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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 34720 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 2726 | . . 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 34721 | . 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 480 | . . 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 480 | . . 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 480 | . . 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 480 | . . 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 480 | . . 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 768 | . . 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 770 | . . 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 34722 | . 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 395 β¨ wo 844 β wcel 2098 βwrex 3064 β wss 3943 π« cpw 4597 class class class wbr 5141 β‘ccnv 5668 ran crn 5670 βΎ cres 5671 β cima 5672 β1-1βwf1 6533 βcfv 6536 Isom wiso 6537 (class class class)co 7404 βcr 11108 1c1 11110 + caddc 11112 Β· cmul 11114 < clt 11249 β€ cle 11250 β cmin 11445 βcn 12213 ...cfz 13487 β―chash 14292 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-oi 9504 df-dju 9895 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-xnn0 12546 df-z 12560 df-uz 12824 df-fz 13488 df-hash 14293 |
This theorem is referenced by: (None) |
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