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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16132. Successor value for the interval sequence. (Contributed by Mario Carneiro, 28-May-2014.) |
Ref | Expression |
---|---|
ruc.1 | β’ (π β πΉ:ββΆβ) |
ruc.2 | β’ (π β π· = (π₯ β (β Γ β), π¦ β β β¦ β¦(((1st βπ₯) + (2nd βπ₯)) / 2) / πβ¦if(π < π¦, β¨(1st βπ₯), πβ©, β¨((π + (2nd βπ₯)) / 2), (2nd βπ₯)β©))) |
ruc.4 | β’ πΆ = ({β¨0, β¨0, 1β©β©} βͺ πΉ) |
ruc.5 | β’ πΊ = seq0(π·, πΆ) |
Ref | Expression |
---|---|
ruclem7 | β’ ((π β§ π β β0) β (πΊβ(π + 1)) = ((πΊβπ)π·(πΉβ(π + 1)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 486 | . . . . 5 β’ ((π β§ π β β0) β π β β0) | |
2 | nn0uz 12812 | . . . . 5 β’ β0 = (β€β₯β0) | |
3 | 1, 2 | eleqtrdi 2848 | . . . 4 β’ ((π β§ π β β0) β π β (β€β₯β0)) |
4 | seqp1 13928 | . . . 4 β’ (π β (β€β₯β0) β (seq0(π·, πΆ)β(π + 1)) = ((seq0(π·, πΆ)βπ)π·(πΆβ(π + 1)))) | |
5 | 3, 4 | syl 17 | . . 3 β’ ((π β§ π β β0) β (seq0(π·, πΆ)β(π + 1)) = ((seq0(π·, πΆ)βπ)π·(πΆβ(π + 1)))) |
6 | ruc.5 | . . . 4 β’ πΊ = seq0(π·, πΆ) | |
7 | 6 | fveq1i 6848 | . . 3 β’ (πΊβ(π + 1)) = (seq0(π·, πΆ)β(π + 1)) |
8 | 6 | fveq1i 6848 | . . . 4 β’ (πΊβπ) = (seq0(π·, πΆ)βπ) |
9 | 8 | oveq1i 7372 | . . 3 β’ ((πΊβπ)π·(πΆβ(π + 1))) = ((seq0(π·, πΆ)βπ)π·(πΆβ(π + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2802 | . 2 β’ ((π β§ π β β0) β (πΊβ(π + 1)) = ((πΊβπ)π·(πΆβ(π + 1)))) |
11 | nn0p1nn 12459 | . . . . . . 7 β’ (π β β0 β (π + 1) β β) | |
12 | 11 | adantl 483 | . . . . . 6 β’ ((π β§ π β β0) β (π + 1) β β) |
13 | 12 | nnne0d 12210 | . . . . 5 β’ ((π β§ π β β0) β (π + 1) β 0) |
14 | 13 | necomd 3000 | . . . 4 β’ ((π β§ π β β0) β 0 β (π + 1)) |
15 | ruc.4 | . . . . . . 7 β’ πΆ = ({β¨0, β¨0, 1β©β©} βͺ πΉ) | |
16 | 15 | equncomi 4120 | . . . . . 6 β’ πΆ = (πΉ βͺ {β¨0, β¨0, 1β©β©}) |
17 | 16 | fveq1i 6848 | . . . . 5 β’ (πΆβ(π + 1)) = ((πΉ βͺ {β¨0, β¨0, 1β©β©})β(π + 1)) |
18 | fvunsn 7130 | . . . . 5 β’ (0 β (π + 1) β ((πΉ βͺ {β¨0, β¨0, 1β©β©})β(π + 1)) = (πΉβ(π + 1))) | |
19 | 17, 18 | eqtrid 2789 | . . . 4 β’ (0 β (π + 1) β (πΆβ(π + 1)) = (πΉβ(π + 1))) |
20 | 14, 19 | syl 17 | . . 3 β’ ((π β§ π β β0) β (πΆβ(π + 1)) = (πΉβ(π + 1))) |
21 | 20 | oveq2d 7378 | . 2 β’ ((π β§ π β β0) β ((πΊβπ)π·(πΆβ(π + 1))) = ((πΊβπ)π·(πΉβ(π + 1)))) |
22 | 10, 21 | eqtrd 2777 | 1 β’ ((π β§ π β β0) β (πΊβ(π + 1)) = ((πΊβπ)π·(πΉβ(π + 1)))) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 β¦csb 3860 βͺ cun 3913 ifcif 4491 {csn 4591 β¨cop 4597 class class class wbr 5110 Γ cxp 5636 βΆwf 6497 βcfv 6501 (class class class)co 7362 β cmpo 7364 1st c1st 7924 2nd c2nd 7925 βcr 11057 0cc0 11058 1c1 11059 + caddc 11061 < clt 11196 / cdiv 11819 βcn 12160 2c2 12215 β0cn0 12420 β€β₯cuz 12770 seqcseq 13913 |
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 2708 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 |
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 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-en 8891 df-dom 8892 df-sdom 8893 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-n0 12421 df-z 12507 df-uz 12771 df-seq 13914 |
This theorem is referenced by: ruclem8 16126 ruclem9 16127 ruclem12 16130 |
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