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Type | Label | Description |
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Statement | ||
Theorem | vdwapf 15001 | The arithmetic progression function is a function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
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Theorem | vdwapval 15002* | Value of the arithmetic progression function. (Contributed by Mario Carneiro, 18-Aug-2014.) |
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Theorem | vdwapun 15003 | Remove the first element of an arithmetic progression. (Contributed by Mario Carneiro, 11-Sep-2014.) |
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Theorem | vdwapid1 15004 | The first element of an arithmetic progression. (Contributed by Mario Carneiro, 12-Sep-2014.) |
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Theorem | vdwap0 15005 | Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
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Theorem | vdwap1 15006 | Value of a length-1 arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
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Theorem | vdwmc 15007* |
The predicate " The ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwmc2 15008* | Expand out the definition of an arithmetic progression. (Contributed by Mario Carneiro, 18-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwpc 15009* |
The predicate " The coloring ![]() ![]() ![]() ![]() |
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Theorem | vdwlem1 15010* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem2 15011* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem3 15012 | Lemma for vdw 15023. (Contributed by Mario Carneiro, 13-Sep-2014.) |
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Theorem | vdwlem4 15013* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
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Theorem | vdwlem5 15014* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
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Theorem | vdwlem6 15015* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 13-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem7 15016* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem8 15017* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 18-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem9 15018* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 12-Sep-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem10 15019* |
Lemma for vdw 15023. Set up secondary induction on ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem11 15020* | Lemma for vdw 15023. (Contributed by Mario Carneiro, 18-Aug-2014.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem12 15021 |
Lemma for vdw 15023. ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwlem13 15022* |
Lemma for vdw 15023. Main induction on ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdw 15023* |
Van der Waerden's theorem. For any finite coloring ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwnnlem1 15024* |
Corollary of vdw 15023, and lemma for vdwnn 15027. If ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwnnlem2 15025* |
Lemma for vdwnn 15027. The set of all "bad" ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwnnlem3 15026* | Lemma for vdwnn 15027. (Contributed by Mario Carneiro, 13-Sep-2014.) (Proof shortened by AV, 27-Sep-2020.) |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | vdwnn 15027* |
Van der Waerden's theorem, infinitary version. For any finite coloring
![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Syntax | cram 15028 | Extend class notation with the Ramsey number function. |
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Syntax | cramold 15029 | Extend class notation with the Ramsey number function (old version). |
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Theorem | ramtlecl 15030* |
The set ![]() |
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Definition | df-ram 15031* |
Define the Ramsey number function. The input is a number ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Definition | df-ramOLD 15032* |
Define the Ramsey number function. The input is a number ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | hashbcval 15033* |
Value of the "binomial set", the set of all ![]() ![]() |
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | ||
Theorem | hashbccl 15034* | The binomial set is a finite set. (Contributed by Mario Carneiro, 20-Apr-2015.) |
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Theorem | hashbcss 15035* | Subset relation for the binomial set. (Contributed by Mario Carneiro, 20-Apr-2015.) |
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Theorem | hashbc0 15036* | The set of subsets of size zero is the singleton of the empty set. (Contributed by Mario Carneiro, 22-Apr-2015.) |
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Theorem | hashbc2 15037* | The size of the binomial set is the binomial coefficient. (Contributed by Mario Carneiro, 20-Apr-2015.) |
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Theorem | 0hashbc 15038* | There are no subsets of the empty set with size greater than zero. (Contributed by Mario Carneiro, 22-Apr-2015.) |
