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Type | Label | Description |
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Statement | ||
Theorem | f1oeq2d 37501 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rnresun 37502 | Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | f1oeq1d 37503 | Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | dffo3f 37504* | An onto mapping expressed in terms of function values. As dffo3 6065 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rnresss 37505 | The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | mpteq1i 37506* | An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | elrnmptd 37507* | The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | elrnmptf 37508 | The range of a function in maps-to notation. Same as elrnmpt 5103, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | mptss 37509* | Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rnmptssrn 37510* | Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | disjf1 37511* | A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | rnsnf 37512 | The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | wessf1ornlem 37513* |
Given a function ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | wessf1orn 37514* |
Given a function ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | foelrnf 37515* | Property of a surjective function. As foelrn 6069 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | nelrnres 37516 |
If ![]() |
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Theorem | disjrnmpt2 37517* | Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | elrnmpt1sf 37518* | Elementhood in an image set. Same as elrnmpt1s 5104, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | founiiun0 37519* | Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | disjf1o 37520* | A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | fompt 37521* | Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
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Theorem | disjinfi 37522* |
Only a finite number of disjoint sets can have a non empty intersection
with a finite set ![]() |
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Theorem | fvovco 37523 | Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | ssnnf1octb 37524* |
There exists a bijection between a subset of ![]() |
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Theorem | mapdm0 37525 | The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | nnf1oxpnn 37526 | There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | rnmptssd 37527* | The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | projf1o 37528* | A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.) |
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Theorem | fvmap 37529 | Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.) |
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Theorem | mapsnd 37530* | The value of set exponentiation with a singleton exponent. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | mptexd 37531* | If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | fvixp2 37532* | Projection of a factor of an indexed Cartesian product. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | fidmfisupp 37533 | A function with a finite domain is finitely supported. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | mapsnend 37534 | Set exponentiation to a singleton exponent is equinumerous to its base. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | choicefi 37535* | For a finite set, a choice function exists, without using the axiom of choice. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | mpct 37536 | The exponentiation of a countable set to a finite set is countable. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
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Theorem | cnmetcoval 37537 | Value of the distance function of the metric space of complex numbers, composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | fcomptss 37538* | Express composition of two functions as a maps-to applying both in sequence. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | elmapsnd 37539 | Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | mapss2 37540 | Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | fsneq 37541 | Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | mpbirand 37542 | Detach truth from conjunction in biconditional. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | difmap 37543 | Difference of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | unirnmap 37544 | Given a subset of a set exponentiation, the base set can be restricted. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | ffrn 37545 | A function maps to its range. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | ffund 37546 | A mapping is a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | inmap 37547 | Intersection of two sets exponentiations. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | fcoss 37548 |
Composition of two mappings. Similar to fco 5766, but with a weaker
condition on the domain of ![]() |
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Theorem | fsneqrn 37549 | Equality condition for two functions defined on a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | difmapsn 37550 | Difference of two sets exponentiatiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | mapssbi 37551 | Subset inheritance for set exponentiation. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | unirnmapsn 37552 | Equality theorem for a subset of a set exponentiation, where the exponent is a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | iunmapss 37553* | The indexed union of set exponentiations is a subset of the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | ssmapsn 37554* |
A subset ![]() ![]() |
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Theorem | iunmapsn 37555* | The indexed union of set exponentiations to a singleton is equal to the set exponentiation of the indexed union. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | absfico 37556 | Mapping domain and codomain of the absolute value function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | icof 37557 | The set of left-closed right-open intervals of extended reals maps to subsets of extended reals. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
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Theorem | sub2times 37558 | Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | xrltled 37559 | 'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | abssubrp 37560 | The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | elfzfzo 37561 | Relationship between membership in a half open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | oddfl 37562 | Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | abscosbd 37563 | Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | mul13d 37564 | Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | negpilt0 37565 |
Negative ![]() |
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Theorem | dstregt0 37566* |
A complex number ![]() ![]() |
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Theorem | subadd4b 37567 | Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | xrlttri5d 37568 | Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | neglt 37569 | The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | zltlesub 37570 |
If an integer ![]() ![]() ![]() |
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Theorem | divlt0gt0d 37571 | The ratio of a negative numerator and a positive denominator is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | 3rp 37572 | 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | leimnltdOLD 37573 | 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete as of 18-Jul-2020. Use lensymd 9817 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | lt2addmuld 37574 | If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | subsub23d 37575 | Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | 2timesgt 37576 | Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | reopn 37577 | The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | elfzop1le2 37578 | A member in a half-open integer interval plus 1 is less or equal than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | sub31 37579 | Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | nnne1ge2 37580 | A positive integer which is not 1 is greater than or equal to 2. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | lefldiveq 37581 |
A closed enough, smaller real ![]() ![]() ![]() |
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Theorem | negsubdi3d 37582 | Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltaddneg 37583 | Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | ltdiv2dd 37584 | Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | adddirp1d 37585 | Distributive law, plus 1 version. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | absnpncand 37586 | Triangular inequality, combined with cancellation law for subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by abs3difd 13577, and absnpncand 37586 should be deleted afterwards. |
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Theorem | abssinbd 37587 | Bound for the absolute value of the sine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | halffl 37588 |
Floor of ![]() ![]() ![]() ![]() ![]() |
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Theorem | monoords 37589* | Ordering relation for a strictly monotonic sequence, increasing case. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | hashssle 37590 | The size of a subset of a finite set is less than the size of the containing set. (Contributed by Glauco Siliprandi, 11-Dec-2019.) TODO (NM): usage (2 times) should be replaced by hashss 12624, and hashssle 37590 should be deleted afterwards. |
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Theorem | flltnz 37591 | If A is not an integer, then the floor of A is less than A. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | lttri5d 37592 | Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fzisoeu 37593* | A finite ordered set has a unique order isomorphism to a generic finite sequence of integers. This theorem generalizes fz1iso 12664 for the base index and also states the uniqueness condition. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | lt3addmuld 37594 | If three real numbers are less than a fourth real number, the sum of the three real numbers is less than three times the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | absnpncan2d 37595 | Triangular inequality, combined with cancellation law for subtraction (applied twice). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fperiodmullem 37596* | A function with period T is also periodic with period nonnegative multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | fperiodmul 37597* | A function with period T is also periodic with period multiple of T. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | upbdrech 37598* | Choice of an upper bound for a non empty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | lt4addmuld 37599 | If four real numbers are less than a fifth real number, the sum of the four real numbers is less than four times the fifth real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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Theorem | absnpncan3d 37600 | Triangular inequality, combined with cancellation law for subtraction (applied three times). (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
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