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Theorem List for Metamath Proof Explorer - 26101-26200   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremordcmp 26101 An ordinal topology is compact iff the underlying set is its supremum (union) only when the ordinal is . (Contributed by Chen-Pang He, 1-Nov-2015.)

Theoremssoninhaus 26102 The ordinal topologies and are Hausdorff. (Contributed by Chen-Pang He, 10-Nov-2015.)

Theoremonint1 26103 The ordinal T1 spaces are and , proven without the Axiom of Regularity. (Contributed by Chen-Pang He, 9-Nov-2015.)

Theoremoninhaus 26104 The ordinal Hausdorff spaces are and . (Contributed by Chen-Pang He, 10-Nov-2015.)

19.10  Mathbox for Jeff Hoffman

19.10.1  Inferences for finite induction on generic function values

Theoremfindabrcl 26108* Please add description here. (Contributed by Jeff Hoffman, 16-Feb-2008.) (Revised by Mario Carneiro, 11-Sep-2015.)

19.10.2  gdc.mm

Theoremnnssi2 26109 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremnnssi3 26110 Convert a theorem for real/complex numbers into one for natural numbers. (Contributed by Jeff Hoffman, 17-Jun-2008.)

Theoremnndivlub 26112 A factor of a natural number cannot exceed it. (Contributed by Jeff Hoffman, 17-Jun-2008.)

SyntaxcgcdOLD 26113 Extend class notation to include the gdc function. (New usage is discouraged.)

Definitiondf-gcdOLD 26114* is the largest natural number that evenly divides both and . (Contributed by Jeff Hoffman, 17-Jun-2008.) (New usage is discouraged.)

Theoremee7.2aOLD 26115 Lemma for Euclid's Elements, Book 7, proposition 2. The original mentions the smaller measure being 'continually subtracted' from the larger. Many authors interpret this phrase as mod . Here, just one subtraction step is proved to preserve the . The function will be used in other proofs for iterated subtraction. (Contributed by Jeff Hoffman, 17-Jun-2008.) (Proof modification is discouraged.) (New usage is discouraged.)

19.11  Mathbox for Wolf Lammen

Most of the theorems in the section "Logical implication" are about handling chains of implications: . With respect to chains, a rich set of rules clarify

- how to swap antecedents (com12, ...);

- how to drop antecedents (ax-mp, pm2.43, ...);

- how to add antecedents (a1i, ...)

- how to replace an antecedent (syl, ...);

- how to replace a consequent (ax-mp, syl, ...);

- what is, when an antecedent equals the consequent (ax-1, id, ...).

In all these cases, the operands of the chain have no inner structure, or it is of no importance. These chains are called "simple" here.

There is less support, when the operands are structured themselves. Some kinds of inner structure involving the operator are best handled by the symmetric operators and . But a nested, simple chain has no such convenient replacement. I can focus on antecedents here, since a consequent representing a chain is, in conjunction with its antecedents, just an extended simple chain again.

The following theorems show, how operations on nested chains appear somehow mirrored: The minor premises of the syllogisms look reverted, in comparison to their normal counterparts, and while adding an antecedent to a chain via a1i 11 is easy, in nested chains they can be easily dropped.

Theoremwl-jarri 26116 Dropping a nested antecedent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2212 is an instance of this idea.

Theoremwl-jarli 26117 Dropping a nested consequent. This theorem is one of two reversions of ja 155. Since ja 155 is reversible, one can conclude, that a nested (chain of) implication(s) is just a packed notation of two or more theorems/ hypotheses with a common consequent. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

ax46 2212 is an instance of this idea.

