HomeHome Metamath Proof Explorer
Theorem List (p. 132 of 470)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-29646)
  Hilbert Space Explorer  Hilbert Space Explorer
(29647-31169)
  Users' Mathboxes  Users' Mathboxes
(31170-46966)
 

Theorem List for Metamath Proof Explorer - 13101-13200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxleadd1a 13101 Extended real version of leadd1 11557; note that the converse implication is not true, unlike the real version (for example 0 < 1 but (1 +𝑒 +∞) ≀ (0 +𝑒 +∞)). (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ 𝐴 ≀ 𝐡) β†’ (𝐴 +𝑒 𝐢) ≀ (𝐡 +𝑒 𝐢))
 
Theoremxleadd2a 13102 Commuted form of xleadd1a 13101. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ 𝐴 ≀ 𝐡) β†’ (𝐢 +𝑒 𝐴) ≀ (𝐢 +𝑒 𝐡))
 
Theoremxleadd1 13103 Weakened version of xleadd1a 13101 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) β†’ (𝐴 ≀ 𝐡 ↔ (𝐴 +𝑒 𝐢) ≀ (𝐡 +𝑒 𝐢)))
 
Theoremxltadd1 13104 Extended real version of ltadd1 11556. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (𝐴 +𝑒 𝐢) < (𝐡 +𝑒 𝐢)))
 
Theoremxltadd2 13105 Extended real version of ltadd2 11193. (Contributed by Mario Carneiro, 23-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) β†’ (𝐴 < 𝐡 ↔ (𝐢 +𝑒 𝐴) < (𝐢 +𝑒 𝐡)))
 
Theoremxaddge0 13106 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (0 ≀ 𝐴 ∧ 0 ≀ 𝐡)) β†’ 0 ≀ (𝐴 +𝑒 𝐡))
 
Theoremxle2add 13107 Extended real version of le2add 11571. (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) β†’ ((𝐴 ≀ 𝐢 ∧ 𝐡 ≀ 𝐷) β†’ (𝐴 +𝑒 𝐡) ≀ (𝐢 +𝑒 𝐷)))
 
Theoremxlt2add 13108 Extended real version of lt2add 11574. Note that ltleadd 11572, which has weaker assumptions, is not true for the extended reals (since 0 + +∞ < 1 + +∞ fails). (Contributed by Mario Carneiro, 23-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (𝐢 ∈ ℝ* ∧ 𝐷 ∈ ℝ*)) β†’ ((𝐴 < 𝐢 ∧ 𝐡 < 𝐷) β†’ (𝐴 +𝑒 𝐡) < (𝐢 +𝑒 𝐷)))
 
Theoremxsubge0 13109 Extended real version of subge0 11602. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (0 ≀ (𝐴 +𝑒 -𝑒𝐡) ↔ 𝐡 ≀ 𝐴))
 
Theoremxposdif 13110 Extended real version of posdif 11582. (Contributed by Mario Carneiro, 24-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 < 𝐡 ↔ 0 < (𝐡 +𝑒 -𝑒𝐴)))
 
Theoremxlesubadd 13111 Under certain conditions, the conclusion of lesubadd 11561 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ (0 ≀ 𝐴 ∧ 𝐡 β‰  -∞ ∧ 0 ≀ 𝐢)) β†’ ((𝐴 +𝑒 -𝑒𝐡) ≀ 𝐢 ↔ 𝐴 ≀ (𝐢 +𝑒 𝐡)))
 
Theoremxmullem 13112 Lemma for rexmul 13119. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) ∧ Β¬ (𝐴 = 0 ∨ 𝐡 = 0)) ∧ Β¬ (((0 < 𝐡 ∧ 𝐴 = +∞) ∨ (𝐡 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐡 = +∞) ∨ (𝐴 < 0 ∧ 𝐡 = -∞)))) ∧ Β¬ (((0 < 𝐡 ∧ 𝐴 = -∞) ∨ (𝐡 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐡 = -∞) ∨ (𝐴 < 0 ∧ 𝐡 = +∞)))) β†’ 𝐴 ∈ ℝ)
 
