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Theorem List for Metamath Proof Explorer - 10401-10500   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmax0sub 10401 Decompose a real number into positive and negative parts. (Contributed by Mario Carneiro, 6-Aug-2014.)

Theoremifle 10402 An if statement transforms an implication into an inequality of terms. (Contributed by Mario Carneiro, 31-Aug-2014.)

Theoremz2ge 10403* There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005.)

Theoremqbtwnre 10404* The rational numbers are dense in : any two real numbers have a rational between them. Exercise 6 of [Apostol] p. 28. (Contributed by NM, 18-Nov-2004.) (Proof shortened by Mario Carneiro, 13-Jun-2014.)

Theoremqbtwnxr 10405* The rational numbers are dense in : any two extended real numbers have a rational between them. (Contributed by NM, 6-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.)

Theoremqsqueeze 10406* If a nonnegative real is less than any positive rational, it is zero. (Contributed by NM, 6-Feb-2007.)

Theoremqextltlem 10407* Lemma for qextlt 10408 and qextle . (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremqextlt 10408* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremqextle 10409* An extensionality-like property for extended real ordering. (Contributed by Mario Carneiro, 3-Oct-2014.)

Theoremxralrple 10410* Show that is less than by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremalrple 10411* Show that is less than by showing that there is no positive bound on the difference. (Contributed by Mario Carneiro, 12-Jun-2014.)

Theoremxnegeq 10412 Equality of two extended numbers with in front of them. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxnegex 10413 A negative extended real exists as a set. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegpnf 10414 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.)

Theoremxnegmnf 10415 Minus . Remark of [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Revised by Mario Carneiro, 20-Aug-2015.)

Theoremrexneg 10416 Minus a real number. Remark [BourbakiTop1] p. IV.15. (Contributed by FL, 26-Dec-2011.) (Proof shortened by Mario Carneiro, 20-Aug-2015.)

Theoremxneg0 10417 The negative of zero. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcl 10418 Closure of extended real negative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegneg 10419 Extended real version of negneg 8977. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxneg11 10420 Extended real version of neg11 8978. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltnegi 10421 Forward direction of xltneg 10422. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltneg 10422 Extended real version of ltneg 9154. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleneg 10423 Extended real version of leneg 9157. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg1 10424 Extended real version of lt0neg1 9160. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlt0neg2 10425 Extended real version of lt0neg2 9161. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxle0neg1 10426 Extended real version of le0neg1 9162. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxle0neg2 10427 Extended real version of le0neg2 9163. (Contributed by Mario Carneiro, 9-Sep-2015.)

Theoremxaddval 10428 Value of the extended real addition operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddf 10429 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulval 10430 Value of the extended real multiplication operation. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf1 10431 Addition of positive infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddpnf2 10432 Addition of positive infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf1 10433 Addition of negative infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddmnf2 10434 Addition of negative infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theorempnfaddmnf 10435 Addition of positive and negative infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremmnfaddpnf 10436 Addition of negative and positive infinity. This is often taken to be a "null" value or out of the domain, but we define it (somewhat arbitrarily) to be zero so that the resulting function is total, which simplifies proofs. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexadd 10437 The extended real addition operation when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexsub 10438 Extended real subtraction when both arguments are real. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddnemnf 10439 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddnepnf 10440 Closure of extended real addition in the subset . (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegid 10441 Extended real version of negid 8974. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcl 10442 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddcom 10443 The extended real addition operation is commutative. (Contributed by NM, 26-Dec-2011.)

