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Type | Label | Description |
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Statement | ||
Theorem | fmul01lt1lem2 44601* | Given a finite multiplication of values betweeen 0 and 1, a value πΈ larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²ππ΅ & β’ β²ππ & β’ π΄ = seqπΏ( Β· , π΅) & β’ (π β πΏ β β€) & β’ (π β π β (β€β₯βπΏ)) & β’ ((π β§ π β (πΏ...π)) β (π΅βπ) β β) & β’ ((π β§ π β (πΏ...π)) β 0 β€ (π΅βπ)) & β’ ((π β§ π β (πΏ...π)) β (π΅βπ) β€ 1) & β’ (π β πΈ β β+) & β’ (π β π½ β (πΏ...π)) & β’ (π β (π΅βπ½) < πΈ) β β’ (π β (π΄βπ) < πΈ) | ||
Theorem | fmul01lt1 44602* | Given a finite multiplication of values betweeen 0 and 1, a value E larger than any multiplicand, is larger than the whole multiplication. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
β’ β²ππ΅ & β’ β²ππ & β’ β²ππ΄ & β’ π΄ = seq1( Β· , π΅) & β’ (π β π β β) & β’ (π β π΅:(1...π)βΆβ) & β’ ((π β§ π β (1...π)) β 0 β€ (π΅βπ)) & β’ ((π β§ π β (1...π)) β (π΅βπ) β€ 1) & β’ (π β πΈ β β+) & β’ (π β βπ β (1...π)(π΅βπ) < πΈ) β β’ (π β (π΄βπ) < πΈ) | ||
Theorem | cncfmptss 44603* | A continuous complex function restricted to a subset is continuous, using maps-to notation. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²π₯πΉ & β’ (π β πΉ β (π΄βcnβπ΅)) & β’ (π β πΆ β π΄) β β’ (π β (π₯ β πΆ β¦ (πΉβπ₯)) β (πΆβcnβπ΅)) | ||
Theorem | rrpsscn 44604 | The positive reals are a subset of the complex numbers. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β+ β β | ||
Theorem | mulc1cncfg 44605* | A version of mulc1cncf 24646 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 30-Jun-2017.) |
β’ β²π₯πΉ & β’ β²π₯π & β’ (π β πΉ β (π΄βcnββ)) & β’ (π β π΅ β β) β β’ (π β (π₯ β π΄ β¦ (π΅ Β· (πΉβπ₯))) β (π΄βcnββ)) | ||
Theorem | infrglb 44606* | The infimum of a nonempty bounded set of reals is the greatest lower bound. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
β’ (((π΄ β β β§ π΄ β β β§ βπ₯ β β βπ¦ β π΄ π₯ β€ π¦) β§ π΅ β β) β (inf(π΄, β, < ) < π΅ β βπ§ β π΄ π§ < π΅)) | ||
Theorem | expcnfg 44607* | If πΉ is a complex continuous function and N is a fixed number, then F^N is continuous too. A generalization of expcncf 24668. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²π₯πΉ & β’ (π β πΉ β (π΄βcnββ)) & β’ (π β π β β0) β β’ (π β (π₯ β π΄ β¦ ((πΉβπ₯)βπ)) β (π΄βcnββ)) | ||
Theorem | prodeq2ad 44608* | Equality deduction for product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ (π β π΅ = πΆ) β β’ (π β βπ β π΄ π΅ = βπ β π΄ πΆ) | ||
Theorem | fprodsplit1 44609* | Separate out a term in a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) & β’ (π β πΆ β π΄) & β’ ((π β§ π = πΆ) β π΅ = π·) β β’ (π β βπ β π΄ π΅ = (π· Β· βπ β (π΄ β {πΆ})π΅)) | ||
Theorem | fprodexp 44610* | Positive integer exponentiation of a finite product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ β²ππ & β’ (π β π β β0) & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) β β’ (π β βπ β π΄ (π΅βπ) = (βπ β π΄ π΅βπ)) | ||
Theorem | fprodabs2 44611* | The absolute value of a finite product . (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) β β’ (π β (absββπ β π΄ π΅) = βπ β π΄ (absβπ΅)) | ||
Theorem | fprod0 44612* | A finite product with a zero term is zero. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ β²ππ & β’ β²ππΆ & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β π΅ β β) & β’ (π = πΎ β π΅ = πΆ) & β’ (π β πΎ β π΄) & β’ (π β πΆ = 0) β β’ (π β βπ β π΄ π΅ = 0) | ||
Theorem | mccllem 44613* | * Induction step for mccl 44614. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ (π β π΄ β Fin) & β’ (π β πΆ β π΄) & β’ (π β π· β (π΄ β πΆ)) & β’ (π β π΅ β (β0 βm (πΆ βͺ {π·}))) & β’ (π β βπ β (β0 βm πΆ)((!βΞ£π β πΆ (πβπ)) / βπ β πΆ (!β(πβπ))) β β) β β’ (π β ((!βΞ£π β (πΆ βͺ {π·})(π΅βπ)) / βπ β (πΆ βͺ {π·})(!β(π΅βπ))) β β) | ||
Theorem | mccl 44614* | A multinomial coefficient, in its standard domain, is a positive integer. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ β²ππ΅ & β’ (π β π΄ β Fin) & β’ (π β π΅ β (β0 βm π΄)) β β’ (π β ((!βΞ£π β π΄ (π΅βπ)) / βπ β π΄ (!β(π΅βπ))) β β) | ||
