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Theorem List for Metamath Proof Explorer - 28501-28600   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremsgnp 28501 Proof that signum of positive extended real is 1. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnrrp 28502 Proof that signum of positive reals is 1. (Contributed by David A. Wheeler, 18-May-2015.)
sgn

Theoremsgn1 28503 Proof that the signum of 1 is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theoremsgnpnf 28504 Proof that the signum of is 1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

Theoremsgnn 28505 Proof that signum of negative extended real is -1. (Contributed by David A. Wheeler, 15-May-2015.)
sgn

Theoremsgnmnf 28506 Proof that the signum of is -1. (Contributed by David A. Wheeler, 26-Jun-2016.)
sgn

18.24.9  Ceiling function

Syntaxccei 28507 Extend class notation to include the ceiling function.

Definitiondf-ceiling 28508 The ceiling function. Defined in ISO 80000-2:2009(E) operation 2-9.18 and the "NIST Digital Library of Mathematical Functions" , front introduction, "Common Notations and Definitions" section at http://dlmf.nist.gov/front/introduction#Sx4.

By convention metamath users tend to expand this construct directly, instead of using the definition. However, we want to make sure this is separately and formally defined. Proof ceicl 10971 shows that the ceiling function returns an integer when provided a real. Formalized by David A. Wheeler. (Contributed by David A. Wheeler, 19-May-2015.)

Theoremceilingval 28509 The value of the ceiling function. (Contributed by David A. Wheeler, 19-May-2015.)

Theoremceilingcl 28510 Closure of the ceiling function; the real work is in ceicl 10971. (Contributed by David A. Wheeler, 19-May-2015.)

18.24.10  Logarithms generalized to arbitrary base using ` logb `

Theoremene0 28511 is not 0. (Contributed by David A. Wheeler, 17-Oct-2017.)

Theoremene1 28512 is not 1. (Contributed by David A. Wheeler, 17-Oct-2017.)

Theoremelogb 28513 Using as the base is the same as . (Contributed by David A. Wheeler, 17-Oct-2017.)
logb

18.24.11  Logarithm laws generalized to an arbitrary base - log_

Define "log using an arbitrary base" function and then prove some of its properties. This builds on previous work by Stefan O'Rear.

This supports the notational form log_; that looks a little more like traditional notation, but is different from other 2-parameter functions. E.G., log_;;

This form is less convenient to work with inside metamath as compared to the logb form defined separately.

Syntaxclog_ 28514 Extend class notation to include the logarithm generalized to an arbitrary base.
log_

Definitiondf-log_ 28515* Define the log_ operator. This is the logarithm generalized to an arbitrary base. It can be used as log_ for "log base B of X". This formulation suggested by Mario Carneiro. (Contributed by David A. Wheeler, 14-Jul-2017.)
log_

18.24.12  Miscellaneous

Miscellaneous proofs.

Theorem5m4e1 28516 Prove that 5 - 4 = 1. (Contributed by David A. Wheeler, 31-Jan-2017.)

Theorem2p2ne5 28517 Prove that . In George Orwell's "1984", Part One, Chapter Seven, the protagonist Winston notes that, "In the end the Party would announce that two and two made five, and you would have to believe it." http://www.sparknotes.com/lit/1984/section4.rhtml. More generally, the phrase has come to represent an obviously false dogma one may be required to believe. See the Wikipedia article for more about this: https://en.wikipedia.org/wiki/2_%2B_2_%3D_5. Unsurprisingly, we can easily prove that this claim is false. (Contributed by David A. Wheeler, 31-Jan-2017.)

Theoremresolution 28518 Resolution rule. This is the primary inference rule in some automated theorem provers such as prover9. The resolution rule can be traced back to Davis and Putnam (1960). (Contributed by David A. Wheeler, 9-Feb-2017.)

18.25  Mathbox for Alan Sare

Theoremad4ant13 28519 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant14 28520 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant123 28521 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant124 28522 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant134 28523 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant23 28524 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant24 28525 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad4ant234 28526 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant12 28527 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant13 28528 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant14 28529 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant15 28530 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant23 28531 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant24 28532 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant25 28533 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant245 28534 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant234 28535 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant235 28536 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant123 28537 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant124 28538 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant125 28539 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant134 28540 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant135 28541 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant145 28542 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant1345 28543 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

Theoremad5ant2345 28544 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)

18.25.2  Supplementary unification deductions

Theorembiimp 28545 Importation inference similar to imp 418, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi2imp 28546 Importation inference similar to imp 418, except the both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi3impb 28547 Similar to 3impb 1147 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi3impa 28548 Similar to 3impa 1146 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi23impib 28549 3impib 1149 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi13impib 28550 3impib 1149 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi123impib 28551 3impib 1149 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi13impia 28552 3impia 1148 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi123impia 28553 3impia 1148 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi33imp12 28554 3imp 1145 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi23imp13 28555 3imp 1145 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi13imp23 28556 3imp 1145 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi13imp2 28557 Similar to 3imp 1145 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi12imp3 28558 Similar to 3imp 1145 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi23imp1 28559 Similar to 3imp 1145 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

Theorembi123imp0 28560 Similar to 3imp 1145 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)

