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Mirrors > Home > MPE Home > Th. List > ruclem4 | Structured version Visualization version GIF version |
Description: Lemma for ruc 16183. Initial value of 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 |
---|---|
ruclem4 | β’ (π β (πΊβ0) = β¨0, 1β©) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ruc.5 | . . 3 β’ πΊ = seq0(π·, πΆ) | |
2 | 1 | fveq1i 6890 | . 2 β’ (πΊβ0) = (seq0(π·, πΆ)β0) |
3 | 0z 12566 | . . 3 β’ 0 β β€ | |
4 | ruc.4 | . . . . . 6 β’ πΆ = ({β¨0, β¨0, 1β©β©} βͺ πΉ) | |
5 | ruc.1 | . . . . . . . . 9 β’ (π β πΉ:ββΆβ) | |
6 | ffn 6715 | . . . . . . . . 9 β’ (πΉ:ββΆβ β πΉ Fn β) | |
7 | fnresdm 6667 | . . . . . . . . 9 β’ (πΉ Fn β β (πΉ βΎ β) = πΉ) | |
8 | 5, 6, 7 | 3syl 18 | . . . . . . . 8 β’ (π β (πΉ βΎ β) = πΉ) |
9 | dfn2 12482 | . . . . . . . . 9 β’ β = (β0 β {0}) | |
10 | 9 | reseq2i 5977 | . . . . . . . 8 β’ (πΉ βΎ β) = (πΉ βΎ (β0 β {0})) |
11 | 8, 10 | eqtr3di 2788 | . . . . . . 7 β’ (π β πΉ = (πΉ βΎ (β0 β {0}))) |
12 | 11 | uneq2d 4163 | . . . . . 6 β’ (π β ({β¨0, β¨0, 1β©β©} βͺ πΉ) = ({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0})))) |
13 | 4, 12 | eqtrid 2785 | . . . . 5 β’ (π β πΆ = ({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0})))) |
14 | 13 | fveq1d 6891 | . . . 4 β’ (π β (πΆβ0) = (({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0})))β0)) |
15 | c0ex 11205 | . . . . . . 7 β’ 0 β V | |
16 | 15 | a1i 11 | . . . . . 6 β’ (β€ β 0 β V) |
17 | opex 5464 | . . . . . . 7 β’ β¨0, 1β© β V | |
18 | 17 | a1i 11 | . . . . . 6 β’ (β€ β β¨0, 1β© β V) |
19 | eqid 2733 | . . . . . 6 β’ ({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0}))) = ({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0}))) | |
20 | 16, 18, 19 | fvsnun1 7177 | . . . . 5 β’ (β€ β (({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0})))β0) = β¨0, 1β©) |
21 | 20 | mptru 1549 | . . . 4 β’ (({β¨0, β¨0, 1β©β©} βͺ (πΉ βΎ (β0 β {0})))β0) = β¨0, 1β© |
22 | 14, 21 | eqtrdi 2789 | . . 3 β’ (π β (πΆβ0) = β¨0, 1β©) |
23 | 3, 22 | seq1i 13977 | . 2 β’ (π β (seq0(π·, πΆ)β0) = β¨0, 1β©) |
24 | 2, 23 | eqtrid 2785 | 1 β’ (π β (πΊβ0) = β¨0, 1β©) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β€wtru 1543 β wcel 2107 Vcvv 3475 β¦csb 3893 β cdif 3945 βͺ cun 3946 ifcif 4528 {csn 4628 β¨cop 4634 class class class wbr 5148 Γ cxp 5674 βΎ cres 5678 Fn wfn 6536 βΆwf 6537 βcfv 6541 (class class class)co 7406 β cmpo 7408 1st c1st 7970 2nd c2nd 7971 βcr 11106 0cc0 11107 1c1 11108 + caddc 11110 < clt 11245 / cdiv 11868 βcn 12209 2c2 12264 β0cn0 12469 seqcseq 13963 |
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 2704 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-2nd 7973 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-er 8700 df-en 8937 df-dom 8938 df-sdom 8939 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-n0 12470 df-z 12556 df-uz 12820 df-seq 13964 |
This theorem is referenced by: ruclem6 16175 ruclem8 16177 ruclem11 16180 |
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