Theorem | fprodcnlem 44615* | A finite product of functions to complex numbers from a common topological space is continuous. Induction step. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²ππ & β’ πΎ = (TopOpenββfld) & β’ (π β π½ β (TopOnβπ)) & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (π½ Cn πΎ)) & β’ (π β π β π΄) & β’ (π β π β (π΄ β π)) & β’ (π β (π₯ β π β¦ βπ β π π΅) β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ βπ β (π βͺ {π})π΅) β (π½ Cn πΎ)) | ||
Theorem | fprodcn 44616* | A finite product of functions to complex numbers from a common topological space is continuous. The class expression for π΅ normally contains free variables π and π₯ to index it. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²ππ & β’ πΎ = (TopOpenββfld) & β’ (π β π½ β (TopOnβπ)) & β’ (π β π΄ β Fin) & β’ ((π β§ π β π΄) β (π₯ β π β¦ π΅) β (π½ Cn πΎ)) β β’ (π β (π₯ β π β¦ βπ β π΄ π΅) β (π½ Cn πΎ)) | ||
Theorem | clim1fr1 44617* | A class of sequences of fractions that converge to 1. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ πΉ = (π β β β¦ (((π΄ Β· π) + π΅) / (π΄ Β· π))) & β’ (π β π΄ β β) & β’ (π β π΄ β 0) & β’ (π β π΅ β β) β β’ (π β πΉ β 1) | ||
Theorem | isumneg 44618* | Negation of a converging sum. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β Ξ£π β π π΄ β β) & β’ ((π β§ π β π) β (πΉβπ) = π΄) & β’ ((π β§ π β π) β π΄ β β) & β’ (π β seqπ( + , πΉ) β dom β ) β β’ (π β Ξ£π β π -π΄ = -Ξ£π β π π΄) | ||
Theorem | climrec 44619* | Limit of the reciprocal of a converging sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΊ β π΄) & β’ (π β π΄ β 0) & β’ ((π β§ π β π) β (πΊβπ) β (β β {0})) & β’ ((π β§ π β π) β (π»βπ) = (1 / (πΊβπ))) & β’ (π β π» β π) β β’ (π β π» β (1 / π΄)) | ||
Theorem | climmulf 44620* | A version of climmul 15582 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ β²ππ» & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ (π β π» β π) & β’ (π β πΊ β π΅) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΊβπ) β β) & β’ ((π β§ π β π) β (π»βπ) = ((πΉβπ) Β· (πΊβπ))) β β’ (π β π» β (π΄ Β· π΅)) | ||
Theorem | climexp 44621* | The limit of natural powers, is the natural power of the limit. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππ» & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ:πβΆβ) & β’ (π β πΉ β π΄) & β’ (π β π β β0) & β’ (π β π» β π) & β’ ((π β§ π β π) β (π»βπ) = ((πΉβπ)βπ)) β β’ (π β π» β (π΄βπ)) | ||
Theorem | climinf 44622* | A bounded monotonic nonincreasing sequence converges to the infimum of its range. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ:πβΆβ) & β’ ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) & β’ (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) β β’ (π β πΉ β inf(ran πΉ, β, < )) | ||
Theorem | climsuselem1 44623* | The subsequence index πΌ has the expected properties: it belongs to the same upper integers as the original index, and it is always greater than or equal to the original index. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β (πΌβπ) β π) & β’ ((π β§ π β π) β (πΌβ(π + 1)) β (β€β₯β((πΌβπ) + 1))) β β’ ((π β§ πΎ β π) β (πΌβπΎ) β (β€β₯βπΎ)) | ||
Theorem | climsuse 44624* | A subsequence πΊ of a converging sequence πΉ, converges to the same limit. πΌ is the strictly increasing and it is used to index the subsequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ β²ππΌ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ (π β πΉ β π΄) & β’ (π β (πΌβπ) β π) & β’ ((π β§ π β π) β (πΌβ(π + 1)) β (β€β₯β((πΌβπ) + 1))) & β’ (π β πΊ β π) & β’ ((π β§ π β π) β (πΊβπ) = (πΉβ(πΌβπ))) β β’ (π β πΊ β π΄) | ||
Theorem | climrecf 44625* | A version of climrec 44619 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΊ & β’ β²ππ» & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΊ β π΄) & β’ (π β π΄ β 0) & β’ ((π β§ π β π) β (πΊβπ) β (β β {0})) & β’ ((π β§ π β π) β (π»βπ) = (1 / (πΊβπ))) & β’ (π β π» β π) β β’ (π β π» β (1 / π΄)) | ||