Theoremxmullem2 13113 Lemma for xmulneg1 13117. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ ((((0 < 𝐡 ∧ 𝐴 = +∞) ∨ (𝐡 < 0 ∧ 𝐴 = -∞)) ∨ ((0 < 𝐴 ∧ 𝐡 = +∞) ∨ (𝐴 < 0 ∧ 𝐡 = -∞))) β†’ Β¬ (((0 < 𝐡 ∧ 𝐴 = -∞) ∨ (𝐡 < 0 ∧ 𝐴 = +∞)) ∨ ((0 < 𝐴 ∧ 𝐡 = -∞) ∨ (𝐴 < 0 ∧ 𝐡 = +∞)))))
 
Theoremxmulcom 13114 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 Β·e 𝐡) = (𝐡 Β·e 𝐴))
 
Theoremxmul01 13115 Extended real version of mul01 11268. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (𝐴 Β·e 0) = 0)
 
Theoremxmul02 13116 Extended real version of mul02 11267. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (0 Β·e 𝐴) = 0)
 
Theoremxmulneg1 13117 Extended real version of mulneg1 11525. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (-𝑒𝐴 Β·e 𝐡) = -𝑒(𝐴 Β·e 𝐡))
 
Theoremxmulneg2 13118 Extended real version of mulneg2 11526. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 Β·e -𝑒𝐡) = -𝑒(𝐴 Β·e 𝐡))
 
Theoremrexmul 13119 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ) β†’ (𝐴 Β·e 𝐡) = (𝐴 Β· 𝐡))
 
Theoremxmulf 13120 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)
Β·e :(ℝ* Γ— ℝ*)βŸΆβ„*
 
Theoremxmulcl 13121 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐴 Β·e 𝐡) ∈ ℝ*)
 
Theoremxmulpnf1 13122 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) β†’ (𝐴 Β·e +∞) = +∞)
 
Theoremxmulpnf2 13123 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) β†’ (+∞ Β·e 𝐴) = +∞)
 
Theoremxmulmnf1 13124 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) β†’ (𝐴 Β·e -∞) = -∞)
 
Theoremxmulmnf2 13125 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 0 < 𝐴) β†’ (-∞ Β·e 𝐴) = -∞)
 
Theoremxmulpnf1n 13126 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐴 < 0) β†’ (𝐴 Β·e +∞) = -∞)
 
Theoremxmulid1 13127 Extended real version of mulid1 11087. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (𝐴 Β·e 1) = 𝐴)
 
Theoremxmulid2 13128 Extended real version of mulid2 11088. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (1 Β·e 𝐴) = 𝐴)
 
Theoremxmulm1 13129 Extended real version of mulm1 11530. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (-1 Β·e 𝐴) = -𝑒𝐴)
 
Theoremxmulasslem2 13130 Lemma for xmulass 13135. (Contributed by Mario Carneiro, 20-Aug-2015.)
((0 < 𝐴 ∧ 𝐴 = -∞) β†’ πœ‘)
 
Theoremxmulgt0 13131 Extended real version of mulgt0 11166. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐡 ∈ ℝ* ∧ 0 < 𝐡)) β†’ 0 < (𝐴 Β·e 𝐡))
 
Theoremxmulge0 13132 Extended real version of mulge0 11607. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡)) β†’ 0 ≀ (𝐴 Β·e 𝐡))
 
Theoremxmulasslem 13133* Lemma for xmulass 13135. (Contributed by Mario Carneiro, 20-Aug-2015.)
(π‘₯ = 𝐷 β†’ (πœ“ ↔ 𝑋 = π‘Œ))    &   (π‘₯ = -𝑒𝐷 β†’ (πœ“ ↔ 𝐸 = 𝐹))    &   (πœ‘ β†’ 𝑋 ∈ ℝ*)    &   (πœ‘ β†’ π‘Œ ∈ ℝ*)    &   (πœ‘ β†’ 𝐷 ∈ ℝ*)    &   ((πœ‘ ∧ (π‘₯ ∈ ℝ* ∧ 0 < π‘₯)) β†’ πœ“)    &   (πœ‘ β†’ (π‘₯ = 0 β†’ πœ“))    &   (πœ‘ β†’ 𝐸 = -𝑒𝑋)    &   (πœ‘ β†’ 𝐹 = -π‘’π‘Œ)    β‡’   (πœ‘ β†’ 𝑋 = π‘Œ)
 