Theoremwl-mps 26118 Replacing a nested consequent. A sort of modus ponens in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-syls1 26119 Replacing a nested consequent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-syls2 26120 Replacing a nested antecedent. A sort of syllogism in antecedent position. (Contributed by Wolf Lammen, 20-Sep-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestant 26121 A true wff can always be added as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 26122) (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestantALT 26122 Proof of wl-adnestant 26121 not based on ax-3 7. (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-adnestantd 26123 Deduction version of wl-adnestant 26121. Generalization of a2i 13, imim12i 55, imim1i 56 and imim2i 14, which can be proved by specializing its hypotheses, and some trivial rearrangements. This theorem clarifies in a more general way, under what conditions a wff may be introduced as a nested antecedent to an antecedent. Note: this theorem is intuitionistically valid (see wl-adnestantALT 26122). (Contributed by Wolf Lammen, 4-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitr1 26124 Closed form of bitri 241. Place before bitri 241. [ +33] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitri 26125 An inference from transitive law for logical equivalence. [ -5] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitrd 26126 Deduction form of bitri 241. [ -7] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1 26127 Theorem *4.86 of [WhiteheadRussell] p. 122. Place this (and the following theorems) after bitr1. [ +22] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1i 26128 Inference adding a biconditional to the right in an equivalence. Move after bibi1. [ -8] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi1d 26129 Deduction adding a biconditional to the right in an equivalence. Move after bibi1i. [ -9] (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bibi2d 26130 Deduction adding a biconditional to the left in an equivalence. Move after bibi1d. [ -25] (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.74lem 26131 Moving a common antecedent on one side of an equivalence. Place before pm5.74 236. [ +25] (Contributed by Wolf Lammen, 5-Oct-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.74 26132 Distribution of implication over biconditional. Theorem *5.74 of [ WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.) Replace and move biimt 326.. albi 1570 before it. [ -22] (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm5.32 26133 Distribution of implication over biconditional. Theorem *5.32 of [ WhiteheadRussell] p. 125. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 5-Oct-2013.) Replace. [ -43] (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-bitr 26134 Theorem *4.22 of [WhiteheadRussell] p. 117. Replace. [ -4] (Contributed by NM, 3-Jan-2005.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-pm2.86i 26135 Inference based on pm2.86 96. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Apr-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Theoremwl-dedlem0a 26136 Lemma for an alternate version of weak deduction theorem. (Contributed by NM, 2-Apr-1994.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 4-Dec-2012.) (Proof modification is discouraged.) (New usage is discouraged.)

19.12  Mathbox for Brendan Leahy

Theoremsupaddc 26137* The supremum function distributes over addition in a sense similar to that in supmul1 9929. (Contributed by Brendan Leahy, 25-Sep-2017.)

Theoremsupadd 26138* The supremum function distributes over addition in a sense similar to that in supmul 9932. (Contributed by Brendan Leahy, 26-Sep-2017.)

Theoremrabiun2 26139* Abstraction restricted to an indexed union. (Contributed by Brendan Leahy, 26-Oct-2017.)

Theoremltflcei 26140 Theorem to move the floor function across a strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)

Theoremleceifl 26141 Theorem to move the floor function across a non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)

Theoremlxflflp1 26142 Theorem to move floor function between strict and non-strict inequality. (Contributed by Brendan Leahy, 25-Oct-2017.)

Theoremmblfinlem 26143* Lemma for ismblfin 26146, effectively one direction of the same fact for open sets, made necessary by Viaclovsky's slightly different defintion of outer measure. Note that unlike the main theorem, this holds for sets of infinite measure. (Contributed by Brendan Leahy, 21-Feb-2018.)

Theoremmblfinlem2 26144* The difference between two sets measurable by the criterion in ismblfin 26146 is itself measurable by the same. Proposition 0.3 of [Viaclovsky7] p. 3. (Contributed by Brendan Leahy, 25-Mar-2018.)

Theoremmblfinlem3 26145* Backward direction of ismblfin 26146. (Contributed by Brendan Leahy, 28-Mar-2018.)

Theoremismblfin 26146* Measurability in terms of inner and outer measure. Proposition 7 of [Viaclovsky8] p. 3. (Contributed by Brendan Leahy, 4-Mar-2018.) (Revised by Brendan Leahy, 28-Mar-2018.)

Theoremovoliunnfl 26147* ovoliun 19354 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 21-Nov-2017.)

Theoremex-ovoliunnfl 26148* Demonstration of ovoliunnfl 26147. (Contributed by Brendan Leahy, 21-Nov-2017.)