Theorem | climneg 44626* | Complex limit of the negative of a sequence. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ ((π β§ π β π) β (πΉβπ) β β) β β’ (π β (π β π β¦ -(πΉβπ)) β -π΄) | ||
Theorem | climinff 44627* | A version of climinf 44622 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 29-Jun-2017.) (Revised by AV, 15-Sep-2020.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ:πβΆβ) & β’ ((π β§ π β π) β (πΉβ(π + 1)) β€ (πΉβπ)) & β’ (π β βπ₯ β β βπ β π π₯ β€ (πΉβπ)) β β’ (π β πΉ β inf(ran πΉ, β, < )) | ||
Theorem | climdivf 44628* | Limit of the ratio of two converging sequences. (Contributed by Glauco Siliprandi, 29-Jun-2017.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ β²ππ» & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ (π β π» β π) & β’ (π β πΊ β π΅) & β’ (π β π΅ β 0) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΊβπ) β (β β {0})) & β’ ((π β§ π β π) β (π»βπ) = ((πΉβπ) / (πΊβπ))) β β’ (π β π» β (π΄ / π΅)) | ||
Theorem | climreeq 44629 | If πΉ is a real function, then πΉ converges to π΄ with respect to the standard topology on the reals if and only if it converges to π΄ with respect to the standard topology on complex numbers. In the theorem, π is defined to be convergence w.r.t. the standard topology on the reals and then πΉπ π΄ represents the statement "πΉ converges to π΄, with respect to the standard topology on the reals". Notice that there is no need for the hypothesis that π΄ is a real number. (Contributed by Glauco Siliprandi, 2-Jul-2017.) |
β’ π = (βπ‘β(topGenβran (,))) & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ:πβΆβ) β β’ (π β (πΉπ π΄ β πΉ β π΄)) | ||
Theorem | ellimciota 44630* | An explicit value for the limit, when the limit exists at a limit point. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β π΄ β β) & β’ (π β π΅ β ((limPtβπΎ)βπ΄)) & β’ (π β (πΉ limβ π΅) β β ) & β’ πΎ = (TopOpenββfld) β β’ (π β (β©π₯π₯ β (πΉ limβ π΅)) β (πΉ limβ π΅)) | ||
Theorem | climaddf 44631* | A version of climadd 15581 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ β²ππ» & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ (π β π» β π) & β’ (π β πΊ β π΅) & β’ ((π β§ π β π) β (πΉβπ) β β) & β’ ((π β§ π β π) β (πΊβπ) β β) & β’ ((π β§ π β π) β (π»βπ) = ((πΉβπ) + (πΊβπ))) β β’ (π β π» β (π΄ + π΅)) | ||
Theorem | mullimc 44632* | Limit of the product of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ πΆ) & β’ π» = (π₯ β π΄ β¦ (π΅ Β· πΆ)) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ ((π β§ π₯ β π΄) β πΆ β β) & β’ (π β π β (πΉ limβ π·)) & β’ (π β π β (πΊ limβ π·)) β β’ (π β (π Β· π) β (π» limβ π·)) | ||
Theorem | ellimcabssub0 44633* | An equivalent condition for being a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ (π΅ β πΆ)) & β’ (π β π΄ β β) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ (π β π· β β) & β’ (π β πΆ β β) β β’ (π β (πΆ β (πΉ limβ π·) β 0 β (πΊ limβ π·))) | ||
Theorem | limcdm0 44634 | If a function has empty domain, every complex number is a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:β βΆβ) & β’ (π β π΅ β β) β β’ (π β (πΉ limβ π΅) = β) | ||
Theorem | islptre 44635* | An equivalence condition for a limit point w.r.t. the standard topology on the reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (topGenβran (,)) & β’ (π β π΄ β β) & β’ (π β π΅ β β) β β’ (π β (π΅ β ((limPtβπ½)βπ΄) β βπ β β* βπ β β* (π΅ β (π(,)π) β ((π(,)π) β© (π΄ β {π΅})) β β ))) | ||
Theorem | limccog 44636 | Limit of the composition of two functions. If the limit of πΉ at π΄ is π΅ and the limit of πΊ at π΅ is πΆ, then the limit of πΊ β πΉ at π΄ is πΆ. With respect to limcco 25643 and limccnp 25641, here we drop continuity assumptions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β ran πΉ β (dom πΊ β {π΅})) & β’ (π β π΅ β (πΉ limβ π΄)) & β’ (π β πΆ β (πΊ limβ π΅)) β β’ (π β πΆ β ((πΊ β πΉ) limβ π΄)) | ||