Theoremxmulasslem3 13134 Lemma for xmulass 13135. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 < 𝐴) ∧ (𝐡 ∈ ℝ* ∧ 0 < 𝐡) ∧ (𝐢 ∈ ℝ* ∧ 0 < 𝐢)) β†’ ((𝐴 Β·e 𝐡) Β·e 𝐢) = (𝐴 Β·e (𝐡 Β·e 𝐢)))
 
Theoremxmulass 13135 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 13097 which has to avoid the "undefined" combinations +∞ +𝑒 -∞ and -∞ +𝑒 +∞. The equivalent "undefined" expression here would be 0 Β·e +∞, but since this is defined to equal 0 any zeroes in the expression make the whole thing evaluate to zero (on both sides), thus establishing the identity in this case. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ ((𝐴 Β·e 𝐡) Β·e 𝐢) = (𝐴 Β·e (𝐡 Β·e 𝐢)))
 
Theoremxlemul1a 13136 Extended real version of lemul1a 11943. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (𝐢 ∈ ℝ* ∧ 0 ≀ 𝐢)) ∧ 𝐴 ≀ 𝐡) β†’ (𝐴 Β·e 𝐢) ≀ (𝐡 Β·e 𝐢))
 
Theoremxlemul2a 13137 Extended real version of lemul2a 11944. (Contributed by Mario Carneiro, 8-Sep-2015.)
(((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ (𝐢 ∈ ℝ* ∧ 0 ≀ 𝐢)) ∧ 𝐴 ≀ 𝐡) β†’ (𝐢 Β·e 𝐴) ≀ (𝐢 Β·e 𝐡))
 
Theoremxlemul1 13138 Extended real version of lemul1 11941. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ+) β†’ (𝐴 ≀ 𝐡 ↔ (𝐴 Β·e 𝐢) ≀ (𝐡 Β·e 𝐢)))
 
Theoremxlemul2 13139 Extended real version of lemul2 11942. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ+) β†’ (𝐴 ≀ 𝐡 ↔ (𝐢 Β·e 𝐴) ≀ (𝐢 Β·e 𝐡)))
 
Theoremxltmul1 13140 Extended real version of ltmul1 11939. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ+) β†’ (𝐴 < 𝐡 ↔ (𝐴 Β·e 𝐢) < (𝐡 Β·e 𝐢)))
 
Theoremxltmul2 13141 Extended real version of ltmul2 11940. (Contributed by Mario Carneiro, 8-Sep-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ+) β†’ (𝐴 < 𝐡 ↔ (𝐢 Β·e 𝐴) < (𝐢 Β·e 𝐡)))
 
Theoremxadddilem 13142 Lemma for xadddi 13143. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) ∧ 0 < 𝐴) β†’ (𝐴 Β·e (𝐡 +𝑒 𝐢)) = ((𝐴 Β·e 𝐡) +𝑒 (𝐴 Β·e 𝐢)))
 
Theoremxadddi 13143 Distributive property for extended real addition and multiplication. Like xaddass 13097, this has an unusual domain of correctness due to counterexamples like (+∞ Β· (2 βˆ’ 1)) = -∞ β‰  ((+∞ Β· 2) βˆ’ (+∞ Β· 1)) = (+∞ βˆ’ +∞) = 0. In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ*) β†’ (𝐴 Β·e (𝐡 +𝑒 𝐢)) = ((𝐴 Β·e 𝐡) +𝑒 (𝐴 Β·e 𝐢)))
 
Theoremxadddir 13144 Commuted version of xadddi 13143. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ 𝐡 ∈ ℝ* ∧ 𝐢 ∈ ℝ) β†’ ((𝐴 +𝑒 𝐡) Β·e 𝐢) = ((𝐴 Β·e 𝐢) +𝑒 (𝐡 Β·e 𝐢)))
 