Theoremvoliunnfl 26149* voliun 19401 is incompatible with the Feferman-Levy model; in that model, therefore, the Lebesgue measure as we've defined it isn't actually a measure. (Contributed by Brendan Leahy, 16-Dec-2017.)
Disj

Theoremvolsupnfl 26150* volsup 19403 is incompatible with the Feferman-Levy model. (Contributed by Brendan Leahy, 2-Jan-2018.)

Theorem0mbf 26151 The empty function is measurable. (Contributed by Brendan Leahy, 28-Mar-2018.)
MblFn

Theoremmbfresfi 26152* Measurability of a piecewise function across arbitrarily many subsets. (Contributed by Brendan Leahy, 31-Mar-2018.)
MblFn              MblFn

Theoremmbfposadd 26153* If the sum of two measurable functions is measurable, the sum of their nonnegative parts is measurable. (Contributed by Brendan Leahy, 2-Apr-2018.)
MblFn              MblFn              MblFn       MblFn

Theoremcnambfre 26154 A real-valued, a.e. continuous function is measurable. (Contributed by Brendan Leahy, 4-Apr-2018.)
t MblFn

Theoremitg2addnclem 26155* An alternate expression for the integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017.) (Revised by Brendan Leahy, 10-Mar-2018.)

Theoremitg2addnclem2 26156* Lemma for itg2addnc 26158. The function described is a simple function. (Contributed by Brendan Leahy, 29-Oct-2017.)
MblFn

Theoremitg2addnclem3 26157* Lemma incomprehensible in isolation split off to shorten proof of itg2addnc 26158. (Contributed by Brendan Leahy, 11-Mar-2018.)
MblFn

Theoremitg2addnc 26158 Alternate proof of itg2add 19604 using the "buffer zone" definition from the first lemma, in which every simple function in the set is divided into to by dividing its buffer by a third and finding the largest allowable function locked to a grid laid out in increments of the new, smaller buffer up to the original simple function. The measurability of this function follows from that of the augend, and subtracting it from the original simple function yields another simple function by i1fsub 19553, which is allowable by the fact that the grid must have a mark between one third and two thirds the original buffer. This has two advantages over the current approach: first, eliminating ax-cc 8271, and second, weakening the measurability hypothesis to only the augend. (Contributed by Brendan Leahy, 31-Oct-2017.) (Revised by Brendan Leahy, 13-Mar-2018.)
MblFn

Theoremitg2gt0cn 26159* itg2gt0 19605 holds on functions continuous on an open interval in the absence of ax-cc 8271. The fourth hypothesis is made unnecessary by the continuity hypothesis. (Contributed by Brendan Leahy, 16-Nov-2017.)

Theoremibladdnclem 26160* Lemma for ibladdnc 26161; cf ibladdlem 19664, whose fifth hypothesis is rendered unnecessary by the weakened hypotheses of itg2addnc 26158. (Contributed by Brendan Leahy, 31-Oct-2017.)
MblFn

Theoremibladdnc 26161* Choice-free analogue of itgadd 19669. A measurability hypothesis is necessitated by the loss of mbfadd 19506; for large classes of functions, such as continuous functions, it should be relatively easy to show. (Contributed by Brendan Leahy, 1-Nov-2017.)
MblFn

MblFn

Theoremitgaddnclem2 26163* Lemma for itgaddnc 26164; cf. itgaddlem2 19668. (Contributed by Brendan Leahy, 10-Nov-2017.) (Revised by Brendan Leahy, 3-Apr-2018.)
MblFn

Theoremitgaddnc 26164* Choice-free analogue of itgadd 19669. (Contributed by Brendan Leahy, 11-Nov-2017.)
MblFn

Theoremiblsubnc 26165* Choice-free analogue of iblsub 19666. (Contributed by Brendan Leahy, 11-Nov-2017.)
MblFn

Theoremitgsubnc 26166* Choice-free analogue of itgsub 19670. (Contributed by Brendan Leahy, 11-Nov-2017.)
MblFn

Theoremiblabsnclem 26167* Lemma for iblabsnc 26168; cf. iblabslem 19672. (Contributed by Brendan Leahy, 7-Nov-2017.)
MblFn