Theorem | limciccioolb 44637 | The limit of a function at the lower bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π΄ < π΅) & β’ (π β πΉ:(π΄[,]π΅)βΆβ) β β’ (π β ((πΉ βΎ (π΄(,)π΅)) limβ π΄) = (πΉ limβ π΄)) | ||
Theorem | climf 44638* | Express the predicate: The limit of complex number sequence πΉ is π΄, or πΉ converges to π΄. Similar to clim 15443, but without the disjoint var constraint πΉπ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππΉ & β’ (π β πΉ β π) & β’ ((π β§ π β β€) β (πΉβπ) = π΅) β β’ (π β (πΉ β π΄ β (π΄ β β β§ βπ₯ β β+ βπ β β€ βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π₯)))) | ||
Theorem | mullimcf 44639* | Limit of the multiplication of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β πΊ:π΄βΆβ) & β’ π» = (π₯ β π΄ β¦ ((πΉβπ₯) Β· (πΊβπ₯))) & β’ (π β π΅ β (πΉ limβ π·)) & β’ (π β πΆ β (πΊ limβ π·)) β β’ (π β (π΅ Β· πΆ) β (π» limβ π·)) | ||
Theorem | constlimc 44640* | Limit of constant function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β πΆ β β) β β’ (π β π΅ β (πΉ limβ πΆ)) | ||
Theorem | rexlim2d 44641* | Inference removing two restricted quantifiers. Same as rexlimdvv 3209, but with bound-variable hypotheses instead of distinct variable restrictions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²π₯π & β’ β²π¦π & β’ (π β ((π₯ β π΄ β§ π¦ β π΅) β (π β π))) β β’ (π β (βπ₯ β π΄ βπ¦ β π΅ π β π)) | ||
Theorem | idlimc 44642* | Limit of the identity function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π΄ β β) & β’ πΉ = (π₯ β π΄ β¦ π₯) & β’ (π β π β β) β β’ (π β π β (πΉ limβ π)) | ||
Theorem | divcnvg 44643* | The sequence of reciprocals of positive integers, multiplied by the factor π΄, converges to zero. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π΄ β β β§ π β β) β (π β (β€β₯βπ) β¦ (π΄ / π)) β 0) | ||
Theorem | limcperiod 44644* | If πΉ is a periodic function with period π, the limit doesn't change if we shift the limiting point by π. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:dom πΉβΆβ) & β’ (π β π΄ β β) & β’ (π β π΄ β dom πΉ) & β’ (π β π β β) & β’ π΅ = {π₯ β β β£ βπ¦ β π΄ π₯ = (π¦ + π)} & β’ (π β π΅ β dom πΉ) & β’ ((π β§ π¦ β π΄) β (πΉβ(π¦ + π)) = (πΉβπ¦)) & β’ (π β πΆ β ((πΉ βΎ π΄) limβ π·)) β β’ (π β πΆ β ((πΉ βΎ π΅) limβ (π· + π))) | ||
Theorem | limcrecl 44645 | If πΉ is a real-valued function, π΅ is a limit point of its domain, and the limit of πΉ at π΅ exists, then this limit is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β π΄ β β) & β’ (π β π΅ β ((limPtβ(TopOpenββfld))βπ΄)) & β’ (π β πΏ β (πΉ limβ π΅)) β β’ (π β πΏ β β) | ||
Theorem | sumnnodd 44646* | A series indexed by β with only odd terms. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:ββΆβ) & β’ ((π β§ π β β β§ (π / 2) β β) β (πΉβπ) = 0) & β’ (π β seq1( + , πΉ) β π΅) β β’ (π β (seq1( + , (π β β β¦ (πΉβ((2 Β· π) β 1)))) β π΅ β§ Ξ£π β β (πΉβπ) = Ξ£π β β (πΉβ((2 Β· π) β 1)))) | ||
Theorem | lptioo2 44647 | The upper bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (topGenβran (,)) & β’ (π β π΄ β β*) & β’ (π β π΅ β β) & β’ (π β π΄ < π΅) β β’ (π β π΅ β ((limPtβπ½)β(π΄(,)π΅))) | ||
Theorem | lptioo1 44648 | The lower bound of an open interval is a limit point of the interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (topGenβran (,)) & β’ (π β π΄ β β) & β’ (π β π΅ β β*) & β’ (π β π΄ < π΅) β β’ (π β π΄ β ((limPtβπ½)β(π΄(,)π΅))) | ||
Theorem | elprn1 44649 | A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π΄ β {π΅, πΆ} β§ π΄ β π΅) β π΄ = πΆ) | ||
Theorem | elprn2 44650 | A member of an unordered pair that is not the "second", must be the "first". (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ ((π΄ β {π΅, πΆ} β§ π΄ β πΆ) β π΄ = π΅) | ||
Theorem | limcmptdm 44651* | The domain of a maps-to function with a limit. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ (π β πΆ β (πΉ limβ π·)) β β’ (π β π΄ β β) | ||
Theorem | clim2f 44652* | Express the predicate: The limit of complex number sequence πΉ is π΄, or πΉ converges to π΄, with more general quantifier restrictions than clim 15443. Similar to clim2 15453, but without the disjoint var constraint πΉπ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) = π΅) β β’ (π β (πΉ β π΄ β (π΄ β β β§ βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π₯)))) | ||
