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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdsze2 | Structured version Visualization version GIF version |
Description: Generalize the statement of the ErdΕs-Szekeres theorem erdsze 34845 to "sequences" indexed by an arbitrary subset of β, which can be infinite. This is part of Metamath 100 proof #73. (Contributed by Mario Carneiro, 22-Jan-2015.) |
Ref | Expression |
---|---|
erdsze2.r | β’ (π β π β β) |
erdsze2.s | β’ (π β π β β) |
erdsze2.f | β’ (π β πΉ:π΄β1-1ββ) |
erdsze2.a | β’ (π β π΄ β β) |
erdsze2.l | β’ (π β ((π β 1) Β· (π β 1)) < (β―βπ΄)) |
Ref | Expression |
---|---|
erdsze2 | β’ (π β βπ β π« π΄((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | erdsze2.r | . . 3 β’ (π β π β β) | |
2 | erdsze2.s | . . 3 β’ (π β π β β) | |
3 | erdsze2.f | . . 3 β’ (π β πΉ:π΄β1-1ββ) | |
4 | erdsze2.a | . . 3 β’ (π β π΄ β β) | |
5 | eqid 2728 | . . 3 β’ ((π β 1) Β· (π β 1)) = ((π β 1) Β· (π β 1)) | |
6 | erdsze2.l | . . 3 β’ (π β ((π β 1) Β· (π β 1)) < (β―βπ΄)) | |
7 | 1, 2, 3, 4, 5, 6 | erdsze2lem1 34846 | . 2 β’ (π β βπ(π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) |
8 | 1 | adantr 479 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β π β β) |
9 | 2 | adantr 479 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β π β β) |
10 | 3 | adantr 479 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β πΉ:π΄β1-1ββ) |
11 | 4 | adantr 479 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β π΄ β β) |
12 | 6 | adantr 479 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β ((π β 1) Β· (π β 1)) < (β―βπ΄)) |
13 | simprl 769 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄) | |
14 | simprr 771 | . . 3 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π)) | |
15 | 8, 9, 10, 11, 5, 12, 13, 14 | erdsze2lem2 34847 | . 2 β’ ((π β§ (π:(1...(((π β 1) Β· (π β 1)) + 1))β1-1βπ΄ β§ π Isom < , < ((1...(((π β 1) Β· (π β 1)) + 1)), ran π))) β βπ β π« π΄((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) |
16 | 7, 15 | exlimddv 1930 | 1 β’ (π β βπ β π« π΄((π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , < (π , (πΉ β π ))) β¨ (π β€ (β―βπ ) β§ (πΉ βΎ π ) Isom < , β‘ < (π , (πΉ β π ))))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 β¨ wo 845 β wcel 2098 βwrex 3067 β wss 3949 π« cpw 4606 class class class wbr 5152 β‘ccnv 5681 ran crn 5683 βΎ cres 5684 β cima 5685 β1-1βwf1 6550 βcfv 6553 Isom wiso 6554 (class class class)co 7426 βcr 11145 1c1 11147 + caddc 11149 Β· cmul 11151 < clt 11286 β€ cle 11287 β cmin 11482 βcn 12250 ...cfz 13524 β―chash 14329 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-dju 9932 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-nn 12251 df-n0 12511 df-xnn0 12583 df-z 12597 df-uz 12861 df-fz 13525 df-hash 14330 |
This theorem is referenced by: (None) |
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