Theoremiblabsnc 26168* Choice-free analogue of iblabs 19673. As with ibladdnc 26161, a measurability hypothesis is needed. (Contributed by Brendan Leahy, 7-Nov-2017.)
MblFn

Theoremiblmulc2nc 26169* Choice-free analogue of iblmulc2 19675. (Contributed by Brendan Leahy, 17-Nov-2017.)
MblFn

Theoremitgmulc2nclem1 26170* Lemma for itgmulc2nc 26172; cf. itgmulc2lem1 19676. (Contributed by Brendan Leahy, 17-Nov-2017.)
MblFn

Theoremitgmulc2nclem2 26171* Lemma for itgmulc2nc 26172; cf. itgmulc2lem2 19677. (Contributed by Brendan Leahy, 19-Nov-2017.)
MblFn

Theoremitgmulc2nc 26172* Choice-free analogue of itgmulc2 19678. (Contributed by Brendan Leahy, 19-Nov-2017.)
MblFn

Theoremitgabsnc 26173* Choice-free analogue of itgabs 19679. (Contributed by Brendan Leahy, 19-Nov-2017.)
MblFn       MblFn

Theorembddiblnc 26174* Choice-free proof of bddibl 19684. (Contributed by Brendan Leahy, 2-Nov-2017.) (Revised by Brendan Leahy, 6-Nov-2017.)
MblFn

Theoremcnicciblnc 26175 Choice-free proof of cniccibl 19685. (Contributed by Brendan Leahy, 2-Nov-2017.)

Theoremitggt0cn 26176* itggt0 19686 holds for continuous functions in the absence of ax-cc 8271. (Contributed by Brendan Leahy, 16-Nov-2017.)

Theoremftc1cnnclem 26177* Lemma for ftc1cnnc 26178; cf. ftc1lem4 19876. The stronger assumptions of ftc1cn 19880 are exploited to make use of weaker theorems. (Contributed by Brendan Leahy, 19-Nov-2017.)

Theoremftc1cnnc 26178* Choice-free proof of ftc1cn 19880. (Contributed by Brendan Leahy, 20-Nov-2017.)

Theoremdvreasin 26179 Real derivative of arcsine. (Contributed by Brendan Leahy, 3-Aug-2017.)
arcsin

Theoremdvreacos 26180 Real derivative of arccosine. (Contributed by Brendan Leahy, 3-Aug-2017.)
arccos

Theoremareacirclem2 26181* Antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)
arcsin

Theoremareacirclem3 26182* Continuity of cross-section of circle. (Contributed by Brendan Leahy, 28-Aug-2017.)

Theoremareacirclem4 26183* Endpoint-inclusive continuity of Cartesian ordinate of circle. (Contributed by Brendan Leahy, 29-Aug-2017.)

Theoremareacirclem1 26184* Integrability of cross-section of circle. (Contributed by Brendan Leahy, 26-Aug-2017.)

Theoremareacirclem5 26185* Endpoint-inclusive continuity of antiderivative of cross-section of circle. (Contributed by Brendan Leahy, 31-Aug-2017.)
arcsin

Theoremareacirclem6 26186* Finding the cross-section of a circle. (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)

Theoremareacirc 26187* The area of a circle of radius is . (Contributed by Brendan Leahy, 31-Aug-2017.) (Revised by Brendan Leahy, 22-Sep-2017.)
area

19.13  Mathbox for Jeff Hankins

19.13.1  Miscellany

Theorema1i13 26188 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i4 26189 Add an antecedent to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i14 26190 Add two antecedents to a wff. (Contributed by Jeff Hankins, 4-Aug-2009.)

Theorema1i24 26191 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)

Theorema1i34 26192 Add two antecedents to a wff. (Contributed by Jeff Hankins, 5-Aug-2009.)

Theoremexp5d 26193 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5g 26194 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5j 26195 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5k 26196 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp5l 26197 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp56 26198 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp58 26199 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

Theoremexp510 26200 An exportation inference. (Contributed by Jeff Hankins, 7-Jul-2009.)

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