Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > absre | GIF version |
Description: Absolute value of a real number. (Contributed by NM, 17-Mar-2005.) |
Ref | Expression |
---|---|
absre | ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴↑2))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recn 7900 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
2 | absval 10958 | . . 3 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴 · (∗‘𝐴)))) |
4 | 1 | sqvald 10599 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = (𝐴 · 𝐴)) |
5 | cjre 10839 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (∗‘𝐴) = 𝐴) | |
6 | 5 | oveq2d 5867 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 · (∗‘𝐴)) = (𝐴 · 𝐴)) |
7 | 4, 6 | eqtr4d 2206 | . . 3 ⊢ (𝐴 ∈ ℝ → (𝐴↑2) = (𝐴 · (∗‘𝐴))) |
8 | 7 | fveq2d 5498 | . 2 ⊢ (𝐴 ∈ ℝ → (√‘(𝐴↑2)) = (√‘(𝐴 · (∗‘𝐴)))) |
9 | 3, 8 | eqtr4d 2206 | 1 ⊢ (𝐴 ∈ ℝ → (abs‘𝐴) = (√‘(𝐴↑2))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∈ wcel 2141 ‘cfv 5196 (class class class)co 5851 ℂcc 7765 ℝcr 7766 · cmul 7772 2c2 8922 ↑cexp 10468 ∗ccj 10796 √csqrt 10953 abscabs 10954 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-n0 9129 df-z 9206 df-uz 9481 df-seqfrec 10395 df-exp 10469 df-cj 10799 df-re 10800 df-im 10801 df-rsqrt 10955 df-abs 10956 |
This theorem is referenced by: absrele 11040 absrei 11086 absred 11119 |
Copyright terms: Public domain | W3C validator |