Theorem | limcicciooub 44653 | The limit of a function at the upper bound of a closed interval only depends on the values in the inner open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β) & β’ (π β π΄ < π΅) & β’ (π β πΉ:(π΄[,]π΅)βΆβ) β β’ (π β ((πΉ βΎ (π΄(,)π΅)) limβ π΅) = (πΉ limβ π΅)) | ||
Theorem | ltmod 44654 | A sufficient condition for a "less than" relationship for the mod operator. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π΄ β β) & β’ (π β π΅ β β+) & β’ (π β πΆ β ((π΄ β (π΄ mod π΅))[,)π΄)) β β’ (π β (πΆ mod π΅) < (π΄ mod π΅)) | ||
Theorem | islpcn 44655* | A characterization for a limit point for the standard topology on the complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β π β β) & β’ (π β π β β) β β’ (π β (π β ((limPtβ(TopOpenββfld))βπ) β βπ β β+ βπ₯ β (π β {π})(absβ(π₯ β π)) < π)) | ||
Theorem | lptre2pt 44656* | If a set in the real line has a limit point than it contains two distinct points that are closer than a given distance. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (topGenβran (,)) & β’ (π β π΄ β β) & β’ (π β ((limPtβπ½)βπ΄) β β ) & β’ (π β πΈ β β+) β β’ (π β βπ₯ β π΄ βπ¦ β π΄ (π₯ β π¦ β§ (absβ(π₯ β π¦)) < πΈ)) | ||
Theorem | limsupre 44657* | If a sequence is bounded, then the limsup is real. (Contributed by Glauco Siliprandi, 11-Dec-2019.) (Revised by AV, 13-Sep-2020.) |
β’ (π β π΅ β β) & β’ (π β sup(π΅, β*, < ) = +β) & β’ (π β πΉ:π΅βΆβ) & β’ (π β βπ β β βπ β β βπ β π΅ (π β€ π β (absβ(πΉβπ)) β€ π)) β β’ (π β (lim supβπΉ) β β) | ||
Theorem | limcresiooub 44658 | The left limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β π΅ β β*) & β’ (π β πΆ β β) & β’ (π β π΅ < πΆ) & β’ (π β (π΅(,)πΆ) β π΄) & β’ (π β π· β β*) & β’ (π β π· β€ π΅) β β’ (π β ((πΉ βΎ (π΅(,)πΆ)) limβ πΆ) = ((πΉ βΎ (π·(,)πΆ)) limβ πΆ)) | ||
Theorem | limcresioolb 44659 | The right limit doesn't change if the function is restricted to a smaller open interval. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ (π β πΉ:π΄βΆβ) & β’ (π β π΅ β β) & β’ (π β πΆ β β*) & β’ (π β π΅ < πΆ) & β’ (π β (π΅(,)πΆ) β π΄) & β’ (π β π· β β*) & β’ (π β πΆ β€ π·) β β’ (π β ((πΉ βΎ (π΅(,)πΆ)) limβ π΅) = ((πΉ βΎ (π΅(,)π·)) limβ π΅)) | ||
Theorem | limcleqr 44660 | If the left and the right limits are equal, the limit of the function exits and the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΎ = (TopOpenββfld) & β’ (π β π΄ β β) & β’ π½ = (topGenβran (,)) & β’ (π β πΉ:π΄βΆβ) & β’ (π β π΅ β β) & β’ (π β πΏ β ((πΉ βΎ (-β(,)π΅)) limβ π΅)) & β’ (π β π β ((πΉ βΎ (π΅(,)+β)) limβ π΅)) & β’ (π β πΏ = π ) β β’ (π β πΏ β (πΉ limβ π΅)) | ||
Theorem | lptioo2cn 44661 | The upper bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (TopOpenββfld) & β’ (π β π΄ β β*) & β’ (π β π΅ β β) & β’ (π β π΄ < π΅) β β’ (π β π΅ β ((limPtβπ½)β(π΄(,)π΅))) | ||
Theorem | lptioo1cn 44662 | The lower bound of an open interval is a limit point of the interval, wirth respect to the standard topology on complex numbers. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ π½ = (TopOpenββfld) & β’ (π β π΅ β β*) & β’ (π β π΄ β β) & β’ (π β π΄ < π΅) β β’ (π β π΄ β ((limPtβπ½)β(π΄(,)π΅))) | ||
Theorem | neglimc 44663* | Limit of the negative function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ -π΅) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ (π β πΆ β (πΉ limβ π·)) β β’ (π β -πΆ β (πΊ limβ π·)) | ||
Theorem | addlimc 44664* | Sum of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ πΆ) & β’ π» = (π₯ β π΄ β¦ (π΅ + πΆ)) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ ((π β§ π₯ β π΄) β πΆ β β) & β’ (π β πΈ β (πΉ limβ π·)) & β’ (π β πΌ β (πΊ limβ π·)) β β’ (π β (πΈ + πΌ) β (π» limβ π·)) | ||