Theoremxadddi2 13145 The assumption that the multiplier be real in xadddi 13143 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)
((𝐴 ∈ ℝ* ∧ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡) ∧ (𝐢 ∈ ℝ* ∧ 0 ≀ 𝐢)) β†’ (𝐴 Β·e (𝐡 +𝑒 𝐢)) = ((𝐴 Β·e 𝐡) +𝑒 (𝐴 Β·e 𝐢)))
 
Theoremxadddi2r 13146 Commuted version of xadddi2 13145. (Contributed by Mario Carneiro, 20-Aug-2015.)
(((𝐴 ∈ ℝ* ∧ 0 ≀ 𝐴) ∧ (𝐡 ∈ ℝ* ∧ 0 ≀ 𝐡) ∧ 𝐢 ∈ ℝ*) β†’ ((𝐴 +𝑒 𝐡) Β·e 𝐢) = ((𝐴 Β·e 𝐢) +𝑒 (𝐡 Β·e 𝐢)))
 
Theoremx2times 13147 Extended real version of 2times 12223. (Contributed by Mario Carneiro, 20-Aug-2015.)
(𝐴 ∈ ℝ* β†’ (2 Β·e 𝐴) = (𝐴 +𝑒 𝐴))
 
Theoremxnegcld 13148 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    β‡’   (πœ‘ β†’ -𝑒𝐴 ∈ ℝ*)
 
Theoremxaddcld 13149 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 +𝑒 𝐡) ∈ ℝ*)
 
Theoremxmulcld 13150 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)
(πœ‘ β†’ 𝐴 ∈ ℝ*)    &   (πœ‘ β†’ 𝐡 ∈ ℝ*)    β‡’   (πœ‘ β†’ (𝐴 Β·e 𝐡) ∈ ℝ*)
 
Theoremxadd4d 13151 Rearrangement of 4 terms in a sum for extended addition, analogous to add4d 11317. (Contributed by Alexander van der Vekens, 21-Dec-2017.)
(πœ‘ β†’ (𝐴 ∈ ℝ* ∧ 𝐴 β‰  -∞))    &   (πœ‘ β†’ (𝐡 ∈ ℝ* ∧ 𝐡 β‰  -∞))    &   (πœ‘ β†’ (𝐢 ∈ ℝ* ∧ 𝐢 β‰  -∞))    &   (πœ‘ β†’ (𝐷 ∈ ℝ* ∧ 𝐷 β‰  -∞))    β‡’   (πœ‘ β†’ ((𝐴 +𝑒 𝐡) +𝑒 (𝐢 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐢) +𝑒 (𝐡 +𝑒 𝐷)))
 
Theoremxnn0add4d 13152 Rearrangement of 4 terms in a sum for extended addition of extended nonnegative integers, analogous to xadd4d 13151. (Contributed by AV, 12-Dec-2020.)
(πœ‘ β†’ 𝐴 ∈ β„•0*)    &   (πœ‘ β†’ 𝐡 ∈ β„•0*)    &   (πœ‘ β†’ 𝐢 ∈ β„•0*)    &   (πœ‘ β†’ 𝐷 ∈ β„•0*)    β‡’   (πœ‘ β†’ ((𝐴 +𝑒 𝐡) +𝑒 (𝐢 +𝑒 𝐷)) = ((𝐴 +𝑒 𝐢) +𝑒 (𝐡 +𝑒 𝐷)))
 
5.5.3  Supremum and infimum on the extended reals
 
Theoremxrsupexmnf 13153* Adding minus infinity to a set does not affect the existence of its supremum. (Contributed by NM, 26-Oct-2005.)
(βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ* (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)) β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ (𝐴 βˆͺ {-∞}) Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ* (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ (𝐴 βˆͺ {-∞})𝑦 < 𝑧)))
 
Theoremxrinfmexpnf 13154* Adding plus infinity to a set does not affect the existence of its infimum. (Contributed by NM, 19-Jan-2006.)
(βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ* (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)) β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ (𝐴 βˆͺ {+∞}) Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ* (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ (𝐴 βˆͺ {+∞})𝑧 < 𝑦)))
 
