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Mirrors > Home > ILE Home > Th. List > abscld | Unicode version |
Description: Real closure of absolute value. (Contributed by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abscld.1 |
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Ref | Expression |
---|---|
abscld |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abscld.1 |
. 2
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2 | abscl 11044 |
. 2
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3 | 1, 2 | syl 14 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4115 ax-sep 4118 ax-nul 4126 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-iinf 4584 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 ax-arch 7921 ax-caucvg 7922 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-tr 4099 df-id 4290 df-po 4293 df-iso 4294 df-iord 4363 df-on 4365 df-ilim 4366 df-suc 4368 df-iom 4587 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-rn 4634 df-res 4635 df-ima 4636 df-iota 5174 df-fun 5214 df-fn 5215 df-f 5216 df-f1 5217 df-fo 5218 df-f1o 5219 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-1st 6135 df-2nd 6136 df-recs 6300 df-frec 6386 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 df-inn 8909 df-2 8967 df-3 8968 df-4 8969 df-n0 9166 df-z 9243 df-uz 9518 df-rp 9641 df-seqfrec 10432 df-exp 10506 df-cj 10835 df-re 10836 df-im 10837 df-rsqrt 10991 df-abs 10992 |
This theorem is referenced by: maxabsle 11197 maxabslemlub 11200 maxabslemval 11201 maxcl 11203 dfabsmax 11210 maxltsup 11211 max0addsup 11212 minabs 11228 bdtrilem 11231 bdtri 11232 mul0inf 11233 climuni 11285 climabs0 11299 mulcn2 11304 reccn2ap 11305 cn1lem 11306 cjcn2 11308 climsqz 11327 climsqz2 11328 climcvg1nlem 11341 fsumabs 11457 iserabs 11467 divcnv 11489 expcnv 11496 explecnv 11497 absltap 11501 absgtap 11502 georeclim 11505 geoisumr 11510 cvgratnnlemnexp 11516 cvgratnnlemmn 11517 cvgratnnlemabsle 11519 cvgratnnlemfm 11521 cvgratnnlemrate 11522 cvgratnn 11523 cvgratz 11524 mertenslemub 11526 mertenslemi1 11527 mertenslem2 11528 fprodabs 11608 efcllemp 11650 efaddlem 11666 eftlub 11682 ef01bndlem 11748 sin01bnd 11749 cos01bnd 11750 absef 11761 dvdsabseq 11836 alzdvds 11843 dvdsbnd 11940 sqnprm 12119 pclemub 12270 mul4sqlem 12374 addcncntoplem 13718 mulcncflem 13757 cnopnap 13761 limcimolemlt 13800 cnplimclemle 13804 limccnp2lem 13812 dveflem 13854 rpabscxpbnd 14026 lgsdirprm 14102 lgsdilem2 14104 lgsne0 14106 lgsabs1 14107 2sqlem1 14117 mul2sq 14119 2sqlem3 14120 qdencn 14431 apdifflemf 14450 apdiff 14452 |
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