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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 12870 | . . . . . . 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 13750 | . . . . . 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 2813 | . . . 4 β’ (π β (π β (1...π) β π β ((1..^π) βͺ {π}))) |
7 | elun 4143 | . . . . 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 4639 | . . . . . 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 912 | . . . 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 6885 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
14 | 13 | breq2d 5153 | . . . . . . 7 β’ (π = π β ((πβ0) < (πβπ) β (πβ0) < (πβπ))) |
15 | 14 | rspccv 3603 | . . . . . 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 46663 | . . . . . 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 46664 | . . . . . . . 8 β’ (π β (πβ0) < (πβπ)) |
20 | 19 | adantl 481 | . . . . . . 7 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
21 | fveq2 6885 | . . . . . . . 8 β’ (π = π β (πβπ) = (πβπ)) | |
22 | 21 | adantr 480 | . . . . . . 7 β’ ((π = π β§ π) β (πβπ) = (πβπ)) |
23 | 20, 22 | breqtrrd 5169 | . . . . . 6 β’ ((π = π β§ π) β (πβ0) < (πβπ)) |
24 | 23 | ex 412 | . . . . 5 β’ (π = π β (π β (πβ0) < (πβπ))) |
25 | 18, 24 | jaoi 854 | . . . 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 395 β¨ wo 844 = wceq 1533 β wcel 2098 βwral 3055 βͺ cun 3941 {csn 4623 class class class wbr 5141 βcfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 < clt 11252 βcn 12216 β€β₯cuz 12826 ...cfz 13490 ..^cfzo 13633 RePartciccp 46653 |
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-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 |
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-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-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-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 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-er 8705 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-nn 12217 df-2 12279 df-n0 12477 df-z 12563 df-uz 12827 df-fz 13491 df-fzo 13634 df-iccp 46654 |
This theorem is referenced by: iccpartgel 46669 |
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