Theoremxrsupsslem 13155* Lemma for xrsupss 13157. (Contributed by NM, 25-Oct-2005.)
((𝐴 βŠ† ℝ* ∧ (𝐴 βŠ† ℝ ∨ +∞ ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ* (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
Theoremxrinfmsslem 13156* Lemma for xrinfmss 13158. (Contributed by NM, 19-Jan-2006.)
((𝐴 βŠ† ℝ* ∧ (𝐴 βŠ† ℝ ∨ -∞ ∈ 𝐴)) β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ* (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))
 
Theoremxrsupss 13157* Any subset of extended reals has a supremum. (Contributed by NM, 25-Oct-2005.)
(𝐴 βŠ† ℝ* β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ* (𝑦 < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦 < 𝑧)))
 
Theoremxrinfmss 13158* Any subset of extended reals has an infimum. (Contributed by NM, 25-Oct-2005.)
(𝐴 βŠ† ℝ* β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ 𝑦 < π‘₯ ∧ βˆ€π‘¦ ∈ ℝ* (π‘₯ < 𝑦 β†’ βˆƒπ‘§ ∈ 𝐴 𝑧 < 𝑦)))
 
Theoremxrinfmss2 13159* Any subset of extended reals has an infimum. (Contributed by Mario Carneiro, 16-Mar-2014.)
(𝐴 βŠ† ℝ* β†’ βˆƒπ‘₯ ∈ ℝ* (βˆ€π‘¦ ∈ 𝐴 Β¬ π‘₯β—‘ < 𝑦 ∧ βˆ€π‘¦ ∈ ℝ* (𝑦◑ < π‘₯ β†’ βˆƒπ‘§ ∈ 𝐴 𝑦◑ < 𝑧)))
 
Theoremxrub 13160* By quantifying only over reals, we can specify any extended real upper bound for any set of extended reals. (Contributed by NM, 9-Apr-2006.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (βˆ€π‘₯ ∈ ℝ (π‘₯ < 𝐡 β†’ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦) ↔ βˆ€π‘₯ ∈ ℝ* (π‘₯ < 𝐡 β†’ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦)))
 
Theoremsupxr 13161* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.) (Revised by Mario Carneiro, 21-Apr-2015.)
(((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (βˆ€π‘₯ ∈ 𝐴 Β¬ 𝐡 < π‘₯ ∧ βˆ€π‘₯ ∈ ℝ (π‘₯ < 𝐡 β†’ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦))) β†’ sup(𝐴, ℝ*, < ) = 𝐡)
 
Theoremsupxr2 13162* The supremum of a set of extended reals. (Contributed by NM, 9-Apr-2006.)
(((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) ∧ (βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ 𝐡 ∧ βˆ€π‘₯ ∈ ℝ (π‘₯ < 𝐡 β†’ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦))) β†’ sup(𝐴, ℝ*, < ) = 𝐡)
 
Theoremsupxrcl 13163 The supremum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 24-Oct-2005.)
(𝐴 βŠ† ℝ* β†’ sup(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoremsupxrun 13164 The supremum of the union of two sets of extended reals equals the largest of their suprema. (Contributed by NM, 19-Jan-2006.)
((𝐴 βŠ† ℝ* ∧ 𝐡 βŠ† ℝ* ∧ sup(𝐴, ℝ*, < ) ≀ sup(𝐡, ℝ*, < )) β†’ sup((𝐴 βˆͺ 𝐡), ℝ*, < ) = sup(𝐡, ℝ*, < ))
 
Theoremsupxrmnf 13165 Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ sup((𝐴 βˆͺ {-∞}), ℝ*, < ) = sup(𝐴, ℝ*, < ))
 
Theoremsupxrpnf 13166 The supremum of a set of extended reals containing plus infinity is plus infinity. (Contributed by NM, 15-Oct-2005.)
((𝐴 βŠ† ℝ* ∧ +∞ ∈ 𝐴) β†’ sup(𝐴, ℝ*, < ) = +∞)
 
Theoremsupxrunb1 13167* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremsupxrunb2 13168* The supremum of an unbounded-above set of extended reals is plus infinity. (Contributed by NM, 19-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ 𝐴 π‘₯ < 𝑦 ↔ sup(𝐴, ℝ*, < ) = +∞))
 