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Theorem | ramval 15039* | The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramvalOLD 15040* | The value of the Ramsey number function. (Contributed by Mario Carneiro, 21-Apr-2015.) Obsolete version of ramval 15039 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramcl2lem 15041* | Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramcl2lemOLD 15042* | Lemma for extended real closure of the Ramsey number function. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2lem 15041 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramtcl 15043* | The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramtclOLD 15044* | The Ramsey number has the Ramsey number property if any number does. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtcl 15043 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramtcl2 15045* | The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramtcl2OLD 15046* | The Ramsey number is an integer iff there is a number with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtcl2 15045 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramtub 15047* | The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramtubOLD 15048* | The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramtub 15047 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramub 15049* | The Ramsey number is a lower bound on the set of all numbers with the Ramsey number property. (Contributed by Mario Carneiro, 22-Apr-2015.) |
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Theorem | ramub2 15050* | It is sufficient to check the Ramsey property on finite sets of size equal to the upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | rami 15051* | The defining property of a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.) |
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Theorem | ramcl2 15052 | The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) (Revised by AV, 14-Sep-2020.) |
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Theorem | ramcl2OLD 15053 | The Ramsey number is either a nonnegative integer or plus infinity. (Contributed by Mario Carneiro, 20-Apr-2015.) Obsolete version of ramcl2 15052 as of 14-Sep-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | ramxrcl 15054 | The Ramsey number is an extended real number. (This theorem does not imply Ramsey's theorem, unlike ramcl 15066.) (Contributed by Mario Carneiro, 20-Apr-2015.) |
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Theorem | ramubcl 15055 | If the Ramsey number is upper bounded, then it is an integer. (Contributed by Mario Carneiro, 20-Apr-2015.) |
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Theorem | ramlb 15056* | Establish a lower bound on a Ramsey number. (Contributed by Mario Carneiro, 22-Apr-2015.) |
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Theorem | 0ram 15057* |
The Ramsey number when ![]() ![]() ![]() |
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Theorem | 0ram2 15058 |
The Ramsey number when ![]() ![]() ![]() |
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Theorem | ram0 15059 |
The Ramsey number when ![]() ![]() ![]() |
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Theorem | 0ramcl 15060 |
Lemma for ramcl 15066: Existence of the Ramsey number when ![]() ![]() ![]() |
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Theorem | ramz2 15061 |
The Ramsey number when ![]() ![]() |
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Theorem | ramz 15062 |
The Ramsey number when ![]() |
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Theorem | ramub1lem1 15063* | Lemma for ramub1 15065. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ramub1lem2 15064* | Lemma for ramub1 15065. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ramub1 15065* | Inductive step for Ramsey's theorem, in the form of an explicit upper bound. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ramcl 15066 | Ramsey's theorem: the Ramsey number is an integer for every finite coloring and set of upper bounds. (Contributed by Mario Carneiro, 23-Apr-2015.) |
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Theorem | ramsey 15067* |
Ramsey's theorem with the definition Ramsey eliminated. If ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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According to Wikipedia "Primorial", https://en.wikipedia.org/wiki/Primorial (28-Aug-2020): "In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying [all] positive integers [less than or equal to a given number], the function only multiplies [the] prime numbers [less than or equal to the given number]. The name "primorial", coined by Harvey Dubner, draws an analogy to primes similar to the way the name "factorial" relates to factors." | ||
Syntax | cprmo 15068 | Extend class notation to include the primorial of nonnegative integers. |
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Definition | df-prmo 15069* |
Define the primorial function on nonnegative integers as the product of
all prime numbers less than or equal to the integer. For example,
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Theorem | prmoval 15070* | Value of the primorial function for nonnegative integers. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmocl 15071 | Closure of the primorial function. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmone0 15072 | The primorial function is nonzero. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmo0 15073 | The primorial of 0. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmo1 15074 | The primorial of 1. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmop1 15075 | The primorial of a successor. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmonn2 15076 | Value of the primorial function expressed recursively. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmo2 15077 | The primorial of 2. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmo3 15078 | The primorial of 3. (Contributed by AV, 28-Aug-2020.) |
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Theorem | prmdvdsprmo 15079* | The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.) |
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Theorem | prmdvdsprmop 15080* | The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 28-Aug-2020.) |
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Theorem | fvprmselelfz 15081* | The value of the prime selection function is in a finite sequence of integers if the argument is in this finite sequence of integers. (Contributed by AV, 19-Aug-2020.) |