Theorem | 0ellimcdiv 44665* | If the numerator converges to 0 and the denominator converges to a nonzero number, then the fraction converges to 0. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ πΆ) & β’ π» = (π₯ β π΄ β¦ (π΅ / πΆ)) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ ((π β§ π₯ β π΄) β πΆ β (β β {0})) & β’ (π β 0 β (πΉ limβ πΈ)) & β’ (π β π· β (πΊ limβ πΈ)) & β’ (π β π· β 0) β β’ (π β 0 β (π» limβ πΈ)) | ||
Theorem | clim2cf 44666* | Express the predicate πΉ converges to π΄. Similar to clim2 15453, but without the disjoint var constraint πΉπ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ (π β π΄ β β) & β’ ((π β§ π β π) β π΅ β β) β β’ (π β (πΉ β π΄ β βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(absβ(π΅ β π΄)) < π₯)) | ||
Theorem | limclner 44667 | For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΎ = (TopOpenββfld) & β’ (π β π΄ β β) & β’ π½ = (topGenβran (,)) & β’ (π β πΉ:π΄βΆβ) & β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (-β(,)π΅)))) & β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (π΅(,)+β)))) & β’ (π β πΏ β ((πΉ βΎ (-β(,)π΅)) limβ π΅)) & β’ (π β π β ((πΉ βΎ (π΅(,)+β)) limβ π΅)) & β’ (π β πΏ β π ) β β’ (π β (πΉ limβ π΅) = β ) | ||
Theorem | sublimc 44668* | Subtraction of two limits. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ πΆ) & β’ π» = (π₯ β π΄ β¦ (π΅ β πΆ)) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ ((π β§ π₯ β π΄) β πΆ β β) & β’ (π β πΈ β (πΉ limβ π·)) & β’ (π β πΌ β (πΊ limβ π·)) β β’ (π β (πΈ β πΌ) β (π» limβ π·)) | ||
Theorem | reclimc 44669* | Limit of the reciprocal of a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ (1 / π΅)) & β’ ((π β§ π₯ β π΄) β π΅ β (β β {0})) & β’ (π β πΆ β (πΉ limβ π·)) & β’ (π β πΆ β 0) β β’ (π β (1 / πΆ) β (πΊ limβ π·)) | ||
Theorem | clim0cf 44670* | Express the predicate πΉ converges to 0. Similar to clim 15443, but without the disjoint var constraint πΉπ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ ((π β§ π β π) β π΅ β β) β β’ (π β (πΉ β 0 β βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(absβπ΅) < π₯)) | ||
Theorem | limclr 44671 | For a limit point, both from the left and from the right, of the domain, the limit of the function exits only if the left and the right limits are equal. In this case, the three limits coincide. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΎ = (TopOpenββfld) & β’ (π β π΄ β β) & β’ π½ = (topGenβran (,)) & β’ (π β πΉ:π΄βΆβ) & β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (-β(,)π΅)))) & β’ (π β π΅ β ((limPtβπ½)β(π΄ β© (π΅(,)+β)))) & β’ (π β πΏ β ((πΉ βΎ (-β(,)π΅)) limβ π΅)) & β’ (π β π β ((πΉ βΎ (π΅(,)+β)) limβ π΅)) β β’ (π β (((πΉ limβ π΅) β β β πΏ = π ) β§ (πΏ = π β πΏ β (πΉ limβ π΅)))) | ||
Theorem | divlimc 44672* | Limit of the quotient of two functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ πΉ = (π₯ β π΄ β¦ π΅) & β’ πΊ = (π₯ β π΄ β¦ πΆ) & β’ π» = (π₯ β π΄ β¦ (π΅ / πΆ)) & β’ ((π β§ π₯ β π΄) β π΅ β β) & β’ ((π β§ π₯ β π΄) β πΆ β (β β {0})) & β’ (π β π β (πΉ limβ π·)) & β’ (π β π β (πΊ limβ π·)) & β’ (π β π β 0) & β’ ((π β§ π₯ β π΄) β πΆ β 0) β β’ (π β (π / π) β (π» limβ π·)) | ||
Theorem | expfac 44673* | Factorial grows faster than exponential. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
β’ πΉ = (π β β0 β¦ ((π΄βπ) / (!βπ))) β β’ (π΄ β β β πΉ β 0) | ||
Theorem | climconstmpt 44674* | A constant sequence converges to its value. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ (π β π΄ β β) β β’ (π β (π₯ β π β¦ π΄) β π΄) | ||
Theorem | climresmpt 44675* | A function restricted to upper integers converges iff the original function converges. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ π = (β€β₯βπ) & β’ πΉ = (π₯ β π β¦ π΄) & β’ (π β π β π) & β’ πΊ = (π₯ β (β€β₯βπ) β¦ π΄) β β’ (π β (πΊ β π΅ β πΉ β π΅)) | ||
Theorem | climsubmpt 44676* | Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²ππ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ ((π β§ π β π) β π΄ β β) & β’ ((π β§ π β π) β π΅ β β) & β’ (π β (π β π β¦ π΄) β πΆ) & β’ (π β (π β π β¦ π΅) β π·) β β’ (π β (π β π β¦ (π΄ β π΅)) β (πΆ β π·)) | ||
