![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rpnnen2lem6 | Structured version Visualization version GIF version |
Description: Lemma for rpnnen2 16188. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 30-Apr-2014.) |
Ref | Expression |
---|---|
rpnnen2.1 | β’ πΉ = (π₯ β π« β β¦ (π β β β¦ if(π β π₯, ((1 / 3)βπ), 0))) |
Ref | Expression |
---|---|
rpnnen2lem6 | β’ ((π΄ β β β§ π β β) β Ξ£π β (β€β₯βπ)((πΉβπ΄)βπ) β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2727 | . 2 β’ (β€β₯βπ) = (β€β₯βπ) | |
2 | nnz 12595 | . . 3 β’ (π β β β π β β€) | |
3 | 2 | adantl 481 | . 2 β’ ((π΄ β β β§ π β β) β π β β€) |
4 | eqidd 2728 | . 2 β’ (((π΄ β β β§ π β β) β§ π β (β€β₯βπ)) β ((πΉβπ΄)βπ) = ((πΉβπ΄)βπ)) | |
5 | rpnnen2.1 | . . . . 5 β’ πΉ = (π₯ β π« β β¦ (π β β β¦ if(π β π₯, ((1 / 3)βπ), 0))) | |
6 | 5 | rpnnen2lem2 16177 | . . . 4 β’ (π΄ β β β (πΉβπ΄):ββΆβ) |
7 | 6 | ad2antrr 725 | . . 3 β’ (((π΄ β β β§ π β β) β§ π β (β€β₯βπ)) β (πΉβπ΄):ββΆβ) |
8 | eluznn 12918 | . . . 4 β’ ((π β β β§ π β (β€β₯βπ)) β π β β) | |
9 | 8 | adantll 713 | . . 3 β’ (((π΄ β β β§ π β β) β§ π β (β€β₯βπ)) β π β β) |
10 | 7, 9 | ffvelcdmd 7089 | . 2 β’ (((π΄ β β β§ π β β) β§ π β (β€β₯βπ)) β ((πΉβπ΄)βπ) β β) |
11 | 5 | rpnnen2lem5 16180 | . 2 β’ ((π΄ β β β§ π β β) β seqπ( + , (πΉβπ΄)) β dom β ) |
12 | 1, 3, 4, 10, 11 | isumrecl 15729 | 1 β’ ((π΄ β β β§ π β β) β Ξ£π β (β€β₯βπ)((πΉβπ΄)βπ) β β) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 395 = wceq 1534 β wcel 2099 β wss 3944 ifcif 4524 π« cpw 4598 β¦ cmpt 5225 βΆwf 6538 βcfv 6542 (class class class)co 7414 βcr 11123 0cc0 11124 1c1 11125 / cdiv 11887 βcn 12228 3c3 12284 β€cz 12574 β€β₯cuz 12838 βcexp 14044 Ξ£csu 15650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 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 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7863 df-1st 7985 df-2nd 7986 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-er 8716 df-pm 8837 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-sup 9451 df-inf 9452 df-oi 9519 df-card 9948 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-n0 12489 df-z 12575 df-uz 12839 df-rp 12993 df-ico 13348 df-fz 13503 df-fzo 13646 df-fl 13775 df-seq 13985 df-exp 14045 df-hash 14308 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-sum 15651 |
This theorem is referenced by: rpnnen2lem10 16185 rpnnen2lem11 16186 rpnnen2lem12 16187 |
Copyright terms: Public domain | W3C validator |