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Theorem | fvprmselgcd1 15082* | The greatest common divisor of two values of the prime selection function for different arguments is 1. (Contributed by AV, 19-Aug-2020.) |
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Theorem | prmolefac 15083 | The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
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Theorem | prmodvdslcmf 15084 | The primorial of a nonnegative integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
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Theorem | prmolelcmf 15085 | The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 29-Aug-2020.) |
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According to Wikipedia "Prime gap", https://en.wikipedia.org/wiki/Prime_gap (16-Aug-2020): "A prime gap is the difference between two successive prime numbers. The n-th prime gap, denoted gn or g(pn) is the difference between the (n+1)-th and the n-th prime numbers, i.e. gn = pn+1 - pn . We have g1 = 1, g2 = g3 = 2, and g4 = 4." It can be proven that there are arbitrary large gaps, usually by showing that "in the sequence n!+2, n!+3, ..., n!+n the first term is divisible by 2, the second term is divisible by 3, and so on. Thus, this is a sequence of n-1 consecutive composite integers, and it must belong to a gap between primes having length at least n.", see prmgap 15108. Instead of using the factorial of n (see df-fac 12498), any function can be chosen for which f(n) is not relatively prime to the integers 2, 3, ..., n. For example, the least common multiple of the integers 2, 3, ..., n, see prmgaplcm 15110, or the primorial n# (the product of all prime numbers less than or equal to n), see prmgapprmo 15112, are such functions, which provide smaller values than the factorial function (see lcmflefac 14700 and prmolefac 15083 resp. prmolelcmf 15085): "For instance, the first prime gap of size larger than 14 occurs between the primes 523 and 541, while 15! is the vastly larger number 1307674368000." But the least common multiple of the integers 2, 3, ..., 15 is 360360, and 15# is 30030 (p3248 = 30029 and P3249 = 30047, so g3248 = 18). | ||
Theorem | prmgaplem1 15086 | Lemma for prmgap 15108: The factorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 13-Aug-2020.) |
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Theorem | prmgaplem2 15087 | Lemma for prmgap 15108: The factorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 13-Aug-2020.) |
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Theorem | prmgaplcmlem1 15088 | Lemma for prmgaplcm 15110: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
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Theorem | prmgaplcmlem2 15089 | Lemma for prmgaplcm 15110: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) (Revised by AV, 27-Aug-2020.) |
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Theorem | lcmsmapnnOLD 15090* | The function mapping a positive integer to the least common multiple of all positive integers less than or equal to the integer is a mapping from the positive integers to the positive integers. (Contributed by AV, 14-Aug-2020.) Obsolete as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmgaplcmlem1OLD 15091* | Lemma for prmgaplcmOLD 15109: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number is divisible by that integer. (Contributed by AV, 14-Aug-2020.) Obsolete version of prmgaplcmlem1 15088 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmgaplcmlem2OLD 15092* | Lemma for prmgaplcmOLD 15109: The least common multiple of all positive integers less than or equal to a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 14-Aug-2020.) Obsolete version of prmgaplcmlem2 15089 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmormapnnOLD 15093* |
The primorial function is a mapping from the positive integers to the
positive integers. In contrast to prmorcht 24184, where the primorial is
defined by using the sequence builder (![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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Theorem | prmdvdsprmorOLD 15094* | The primorial of a number is divisible by each prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmdvdsprmo 15079 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmdvdsprmorpOLD 15095* | The primorial of a number plus an integer greater than 1 and less then or equal to the number is divisible by a prime less then or equal to the number. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmdvdsprmop 15080 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmgapprmorlemOLD 15096* | Lemma for prmgapprmorOLD 15113: The primorial of a number plus an integer greater than 1 and less then or equal to the number are not coprime. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmgapprmolem 15111 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmorlefacOLD 15097* | The primorial of a positive integer is less than or equal to the factorial of the integer. (Contributed by AV, 15-Aug-2020.) Obsolete version of prmolefac 15083 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmordvdslcmfOLD 15098* | The primorial of a positive integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 27-Aug-2020.) Obsolete version of prmodvdslcmf 15084 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmorlelcmfOLD 15099* | The primorial of a positive integer is less than or equal to the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) (Revised by AV, 27-Aug-2020.) Obsolete version of prmolelcmf 15085 as of 29-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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Theorem | prmordvdslcmsOLDOLD 15100* | The primorial of a positive integer divides the least common multiple of all positive integers less than or equal to the integer. (Contributed by AV, 19-Aug-2020.) Obsolete version of prmordvdslcmfOLD 15098 as of 27-Aug-2020. (New usage is discouraged.) (Proof modification is discouraged.) |
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