Theoremxaddid1 10444 Extended real version of addid1 8872. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddid2 10445 Extended real version of addid2 8875. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegdi 10446 Extended real version of xnegdi 10446. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass 10447 Associativity of extended real addition. The correct condition here is "it is not the case that both and appear as one of , i.e. ", but this condition is difficult to work with, so we break the theorem into two parts: this one, where is not present in , and xaddass2 10448, where is not present. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxaddass2 10448 Associativity of extended real addition. See xaddass 10447 for notes on the hypotheses. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxpncan 10449 Extended real version of pncan 8937. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnpcan 10450 Extended real version of npcan 8940. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1a 10451 Extended real version of leadd1 9122; note that the converse implication is not true, unlike the real version (for example but ). (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd2a 10452 Commuted form of xleadd1a 10451. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxleadd1 10453 Weakened version of xleadd1a 10451 under which the reverse implication is true. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltadd1 10454 Extended real version of ltadd1 9121. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxltadd2 10455 Extended real version of ltadd2 8804. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxaddge0 10456 The sum of nonnegative extended reals is nonnegative. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxle2add 10457 Extended real version of le2add 9136. (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxlt2add 10458 Extended real version of lt2add 9139. Note that ltleadd 9137, which has weaker assumptions, is not true for the extended reals (since fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Theoremxsubge0 10459 Extended real version of subge0 9167. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxposdif 10460 Extended real version of posdif 9147. (Contributed by Mario Carneiro, 24-Aug-2015.)

Theoremxlesubadd 10461 Under certain conditions, the conclusion of lesubadd 9126 is true even in the extended reals. (Contributed by Mario Carneiro, 4-Sep-2015.)

Theoremxmullem 10462 Lemma for rexmul 10469. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmullem2 10463 Lemma for xmulneg1 10467. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulcom 10464 Extended real multiplication is commutative. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul01 10465 Extended real version of mul01 8871. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmul02 10466 Extended real version of mul02 8870. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg1 10467 Extended real version of mulneg1 9096. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulneg2 10468 Extended real version of mulneg2 9097. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremrexmul 10469 The extended real multiplication when both arguments are real. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulf 10470 The extended real multiplication operation is closed in extended reals. (Contributed by Mario Carneiro, 21-Aug-2015.)

Theoremxmulcl 10471 Closure of extended real multiplication. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1 10472 Multiplication by plus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf2 10473 Multiplication by plus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf1 10474 Multiplication by minus infinity on the right. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulmnf2 10475 Multiplication by minus infinity on the left. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulpnf1n 10476 Multiplication by plus infinity on the right, for negative input. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid1 10477 Extended real version of mulid1 8714. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulid2 10478 Extended real version of mulid2 8715. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulm1 10479 Extended real version of mulm1 9101. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem2 10480 Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulgt0 10481 Extended real version of mulgt0 8780. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulge0 10482 Extended real version of mulge0 9171. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem 10483* Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulasslem3 10484 Lemma for xmulass 10485. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxmulass 10485 Associativity of the extended real multiplication operation. Surprisingly, there are no restrictions on the values, unlike xaddass 10447 which has to avoid the "undefined" combinations and . The equivalent "undefined" expression here would be , but since this is defined to equal 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.)

Theoremxlemul1a 10486 Extended real version of lemul1a 9490. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2a 10487 Extended real version of lemul2a 9491. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxlemul1 10488 Extended real version of lemul1 9488. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxlemul2 10489 Extended real version of lemul2 9489. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul1 10490 Extended real version of ltmul1 9486. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxltmul2 10491 Extended real version of ltmul2 9487. (Contributed by Mario Carneiro, 8-Sep-2015.)

Theoremxadddi 10493 Distributive property for extended real addition and multiplication. Like xaddass 10447, this has an unusual domain of correctness due to counterexamples like . In this theorem we show that if the multiplier is real then everything works as expected. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddir 10494 Commuted version of xadddi 10493. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2 10495 The assumption that the multiplier be real in xadddi 10493 can be relaxed if the addends have the same sign. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxadddi2r 10496 Commuted version of xadddi2 10495. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremx2times 10497 Extended real version of 2times 9722. (Contributed by Mario Carneiro, 20-Aug-2015.)

Theoremxnegcld 10498 Closure of extended real negative. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxaddcld 10499 The extended real addition operation is closed in extended reals. (Contributed by Mario Carneiro, 28-May-2016.)

Theoremxmulcld 10500 Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016.)

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