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Mirrors > Home > MPE Home > Th. List > rpnnen1 | Structured version Visualization version GIF version |
Description: One half of rpnnen 16116, where we show an injection from the real numbers to sequences of rational numbers. Specifically, we map a real number π₯ to the sequence (πΉβπ₯):ββΆβ (see rpnnen1lem6 12914) such that ((πΉβπ₯)βπ) is the largest rational number with denominator π that is strictly less than π₯. In this manner, we get a monotonically increasing sequence that converges to π₯, and since each sequence converges to a unique real number, this mapping from reals to sequences of rational numbers is injective. Note: The β and β existence hypotheses provide for use with either nnex 12166 and qex 12893, or nnexALT 12162 and qexALT 12896. The proof should not be modified to use any of those 4 theorems. (Contributed by Mario Carneiro, 13-May-2013.) (Revised by Mario Carneiro, 16-Jun-2013.) (Revised by NM, 15-Aug-2021.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rpnnen1.n | β’ β β V |
rpnnen1.q | β’ β β V |
Ref | Expression |
---|---|
rpnnen1 | β’ β βΌ (β βm β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq1 7369 | . . . 4 β’ (π = π β (π / π) = (π / π)) | |
2 | 1 | breq1d 5120 | . . 3 β’ (π = π β ((π / π) < π₯ β (π / π) < π₯)) |
3 | 2 | cbvrabv 3420 | . 2 β’ {π β β€ β£ (π / π) < π₯} = {π β β€ β£ (π / π) < π₯} |
4 | oveq2 7370 | . . . . . . . . 9 β’ (π = π β (π / π) = (π / π)) | |
5 | 4 | breq1d 5120 | . . . . . . . 8 β’ (π = π β ((π / π) < π¦ β (π / π) < π¦)) |
6 | 5 | rabbidv 3418 | . . . . . . 7 β’ (π = π β {π β β€ β£ (π / π) < π¦} = {π β β€ β£ (π / π) < π¦}) |
7 | 6 | supeq1d 9389 | . . . . . 6 β’ (π = π β sup({π β β€ β£ (π / π) < π¦}, β, < ) = sup({π β β€ β£ (π / π) < π¦}, β, < )) |
8 | id 22 | . . . . . 6 β’ (π = π β π = π) | |
9 | 7, 8 | oveq12d 7380 | . . . . 5 β’ (π = π β (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π) = (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π)) |
10 | 9 | cbvmptv 5223 | . . . 4 β’ (π β β β¦ (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π)) = (π β β β¦ (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π)) |
11 | breq2 5114 | . . . . . . . 8 β’ (π¦ = π₯ β ((π / π) < π¦ β (π / π) < π₯)) | |
12 | 11 | rabbidv 3418 | . . . . . . 7 β’ (π¦ = π₯ β {π β β€ β£ (π / π) < π¦} = {π β β€ β£ (π / π) < π₯}) |
13 | 12 | supeq1d 9389 | . . . . . 6 β’ (π¦ = π₯ β sup({π β β€ β£ (π / π) < π¦}, β, < ) = sup({π β β€ β£ (π / π) < π₯}, β, < )) |
14 | 13 | oveq1d 7377 | . . . . 5 β’ (π¦ = π₯ β (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π) = (sup({π β β€ β£ (π / π) < π₯}, β, < ) / π)) |
15 | 14 | mpteq2dv 5212 | . . . 4 β’ (π¦ = π₯ β (π β β β¦ (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π)) = (π β β β¦ (sup({π β β€ β£ (π / π) < π₯}, β, < ) / π))) |
16 | 10, 15 | eqtrid 2789 | . . 3 β’ (π¦ = π₯ β (π β β β¦ (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π)) = (π β β β¦ (sup({π β β€ β£ (π / π) < π₯}, β, < ) / π))) |
17 | 16 | cbvmptv 5223 | . 2 β’ (π¦ β β β¦ (π β β β¦ (sup({π β β€ β£ (π / π) < π¦}, β, < ) / π))) = (π₯ β β β¦ (π β β β¦ (sup({π β β€ β£ (π / π) < π₯}, β, < ) / π))) |
18 | rpnnen1.n | . 2 β’ β β V | |
19 | rpnnen1.q | . 2 β’ β β V | |
20 | 3, 17, 18, 19 | rpnnen1lem6 12914 | 1 β’ β βΌ (β βm β) |
Colors of variables: wff setvar class |
Syntax hints: β wcel 2107 {crab 3410 Vcvv 3448 class class class wbr 5110 β¦ cmpt 5193 (class class class)co 7362 βm cmap 8772 βΌ cdom 8888 supcsup 9383 βcr 11057 < clt 11196 / cdiv 11819 βcn 12160 β€cz 12506 βcq 12880 |
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-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 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 ax-pre-sup 11136 |
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-rmo 3356 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-1st 7926 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-er 8655 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-sup 9385 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 df-nn 12161 df-n0 12421 df-z 12507 df-q 12881 |
This theorem is referenced by: reexALT 12916 rpnnen 16116 |
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