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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > iccpartgtl | Structured version Visualization version GIF version |
Description: If there is a partition, then all intermediate points and the upper bound are strictly greater than the lower bound. (Contributed by AV, 14-Jul-2020.) |
Ref | Expression |
---|---|
iccpartgtprec.m | β’ (π β π β β) |
iccpartgtprec.p | β’ (π β π β (RePartβπ)) |
Ref | Expression |
---|---|
iccpartgtl | β’ (π β βπ β (1...π)(πβ0) < (πβπ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccpartgtprec.m | . . . . . . 7 β’ (π β π β β) | |
2 | elnnuz 12815 | . . . . . . 7 β’ (π β β β π β (β€β₯β1)) | |
3 | 1, 2 | sylib 217 | . . . . . 6 β’ (π β π β (β€β₯β1)) |
4 | fzisfzounsn 13693 | . . . . . 6 β’ (π β (β€β₯β1) β (1...π) = ((1..^π) βͺ {π})) | |
5 | 3, 4 | syl 17 | . . . . 5 β’ (π β (1...π) = ((1..^π) βͺ {π})) |
6 | 5 | eleq2d 2820 | . . . 4 β’ (π β (π β (1...π) β π β ((1..^π) βͺ {π}))) |
7 | elun 4112 | . . . . 5 β’ (π β ((1..^π) βͺ {π}) β (π β (1..^π) β¨ π β {π})) | |
8 | 7 | a1i 11 | . . . 4 β’ (π β (π β ((1..^π) βͺ {π}) β (π β (1..^π) β¨ π β {π}))) |
9 | velsn 4606 | . . . . . 6 β’ (π β {π} β π = π) | |
10 | 9 | a1i 11 | . . . . 5 β’ (π β (π β {π} β π = π)) |
11 | 10 | orbi2d 915 | . . . 4 β’ (π β ((π β (1..^π) β¨ π β {π}) β (π β (1..^π) β¨ π = π))) |
12 | 6, 8, 11 | 3bitrd 305 | . . 3 β’ (π β (π β (1...π) β (π β (1..^π) β¨ π = π))) |
13 | fveq2 6846 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
14 | 13 | breq2d 5121 | . . . . . . 7 β’ (π = π β ((πβ0) < (πβπ) β (πβ0) < (πβπ))) |
15 | 14 | rspccv 3580 | . . . . . 6 β’ (βπ β (1..^π)(πβ0) < (πβπ) β (π β (1..^π) β (πβ0) < (πβπ))) |
16 | iccpartgtprec.p | . . . . . . 7 β’ (π β π β (RePartβπ)) | |
17 | 1, 16 | iccpartigtl 45705 | . . . . . 6 β’ (π β βπ β (1..^π)(πβ0) < (πβπ)) |
18 | 15, 17 | syl11 33 | . . . . 5 β’ (π β (1..^π) β (π β (πβ0) < (πβπ))) |
19 | 1, 16 | iccpartlt 45706 | . . . . . . . 8 β’ (π β (πβ0) < (πβπ)) |
20 | 19 | adantl 483 | . . . . . . 7 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
21 | fveq2 6846 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
22 | 21 | adantr 482 | . . . . . . 7 β’ ((π = π β§ π) β (πβπ) = (πβπ)) |
23 | 20, 22 | breqtrrd 5137 | . . . . . 6 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
24 | 23 | ex 414 | . . . . 5 β’ (π = π β (π β (πβ0) < (πβπ))) |
25 | 18, 24 | jaoi 856 | . . . 4 β’ ((π β (1..^π) β¨ π = π) β (π β (πβ0) < (πβπ))) |
26 | 25 | com12 32 | . . 3 β’ (π β ((π β (1..^π) β¨ π = π) β (πβ0) < (πβπ))) |
27 | 12, 26 | sylbid 239 | . 2 β’ (π β (π β (1...π) β (πβ0) < (πβπ))) |
28 | 27 | ralrimiv 3139 | 1 β’ (π β βπ β (1...π)(πβ0) < (πβπ)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 β¨ wo 846 = wceq 1542 β wcel 2107 βwral 3061 βͺ cun 3912 {csn 4590 class class class wbr 5109 βcfv 6500 (class class class)co 7361 0cc0 11059 1c1 11060 < clt 11197 βcn 12161 β€β₯cuz 12771 ...cfz 13433 ..^cfzo 13576 RePartciccp 45695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-1st 7925 df-2nd 7926 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-er 8654 df-map 8773 df-en 8890 df-dom 8891 df-sdom 8892 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-nn 12162 df-2 12224 df-n0 12422 df-z 12508 df-uz 12772 df-fz 13434 df-fzo 13577 df-iccp 45696 |
This theorem is referenced by: iccpartgel 45711 |
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