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Mirrors > Home > ILE Home > Th. List > absid | Unicode version |
Description: A nonnegative number is its own absolute value. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
absid |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 107 |
. . . 4
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2 | 1 | recnd 7419 |
. . 3
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3 | absval 10261 |
. . 3
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4 | 2, 3 | syl 14 |
. 2
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5 | 1 | cjred 10232 |
. . . . 5
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6 | 5 | oveq2d 5607 |
. . . 4
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7 | 2 | sqvald 9918 |
. . . 4
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8 | 6, 7 | eqtr4d 2118 |
. . 3
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9 | 8 | fveq2d 5257 |
. 2
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10 | sqrtsq 10304 |
. 2
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11 | 4, 9, 10 | 3eqtrd 2119 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-mulrcl 7347 ax-addcom 7348 ax-mulcom 7349 ax-addass 7350 ax-mulass 7351 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-1rid 7355 ax-0id 7356 ax-rnegex 7357 ax-precex 7358 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-apti 7363 ax-pre-ltadd 7364 ax-pre-mulgt0 7365 ax-pre-mulext 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4084 df-po 4087 df-iso 4088 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-1st 5846 df-2nd 5847 df-recs 6002 df-frec 6088 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-reap 7952 df-ap 7959 df-div 8038 df-inn 8317 df-2 8375 df-n0 8566 df-z 8647 df-uz 8915 df-iseq 9741 df-iexp 9792 df-cj 10103 df-re 10104 df-im 10105 df-rsqrt 10258 df-abs 10259 |
This theorem is referenced by: abs1 10332 absnid 10333 leabs 10334 qabsor 10335 sqabs 10342 nn0abscl 10345 ltabs 10347 abslt 10348 absle 10349 absidm 10358 abssubge0 10362 fzomaxdiflem 10372 absidi 10386 absidd 10427 6gcd4e2 10764 lcmgcdnn 10844 ex-gcd 11001 |
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