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Mirrors > Home > MPE Home > Th. List > releabs | Structured version Visualization version GIF version |
Description: The real part of a number is less than or equal to its absolute value. Proposition 10-3.7(d) of [Gleason] p. 133. (Contributed by NM, 1-Apr-2005.) |
Ref | Expression |
---|---|
releabs | ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | recl 15098 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℝ) | |
2 | 1 | recnd 11279 | . . 3 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ∈ ℂ) |
3 | abscl 15266 | . . 3 ⊢ ((ℜ‘𝐴) ∈ ℂ → (abs‘(ℜ‘𝐴)) ∈ ℝ) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘𝐴)) ∈ ℝ) |
5 | abscl 15266 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
6 | leabs 15287 | . . 3 ⊢ ((ℜ‘𝐴) ∈ ℝ → (ℜ‘𝐴) ≤ (abs‘(ℜ‘𝐴))) | |
7 | 1, 6 | syl 17 | . 2 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘(ℜ‘𝐴))) |
8 | absrele 15296 | . 2 ⊢ (𝐴 ∈ ℂ → (abs‘(ℜ‘𝐴)) ≤ (abs‘𝐴)) | |
9 | 1, 4, 5, 7, 8 | letrd 11408 | 1 ⊢ (𝐴 ∈ ℂ → (ℜ‘𝐴) ≤ (abs‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 class class class wbr 5149 ‘cfv 6549 ℂcc 11143 ℝcr 11144 ≤ cle 11286 ℜcre 15085 abscabs 15222 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11201 ax-resscn 11202 ax-1cn 11203 ax-icn 11204 ax-addcl 11205 ax-addrcl 11206 ax-mulcl 11207 ax-mulrcl 11208 ax-mulcom 11209 ax-addass 11210 ax-mulass 11211 ax-distr 11212 ax-i2m1 11213 ax-1ne0 11214 ax-1rid 11215 ax-rnegex 11216 ax-rrecex 11217 ax-cnre 11218 ax-pre-lttri 11219 ax-pre-lttrn 11220 ax-pre-ltadd 11221 ax-pre-mulgt0 11222 ax-pre-sup 11223 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9472 df-pnf 11287 df-mnf 11288 df-xr 11289 df-ltxr 11290 df-le 11291 df-sub 11483 df-neg 11484 df-div 11909 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14008 df-exp 14068 df-cj 15087 df-re 15088 df-im 15089 df-sqrt 15223 df-abs 15224 |
This theorem is referenced by: abstri 15318 sqreulem 15347 releabsi 15390 releabsd 15439 isosctrlem1 26800 sqrtcvallem2 43211 |
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