Theoremsupxrbnd1 13169* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 < π‘₯ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremsupxrbnd2 13170* The supremum of a bounded-above set of extended reals is less than infinity. (Contributed by NM, 30-Jan-2006.)
(𝐴 βŠ† ℝ* β†’ (βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremxrsup0 13171 The supremum of an empty set under the extended reals is minus infinity. (Contributed by NM, 15-Oct-2005.)
sup(βˆ…, ℝ*, < ) = -∞
 
Theoremsupxrub 13172 A member of a set of extended reals is less than or equal to the set's supremum. (Contributed by NM, 7-Feb-2006.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ 𝐴) β†’ 𝐡 ≀ sup(𝐴, ℝ*, < ))
 
Theoremsupxrlub 13173* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by Mario Carneiro, 13-Sep-2015.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐡 < sup(𝐴, ℝ*, < ) ↔ βˆƒπ‘₯ ∈ 𝐴 𝐡 < π‘₯))
 
Theoremsupxrleub 13174* The supremum of a set of extended reals is less than or equal to an upper bound. (Contributed by NM, 22-Feb-2006.) (Revised by Mario Carneiro, 6-Sep-2014.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (sup(𝐴, ℝ*, < ) ≀ 𝐡 ↔ βˆ€π‘₯ ∈ 𝐴 π‘₯ ≀ 𝐡))
 
Theoremsupxrre 13175* The real and extended real suprema match when the real supremum exists. (Contributed by NM, 18-Oct-2005.) (Proof shortened by Mario Carneiro, 7-Sep-2014.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 𝑦 ≀ π‘₯) β†’ sup(𝐴, ℝ*, < ) = sup(𝐴, ℝ, < ))
 
Theoremsupxrbnd 13176 The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ sup(𝐴, ℝ*, < ) < +∞) β†’ sup(𝐴, ℝ*, < ) ∈ ℝ)
 
Theoremsupxrgtmnf 13177 The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ -∞ < sup(𝐴, ℝ*, < ))
 
Theoremsupxrre1 13178 The supremum of a nonempty set of reals is real iff it is less than plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) < +∞))
 
Theoremsupxrre2 13179 The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ…) β†’ (sup(𝐴, ℝ*, < ) ∈ ℝ ↔ sup(𝐴, ℝ*, < ) β‰  +∞))
 
Theoremsupxrss 13180 Smaller sets of extended reals have smaller suprema. (Contributed by Mario Carneiro, 1-Apr-2015.)
((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† ℝ*) β†’ sup(𝐴, ℝ*, < ) ≀ sup(𝐡, ℝ*, < ))
 
Theoreminfxrcl 13181 The infimum of an arbitrary set of extended reals is an extended real. (Contributed by NM, 19-Jan-2006.) (Revised by AV, 5-Sep-2020.)
(𝐴 βŠ† ℝ* β†’ inf(𝐴, ℝ*, < ) ∈ ℝ*)
 
Theoreminfxrlb 13182 A member of a set of extended reals is greater than or equal to the set's infimum. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ 𝐴) β†’ inf(𝐴, ℝ*, < ) ≀ 𝐡)
 
Theoreminfxrgelb 13183* The infimum of a set of extended reals is greater than or equal to a lower bound. (Contributed by Mario Carneiro, 16-Mar-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ 𝐡 ∈ ℝ*) β†’ (𝐡 ≀ inf(𝐴, ℝ*, < ) ↔ βˆ€π‘₯ ∈ 𝐴 𝐡 ≀ π‘₯))
 
Theoreminfxrre 13184* The real and extended real infima match when the real infimum exists. (Contributed by Mario Carneiro, 7-Sep-2014.) (Revised by AV, 5-Sep-2020.)
((𝐴 βŠ† ℝ ∧ 𝐴 β‰  βˆ… ∧ βˆƒπ‘₯ ∈ ℝ βˆ€π‘¦ ∈ 𝐴 π‘₯ ≀ 𝑦) β†’ inf(𝐴, ℝ*, < ) = inf(𝐴, ℝ, < ))
 