Theorem | climsubc2mpt 44677* | Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²ππ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ ((π β§ π β π) β π΄ β β) & β’ (π β (π β π β¦ π΄) β πΆ) & β’ (π β π΅ β β) β β’ (π β (π β π β¦ (π΄ β π΅)) β (πΆ β π΅)) | ||
Theorem | climsubc1mpt 44678* | Limit of the difference of two converging sequences. (Contributed by Glauco Siliprandi, 8-Apr-2021.) |
β’ β²ππ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β π΄ β β) & β’ ((π β§ π β π) β π΅ β β) & β’ (π β (π β π β¦ π΅) β πΆ) β β’ (π β (π β π β¦ (π΄ β π΅)) β (π΄ β πΆ)) | ||
Theorem | fnlimfv 44679* | The value of the limit function πΊ at any point of its domain π·. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯π· & β’ β²π₯πΉ & β’ πΊ = (π₯ β π· β¦ ( β β(π β π β¦ ((πΉβπ)βπ₯)))) & β’ (π β π β π·) β β’ (π β (πΊβπ) = ( β β(π β π β¦ ((πΉβπ)βπ)))) | ||
Theorem | climreclf 44680* | The limit of a convergent real sequence is real. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ ((π β§ π β π) β (πΉβπ) β β) β β’ (π β π΄ β β) | ||
Theorem | climeldmeq 44681* | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π = (β€β₯βπ) & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = (πΊβπ)) β β’ (π β (πΉ β dom β β πΊ β dom β )) | ||
Theorem | climf2 44682* | Express the predicate: The limit of complex number sequence πΉ is π΄, or πΉ converges to π΄. Similar to clim 15443, but without the disjoint var constraint ππ and πΉπ. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ (π β πΉ β π) & β’ ((π β§ π β β€) β (πΉβπ) = π΅) β β’ (π β (πΉ β π΄ β (π΄ β β β§ βπ₯ β β+ βπ β β€ βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π₯)))) | ||
Theorem | fnlimcnv 44683* | The sequence of function values converges to the value of the limit function πΊ at any point of its domain π·. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²π₯πΉ & β’ π· = {π₯ β βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) β£ (π β π β¦ ((πΉβπ)βπ₯)) β dom β } & β’ πΊ = (π₯ β π· β¦ ( β β(π β π β¦ ((πΉβπ)βπ₯)))) & β’ (π β π β π·) β β’ (π β (π β π β¦ ((πΉβπ)βπ)) β (πΊβπ)) | ||
Theorem | climeldmeqmpt 44684* | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ (π β π΄ β π ) & β’ (π β π β π΄) & β’ ((π β§ π β π΄) β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π β πΆ) & β’ ((π β§ π β πΆ) β π· β π) & β’ ((π β§ π β π) β π΅ = π·) β β’ (π β ((π β π΄ β¦ π΅) β dom β β (π β πΆ β¦ π·) β dom β )) | ||
Theorem | climfveq 44685* | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ π = (β€β₯βπ) & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = (πΊβπ)) β β’ (π β ( β βπΉ) = ( β βπΊ)) | ||
Theorem | clim2f2 44686* | Express the predicate: The limit of complex number sequence πΉ is π΄, or πΉ converges to π΄, with more general quantifier restrictions than clim 15443. Similar to clim2 15453, but without the disjoint var constraint πΉπ. (Contributed by Glauco Siliprandi, 11-Dec-2019.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ ((π β§ π β π) β (πΉβπ) = π΅) β β’ (π β (πΉ β π΄ β (π΄ β β β§ βπ₯ β β+ βπ β π βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π₯)))) | ||
Theorem | climfveqmpt 44687* | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ (π β π΄ β π ) & β’ (π β π β π΄) & β’ ((π β§ π β π΄) β π΅ β π) & β’ (π β πΆ β π) & β’ (π β π β πΆ) & β’ ((π β§ π β πΆ) β π· β π) & β’ ((π β§ π β π) β π΅ = π·) β β’ (π β ( β β(π β π΄ β¦ π΅)) = ( β β(π β πΆ β¦ π·))) | ||
Theorem | climd 44688* | Express the predicate: The limit of complex number sequence πΉ is π΄, or πΉ converges to π΄. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β β€) & β’ (π β πΉ β π΄) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ (π β π β β+) β β’ (π β βπ β π βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π)) | ||
Theorem | clim2d 44689* | The limit of complex number sequence πΉ is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ (π β πΉ β π΄) & β’ ((π β§ π β π) β (πΉβπ) = π΅) & β’ (π β π β β+) β β’ (π β βπ β π βπ β (β€β₯βπ)(π΅ β β β§ (absβ(π΅ β π΄)) < π)) | ||
