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Type | Label | Description |
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Statement | ||
Theorem | xadd0ge2 37601 | A number is less than or equal to itself plus a nonnegative extended real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | nepnfltpnf 37602 |
An extended real that is not ![]() ![]() |
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Theorem | ltadd12dd 37603 | Addition to both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | nemnftgtmnft 37604 | An extended real that is not minus infinity, is larger than minus infinity. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | xrgtso 37605 | 'Greater than' is a strict ordering on the extended reals. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | rpex 37606 | The positive reals form a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | xrge0ge0 37607 | A nonnegative extended real is nonnegative. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | xrssre 37608 |
A subset of extended reals that does not contain ![]() ![]() |
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Theorem | ssuzfz 37609 | A finite subset of the upper integers is a subset of a finite set of sequential integers. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | absfun 37610 | The absolute value is a function. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | infrpge 37611* | The infimum of a non empty, bounded subset of extended reals can be approximated from above by an element of the set. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | xrlexaddrp 37612* |
If an extended real number ![]() ![]() ![]() ![]() |
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Theorem | supsubc 37613* | The supremum function distributes over subtraction in a sense similar to that in supaddc 10601. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | xralrple2 37614* |
Show that ![]() ![]() |
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Theorem | nnuzdisj 37615 |
The first ![]() |
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Theorem | ltdivgt1 37616 | Divsion by a number greater than 1. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | xrltned 37617 | 'Less than' implies not equal. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | nnsplit 37618 |
Express the set of positive integers as the disjoint (see nnuzdisj 37615)
union of the first ![]() |
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Theorem | divdiv3d 37619 | Division into a fraction. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | abslt2sqd 37620 | Comparison of the square of two numbers. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | rexaddd 37621 | The extended real addition operation when both arguments are real. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | qenom 37622 | The set of rational numbers is equinumerous to omega (the set of finite ordinal numbers). (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | qct 37623 | The set of rational numbers is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | gtnelioc 37624 | A real number larger than the upper bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioossioc 37625 | An open interval is a subset of its right closure. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioondisj2 37626 | A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioondisj1 37627 | A condition for two open intervals not to be disjoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioosscn 37628 | An open interval is a set of complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioogtlb 37629 | An element of a closed interval is greater than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | evthiccabs 37630* | Extreme Value Theorem on y closed interval, for the absolute value of y continuous function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltnelicc 37631 | A real number smaller than the lower bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliood 37632 | Membership in an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iooabslt 37633 | An upper bound for the distance from the center of an open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | gtnelicc 37634 | A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iooinlbub 37635 | An open interval has empty intersection with its bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iocgtlb 37636 | An element of a left open right closed interval is larger than its lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iocleub 37637 | An element of a left open right closed interval is smaller or equal to its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliccd 37638 | Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccssred 37639 | A closed real interval is a set of reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliccre 37640 | A member of a closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliooshift 37641 | Element of an open interval shifted by a displacement. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliocd 37642 | Membership in a left open, right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | snunioo2 37643 | The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | icoltub 37644 | An element of a left closed right open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | tgiooss 37645 | The restriction of the complex topology to a subset of reals, is a restriction of the standard topology on reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): Use rerest 21870 instead in fourierdlem48 38055, fourierdlem49 38056, fourierdlem62 38069 then delete this. |
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Theorem | eliocre 37646 | A member of a left open, right closed interval of reals is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iooltub 37647 | An element of an open interval is less than its upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioontr 37648 |
The interior of an interval in the standard topology on ![]() |