Theoreminfxrmnf 13185 The infinimum of a set of extended reals containing minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.) (Revised by AV, 28-Sep-2020.)
((𝐴 βŠ† ℝ* ∧ -∞ ∈ 𝐴) β†’ inf(𝐴, ℝ*, < ) = -∞)
 
Theoremxrinf0 13186 The infimum of the empty set under the extended reals is positive infinity. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 5-Sep-2020.)
inf(βˆ…, ℝ*, < ) = +∞
 
Theoreminfxrss 13187 Larger sets of extended reals have smaller infima. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.)
((𝐴 βŠ† 𝐡 ∧ 𝐡 βŠ† ℝ*) β†’ inf(𝐡, ℝ*, < ) ≀ inf(𝐴, ℝ*, < ))
 
Theoremreltre 13188* For all real numbers there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ βˆƒπ‘¦ ∈ ℝ 𝑦 < π‘₯
 
Theoremrpltrp 13189* For all positive real numbers there is a smaller positive real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ+ βˆƒπ‘¦ ∈ ℝ+ 𝑦 < π‘₯
 
Theoremreltxrnmnf 13190* For all extended real numbers not being minus infinity there is a smaller real number. (Contributed by AV, 5-Sep-2020.)
βˆ€π‘₯ ∈ ℝ* (-∞ < π‘₯ β†’ βˆƒπ‘¦ ∈ ℝ 𝑦 < π‘₯)
 
Theoreminfmremnf 13191 The infimum of the reals is minus infinity. (Contributed by AV, 5-Sep-2020.)
inf(ℝ, ℝ*, < ) = -∞
 
Theoreminfmrp1 13192 The infimum of the positive reals is 0. (Contributed by AV, 5-Sep-2020.)
inf(ℝ+, ℝ, < ) = 0
 
5.5.4  Real number intervals
 
Syntaxcioo 13193 Extend class notation with the set of open intervals of extended reals.
class (,)
 
Syntaxcioc 13194 Extend class notation with the set of open-below, closed-above intervals of extended reals.
class (,]
 
Syntaxcico 13195 Extend class notation with the set of closed-below, open-above intervals of extended reals.
class [,)
 
Syntaxcicc 13196 Extend class notation with the set of closed intervals of extended reals.
class [,]
 
Definitiondf-ioo 13197* Define the set of open intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 < 𝑦)})
 
Definitiondf-ioc 13198* Define the set of open-below, closed-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
(,] = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ < 𝑧 ∧ 𝑧 ≀ 𝑦)})
 
Definitiondf-ico 13199* Define the set of closed-below, open-above intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,) = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ ≀ 𝑧 ∧ 𝑧 < 𝑦)})
 
Definitiondf-icc 13200* Define the set of closed intervals of extended reals. (Contributed by NM, 24-Dec-2006.)
[,] = (π‘₯ ∈ ℝ*, 𝑦 ∈ ℝ* ↦ {𝑧 ∈ ℝ* ∣ (π‘₯ ≀ 𝑧 ∧ 𝑧 ≀ 𝑦)})
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42500 426 42501-42600 427 42601-42700 428 42701-42800 429 42801-42900 430 42901-43000 431 43001-43100 432 43101-43200 433 43201-43300 434 43301-43400 435 43401-43500 436 43501-43600 437 43601-43700 438 43701-43800 439 43801-43900 440 43901-44000 441 44001-44100 442 44101-44200 443 44201-44300 444 44301-44400 445 44401-44500 446 44501-44600 447 44601-44700 448 44701-44800 449 44801-44900 450 44901-45000 451 45001-45100 452 45101-45200 453 45201-45300 454 45301-45400 455 45401-45500 456 45501-45600 457 45601-45700 458 45701-45800 459 45801-45900 460 45901-46000 461 46001-46100 462 46101-46200 463 46201-46300 464 46301-46400 465 46401-46500 466 46501-46600 467 46601-46700 468 46701-46800 469 46801-46900 470 46901-46966
  Copyright terms: Public domain < Previous  Next >