Theorem | fnlimfvre 44690* | The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²π₯πΉ & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ) & β’ π· = {π₯ β βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) β£ (π β π β¦ ((πΉβπ)βπ₯)) β dom β } & β’ (π β π β π·) β β’ (π β ( β β(π β π β¦ ((πΉβπ)βπ))) β β) | ||
Theorem | allbutfifvre 44691* | Given a sequence of real-valued functions, and π that belongs to all but finitely many domains, then its function value is ultimately a real number. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ) & β’ π· = βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) & β’ (π β π β π·) β β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ)βπ) β β) | ||
Theorem | climleltrp 44692* | The limit of complex number sequence πΉ is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ π = (β€β₯βπ) & β’ (π β π β π) & β’ ((π β§ π β (β€β₯βπ)) β (πΉβπ) β β) & β’ (π β πΉ β π΄) & β’ (π β πΆ β β) & β’ (π β π΄ β€ πΆ) & β’ (π β π β β+) β β’ (π β βπ β π βπ β (β€β₯βπ)((πΉβπ) β β β§ (πΉβπ) < (πΆ + π))) | ||
Theorem | fnlimfvre2 44693* | The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²π₯πΉ & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ) & β’ π· = {π₯ β βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) β£ (π β π β¦ ((πΉβπ)βπ₯)) β dom β } & β’ πΊ = (π₯ β π· β¦ ( β β(π β π β¦ ((πΉβπ)βπ₯)))) & β’ (π β π β π·) β β’ (π β (πΊβπ) β β) | ||
Theorem | fnlimf 44694* | The limit function of real functions, is a real-valued function. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²π₯πΉ & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ) & β’ π· = {π₯ β βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) β£ (π β π β¦ ((πΉβπ)βπ₯)) β dom β } & β’ πΊ = (π₯ β π· β¦ ( β β(π β π β¦ ((πΉβπ)βπ₯)))) β β’ (π β πΊ:π·βΆβ) | ||
Theorem | fnlimabslt 44695* | A sequence of function values, approximates the corresponding limit function value, all but finitely many times. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²π₯πΉ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β (πΉβπ):dom (πΉβπ)βΆβ) & β’ π· = {π₯ β βͺ π β π β© π β (β€β₯βπ)dom (πΉβπ) β£ (π β π β¦ ((πΉβπ)βπ₯)) β dom β } & β’ πΊ = (π₯ β π· β¦ ( β β(π β π β¦ ((πΉβπ)βπ₯)))) & β’ (π β π β π·) & β’ (π β π β β+) β β’ (π β βπ β π βπ β (β€β₯βπ)(((πΉβπ)βπ) β β β§ (absβ(((πΉβπ)βπ) β (πΊβπ))) < π)) | ||
Theorem | climfveqf 44696* | Two functions that are eventually equal to one another have the same limit. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ π = (β€β₯βπ) & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = (πΊβπ)) β β’ (π β ( β βπΉ) = ( β βπΊ)) | ||
Theorem | climmptf 44697* | Exhibit a function πΊ with the same convergence properties as the not-quite-function πΉ. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ β²ππΉ & β’ (π β π β β€) & β’ (π β πΉ β π) & β’ π = (β€β₯βπ) & β’ πΊ = (π β π β¦ (πΉβπ)) β β’ (π β (πΉ β π΄ β πΊ β π΄)) | ||
Theorem | climfveqmpt3 44698* | Two functions that are eventually equal to one another have the same limit. TODO: this is more general than climfveqmpt 44687 and should replace it. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ β²ππ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ (π β π΄ β π) & β’ (π β πΆ β π) & β’ (π β π β π΄) & β’ (π β π β πΆ) & β’ ((π β§ π β π) β π΅ β π) & β’ ((π β§ π β π) β π΅ = π·) β β’ (π β ( β β(π β π΄ β¦ π΅)) = ( β β(π β πΆ β¦ π·))) | ||
Theorem | climeldmeqf 44699* | Two functions that are eventually equal, either both are convergent or both are divergent. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ β²ππ & β’ β²ππΉ & β’ β²ππΊ & β’ π = (β€β₯βπ) & β’ (π β πΉ β π) & β’ (π β πΊ β π) & β’ (π β π β β€) & β’ ((π β§ π β π) β (πΉβπ) = (πΊβπ)) β β’ (π β (πΉ β dom β β πΊ β dom β )) | ||
Theorem | climreclmpt 44700* | The limit of B convergent real sequence is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
β’ β²ππ & β’ (π β π β β€) & β’ π = (β€β₯βπ) & β’ ((π β§ π β π) β π΄ β β) & β’ (π β (π β π β¦ π΄) β π΅) β β’ (π β π΅ β β) |
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