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Theorem | eliccxr 37649 | A member of a closed interval is a an extended real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | snunioo1 37650 | The closure of one end of an open real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | lbioc 37651 | An left open right closed interval doesn't contain its left endpoint. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioomidp 37652 | The midpoint is an element of the open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccdifioo 37653 | If the open inverval is removed from the closed interval, only the bounds are left. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccdifprioo 37654 | An open interval is the closed interval without the bounds. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ioossioobi 37655 | Biconditional form of ioossioo 11754. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccshift 37656* | A closed interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccsuble 37657 | An upper bound to the distance of two elements in a closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iocopn 37658 | A left open right closed interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | eliccelioc 37659 | Membership in a closed interval and in a left open right closed interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iooshift 37660* | An open interval shifted by a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | iccintsng 37661 | Intersection of two adiacent closed intervals is a singleton. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | icoiccdif 37662 | Left closed, right open interval gotten by a closed iterval taking away the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | icoopn 37663 | A left closed right open interval is an open set of the standard topology restricted to an interval that contains the original interval and has the same lower bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | icoub 37664 | A left-closed, right-open interval does not contain its upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | eliccxrd 37665 | Membership in a closed real interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | pnfel0pnf 37666 |
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Theorem | ge0nemnf2 37667 |
A nonnegative extended real is not ![]() |
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Theorem | eliccnelico 37668 | An element of a closed interval that is not a member of the left closed right open interval, must be the upper bound. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | eliccelicod 37669 | A member of a closed interval that is not the upper bound, is a member of the left-closed, right-open interval. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | ge0xrre 37670 |
A nonnegative extended real that is not ![]() |
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Theorem | ge0lere 37671 | A nonnegative extended Real number smaller than or equal to a Real number, is a Real number. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | elicores 37672* | Membership in a left-closed, right-open interval with real bounds. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | inficc 37673 | The infimum of a nonempty set, included in a closed interval, is a member of the interval. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | qinioo 37674 |
The rational numbers are dense in ![]() |
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Theorem | lenelioc 37675 | A real number smaller than or equal to the lower bound of a left open right closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | ioonct 37676 | C non empty open interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | xrgtnelicc 37677 | A real number greater than the upper bound of a closed interval is not an element of the interval. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | iccdificc 37678 | The difference of two closed intervals with the same lower bound. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | iocnct 37679 | A non empty left-open, right-closed interval is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | iccnct 37680 | A closed interval, with more than one element is uncountable. (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
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Theorem | sumeq2ad 37681* | Equality deduction for sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsumclf 37682* |
Closure of a finite sum of complex numbers ![]() ![]() ![]() ![]() |
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Theorem | fsumsplitf 37683* | Split a sum into two parts. A version of fsumsplit 13854 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsummulc1f 37684* |
Closure of a finite sum of complex numbers ![]() ![]() ![]() ![]() |
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Theorem | sumsnf 37685* | A sum of a singleton is the term. A version of sumsn 13855 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsumsplitsn 37686* | Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsumnncl 37687* | Closure of a non empty, finite sum of positive integers. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsumsplit1 37688* | Separate out a term in a finite sum. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
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Theorem | fsumge0cl 37689* | The finite sum of nonnegative reals is a nonnegative real. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fsumf1of 37690* | Re-index a finite sum using a bijection. Same as fsumf1o 13837, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fsumiunss 37691* |
Sum over a disjoint indexed union, intersected with a finite set
![]() ![]() ![]() |
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Theorem | fsumreclf 37692* | Closure of a finite sum of reals. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | fsumlessf 37693* | A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | fsumsupp0 37694* | Finite sum of function values, for a function of finite support. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | fmul01 37695* | Multiplying a finite number of values in [ 0 , 1 ] , gives the final product itself a number in [ 0 , 1 ]. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
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Theorem | fmulcl 37696* | If ' Y ' is closed under the multiplication of two functions, then Y is closed under the multiplication ( ' X ' ) of a finite number of functions. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
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Theorem | fmuldfeqlem1 37697* | induction step for the proof of fmuldfeq 37698. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
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Theorem | fmuldfeq 37698* | X and Z are two equivalent definitions of the finite product of real functions. Y is a set of real functions from a common domain T, Y is closed under function multiplication and U is a finite sequence of functions in Y. M is the number of functions multiplied together. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
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Theorem | fmul01lt1lem1 37699* | Given a finite multiplication of values betweeen 0 and 1, a value larger than its frist element is larger the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
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Theorem | fmul01lt1lem2 37700* |
Given a finite multiplication of values betweeen 0 and 1, a value ![]() |
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