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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 12896 | . . . . . . 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 13776 | . . . . . 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 2811 | . . . 4 β’ (π β (π β (1...π) β π β ((1..^π) βͺ {π}))) |
7 | elun 4141 | . . . . 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 4640 | . . . . . 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 913 | . . . 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 304 | . . 3 β’ (π β (π β (1...π) β (π β (1..^π) β¨ π = π))) |
13 | fveq2 6892 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
14 | 13 | breq2d 5155 | . . . . . . 7 β’ (π = π β ((πβ0) < (πβπ) β (πβ0) < (πβπ))) |
15 | 14 | rspccv 3598 | . . . . . 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 46826 | . . . . . 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 46827 | . . . . . . . 8 β’ (π β (πβ0) < (πβπ)) |
20 | 19 | adantl 480 | . . . . . . 7 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
21 | fveq2 6892 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
22 | 21 | adantr 479 | . . . . . . 7 β’ ((π = π β§ π) β (πβπ) = (πβπ)) |
23 | 20, 22 | breqtrrd 5171 | . . . . . 6 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
24 | 23 | ex 411 | . . . . 5 β’ (π = π β (π β (πβ0) < (πβπ))) |
25 | 18, 24 | jaoi 855 | . . . 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 3135 | 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 394 β¨ wo 845 = wceq 1533 β wcel 2098 βwral 3051 βͺ cun 3937 {csn 4624 class class class wbr 5143 βcfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 < clt 11278 βcn 12242 β€β₯cuz 12852 ...cfz 13516 ..^cfzo 13659 RePartciccp 46816 |
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 2696 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7991 df-2nd 7992 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-er 8723 df-map 8845 df-en 8963 df-dom 8964 df-sdom 8965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-nn 12243 df-2 12305 df-n0 12503 df-z 12589 df-uz 12853 df-fz 13517 df-fzo 13660 df-iccp 46817 |
This theorem is referenced by: iccpartgel 46832 |
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