18.25.3  Conventional Metamath proofs, some derived from VD proofs

Theoremiidn3 28561 idn3 28692 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee222 28562 e222 28713 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee3bir 28563 Right-biconditional form of e3 28826 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee13 28564 e13 28837 without virtual deduction connectives. Special theorem needed for Alan Sare's virtual deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee121 28565 e121 28733 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee122 28566 e122 28730 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee333 28567 e333 28822 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee323 28568 e323 28855 without virtual deductions. (Contributed by Alan Sare, 17-Apr-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3ornot23 28569 If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 28939. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremorbi1r 28570 orbi1 686 with order of disjuncts reversed. Derived from orbi1rVD 28940. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorembitr3 28571 Closed nested implication form of bitr3i 242. Derived automatically from bitr3VD 28941. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3orbi123 28572 pm4.39 841 with a 3-conjunct antecedent. This proof is 3orbi123VD 28942 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsyl5imp 28573 Closed form of syl5 28. Derived automatically from syl5impVD 28955. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimpexp3a 28574 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. After the User's Proof was completed, it was minimized. The completed User's Proof before minimization is not shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: qed:1:

Theoremcom3rgbi 28575 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: 2:: 3:1,2: 4:: 5:: 6:4,5: qed:3,6:

Theoremimpexp3acom3r 28576 The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
 1:: 2:: qed:1,2:

Theoremee1111 28577 Non-virtual deduction form of e1111 28752. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: h4:: h5:: 6:1,5: 7:6: 8:2,7: 9:8: 10:9: 11:3,10: 12:11: 13:12: 14:4,13: qed:14:

Theorempm2.43bgbi 28578 Logical equivalence of a 2-left-nested implication and a 1-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theorempm2.43cbi 28579 Logical equivalence of a 3-left-nested implication and a 2-left-nested implicated when two antecedents of the former implication are identical. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 1:: 2:: 3:1,2: 4:: 5:3,4: 6:: qed:5,6:

Theoremee233 28580 Non-virtual deduction form of e233 28854. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: h4:: 5:1,4: 6:5: 7:2,6: 8:7: 9:8: 10:9: 11:10: 12:3,11: 13:12: 14:13: qed:14:

Theoremimbi12 28581 Implication form of imbi12i 316. imbi12 28581 is imbi12VD 28965 without virtual deductions and was automatically derived from imbi12VD 28965 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremimbi13 28582 Join three logical equivalences to form equivalence of implications. imbi13 28582 is imbi13VD 28966 without virtual deductions and was automatically derived from imbi13VD 28966 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremee33 28583 Non-virtual deduction form of e33 28823. (Contributed by Alan Sare, 18-Mar-2012.) (Proof modification is discouraged.) (New usage is discouraged.) The following User's Proof is a Virtual Deduction proof completed automatically by the tools program completeusersproof.cmd, which invokes Mel O'Cat's mmj2 and Norm Megill's Metamath Proof Assistant. The completed Virtual Deduction Proof (not shown) was minimized. The minimized proof is shown.
 h1:: h2:: h3:: 4:1,3: 5:4: 6:2,5: 7:6: 8:7: qed:8:

Theoremcon5 28584 Bi-conditional contraposition variation. This proof is con5VD 28992 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon5i 28585 Inference form of con5 28584. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremexlimexi 28586 Inference similar to Theorem 19.23 of [Margaris] p. 90. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsb5ALT 28587* Equivalence for substitution. Alternate proof of sb5 2052. This proof is sb5ALTVD 29005 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst01 28588 exinst01 28702 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremeexinst11 28589 exinst11 28703 without virtual deductions. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremvk15.4j 28590 Excercise 4j of Unit 15 of "Understanding Symbolic Logic", Fifth Edition (2008), by Virginia Klenk. This proof is the minimized Hilbert-style axiomatic version of the Fitch-style Natural Deduction proof found on page 442 of Klenk and was automatically derived from that proof. vk15.4j 28590 is vk15.4jVD 29006 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremnotnot2ALT 28591 Converse of double negation. Alternate proof of notnot2 104. This proof is notnot2ALTVD 29007 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremcon3ALT 28592 Contraposition. Alternate proof of con3 126. This proof is con3ALTVD 29008 automatically translated and minimized. (Contributed by Alan Sare, 21-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremssralv2 28593* Quantification restricted to a subclass for two quantifiers. ssralv 3250 for two quantifiers. The proof of ssralv2 28593 was automatically generated by minimizing the automatically translated proof of ssralv2VD 28958. The automatic translation is by the tools program translatewithout_overwriting.cmd (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbc3org 28594 sbcorg 3049 with a 3-disjuncts. This proof is sbc3orgVD 28943 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremalrim3con13v 28595* Closed form of alrimi 1757 with 2 additional conjuncts having no occurences of the quantifying variable. This proof is 19.21a3con13vVD 28944 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremrspsbc2 28596* rspsbc 3082 with two quantifying variables. This proof is rspsbc2VD 28947 automatically translated and minimized. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremsbcoreleleq 28597* Substitution of a set variable for another set variable in a 3-conjunct formula. Derived automatically from sbcoreleleqVD 28951. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremtratrb 28598* If a class is transitive and any two distinct elements of the class are E-comparable, then every element of that class is transitive. Derived automatically from tratrbVD 28953. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

Theorem3ax5 28599 ax-5 1547 for a 3 element left-nested implication. Derived automatically from 3ax5VD 28954. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)

TheoremordelordALT 28600 An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. This is an alternate proof of ordelord 4430 using the Axiom of Regularity indirectly through dford2 7337. dford2 is a weaker definition of ordinal number. Given the Axiom of Regularity, it need not be assumed that because this is inferred by the Axiom of Regularity. ordelordALT 28600 is ordelordALTVD 28959 without virtual deductions and was automatically derived from ordelordALTVD 28959 using the tools program translate..without..overwriting.cmd and Metamath's minimize command. (Contributed by Alan Sare, 18-Feb-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

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