039anim2i 617 . . . . . . . 8 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
41 elxp7 7956 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ℝ ∧ (2nd𝑧) ∈ ℝ)))
42 elxp7 7956 . . . . . . . 8 (𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ ((1st𝑧)[,)+∞) ∧ (2nd𝑧) ∈ ℝ)))
4340, 41, 423imtr4i 291 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ (((1st𝑧)[,)+∞) × ℝ))
44 xp1st 7953 . . . . . . . 8 (𝑧 ∈ (ℝ × ℝ) → (1st𝑧) ∈ ℝ)
45 oveq1 7364 . . . . . . . . . 10 (𝑒 = (1st𝑧) → (𝑒[,)+∞) = ((1st𝑧)[,)+∞))
4645xpeq1d 5662 . . . . . . . . 9 (𝑒 = (1st𝑧) → ((𝑒[,)+∞) × ℝ) = (((1st𝑧)[,)+∞) × ℝ))
47 ovex 7390 . . . . . . . . . 10 ((1st𝑧)[,)+∞) ∈ V
4847, 11xpex 7687 . . . . . . . . 9 (((1st𝑧)[,)+∞) × ℝ) ∈ V
4946, 3, 48fvmpt 6948 . . . . . . . 8 ((1st𝑧) ∈ ℝ → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5044, 49syl 17 . . . . . . 7 (𝑧 ∈ (ℝ × ℝ) → ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) = (((1st𝑧)[,)+∞) × ℝ))
5143, 50eleqtrrd 2841 . . . . . 6 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)))
52 elfvunirn 6874 . . . . . 6 (𝑧 ∈ ((𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))‘(1st𝑧)) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5351, 52syl 17 . . . . 5 (𝑧 ∈ (ℝ × ℝ) → 𝑧 ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)))
5453ssriv 3948 . . . 4 (ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ))
55 ssun3 4134 . . . 4 ((ℝ × ℝ) ⊆ ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) → (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))))
5654, 55ax-mp 5 . . 3 (ℝ × ℝ) ⊆ ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
57 uniun 4891 . . 3 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = ( ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5856, 57sseqtrri 3981 . 2 (ℝ × ℝ) ⊆ (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞))))
5933, 58eqssi 3960 1 (ran (𝑒 ∈ ℝ ↦ ((𝑒[,)+∞) × ℝ)) ∪ ran (𝑓 ∈ ℝ ↦ (ℝ × (𝑓[,)+∞)))) = (ℝ × ℝ)
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  cun 3908  wss 3910  𝒫 cpw 4560   cuni 4865   class class class wbr 5105  cmpt 5188   × cxp 5631  ran crn 5634  cfv 6496  (class class class)co 7357  1st c1st 7919  2nd c2nd 7920  cr 11050  +∞cpnf 11186  *cxr 11188   < clt 11189  [,)cico 13266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-pre-lttri 11125  ax-pre-lttrn 11126
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-po 5545  df-so 5546  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-ico 13270
This theorem is referenced by:  sxbrsigalem3  32872  sxbrsigalem2  32886
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