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Mirrors > Home > MPE Home > Th. List > ruclem7 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16190. 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 484 | . . . . 5 β’ ((π β§ π β β0) β π β β0) | |
2 | nn0uz 12865 | . . . . 5 β’ β0 = (β€β₯β0) | |
3 | 1, 2 | eleqtrdi 2837 | . . . 4 β’ ((π β§ π β β0) β π β (β€β₯β0)) |
4 | seqp1 13984 | . . . 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 6885 | . . 3 β’ (πΊβ(π + 1)) = (seq0(π·, πΆ)β(π + 1)) |
8 | 6 | fveq1i 6885 | . . . 4 β’ (πΊβπ) = (seq0(π·, πΆ)βπ) |
9 | 8 | oveq1i 7414 | . . 3 β’ ((πΊβπ)π·(πΆβ(π + 1))) = ((seq0(π·, πΆ)βπ)π·(πΆβ(π + 1))) |
10 | 5, 7, 9 | 3eqtr4g 2791 | . 2 β’ ((π β§ π β β0) β (πΊβ(π + 1)) = ((πΊβπ)π·(πΆβ(π + 1)))) |
11 | nn0p1nn 12512 | . . . . . . 7 β’ (π β β0 β (π + 1) β β) | |
12 | 11 | adantl 481 | . . . . . 6 β’ ((π β§ π β β0) β (π + 1) β β) |
13 | 12 | nnne0d 12263 | . . . . 5 β’ ((π β§ π β β0) β (π + 1) β 0) |
14 | 13 | necomd 2990 | . . . 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 4150 | . . . . . 6 β’ πΆ = (πΉ βͺ {β¨0, β¨0, 1β©β©}) |
17 | 16 | fveq1i 6885 | . . . . 5 β’ (πΆβ(π + 1)) = ((πΉ βͺ {β¨0, β¨0, 1β©β©})β(π + 1)) |
18 | fvunsn 7172 | . . . . 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 2778 | . . . 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 7420 | . 2 β’ ((π β§ π β β0) β ((πΊβπ)π·(πΆβ(π + 1))) = ((πΊβπ)π·(πΉβ(π + 1)))) |
22 | 10, 21 | eqtrd 2766 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 β¦csb 3888 βͺ cun 3941 ifcif 4523 {csn 4623 β¨cop 4629 class class class wbr 5141 Γ cxp 5667 βΆwf 6532 βcfv 6536 (class class class)co 7404 β cmpo 7406 1st c1st 7969 2nd c2nd 7970 βcr 11108 0cc0 11109 1c1 11110 + caddc 11112 < clt 11249 / cdiv 11872 βcn 12213 2c2 12268 β0cn0 12473 β€β₯cuz 12823 seqcseq 13969 |
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 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 |
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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-nn 12214 df-n0 12474 df-z 12560 df-uz 12824 df-seq 13970 |
This theorem is referenced by: ruclem8 16184 ruclem9 16185 ruclem12 16188 |
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