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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 109 |
. . . 4
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2 | 1 | recnd 8017 |
. . 3
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3 | absval 11045 |
. . 3
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4 | 2, 3 | syl 14 |
. 2
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5 | 1 | cjred 11015 |
. . . . 5
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6 | 5 | oveq2d 5913 |
. . . 4
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7 | 2 | sqvald 10685 |
. . . 4
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8 | 6, 7 | eqtr4d 2225 |
. . 3
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9 | 8 | fveq2d 5538 |
. 2
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10 | sqrtsq 11088 |
. 2
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11 | 4, 9, 10 | 3eqtrd 2226 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4133 ax-sep 4136 ax-nul 4144 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-iinf 4605 ax-cnex 7933 ax-resscn 7934 ax-1cn 7935 ax-1re 7936 ax-icn 7937 ax-addcl 7938 ax-addrcl 7939 ax-mulcl 7940 ax-mulrcl 7941 ax-addcom 7942 ax-mulcom 7943 ax-addass 7944 ax-mulass 7945 ax-distr 7946 ax-i2m1 7947 ax-0lt1 7948 ax-1rid 7949 ax-0id 7950 ax-rnegex 7951 ax-precex 7952 ax-cnre 7953 ax-pre-ltirr 7954 ax-pre-ltwlin 7955 ax-pre-lttrn 7956 ax-pre-apti 7957 ax-pre-ltadd 7958 ax-pre-mulgt0 7959 ax-pre-mulext 7960 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4311 df-po 4314 df-iso 4315 df-iord 4384 df-on 4386 df-ilim 4387 df-suc 4389 df-iom 4608 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-f1 5240 df-fo 5241 df-f1o 5242 df-fv 5243 df-riota 5852 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-recs 6331 df-frec 6417 df-pnf 8025 df-mnf 8026 df-xr 8027 df-ltxr 8028 df-le 8029 df-sub 8161 df-neg 8162 df-reap 8563 df-ap 8570 df-div 8661 df-inn 8951 df-2 9009 df-n0 9208 df-z 9285 df-uz 9560 df-seqfrec 10479 df-exp 10554 df-cj 10886 df-re 10887 df-im 10888 df-rsqrt 11042 df-abs 11043 |
This theorem is referenced by: abs1 11116 absnid 11117 leabs 11118 qabsor 11119 sqabs 11126 nn0abscl 11129 ltabs 11131 abslt 11132 absle 11133 absidm 11142 abssubge0 11146 fzomaxdiflem 11156 absidi 11170 absidd 11211 geo2lim 11559 geoihalfsum 11565 ege2le3 11714 eirraplem 11819 6gcd4e2 12031 lcmgcdnn 12117 ex-gcd 14961 |
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