Mathbox for Paul Chapman |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > abs2sqlti | Structured version Visualization version GIF version |
Description: The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.) |
Ref | Expression |
---|---|
abs2sqlti.1 | ⊢ 𝐴 ∈ ℂ |
abs2sqlti.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
abs2sqlti | ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | abs2sqlti.1 | . . 3 ⊢ 𝐴 ∈ ℂ | |
2 | 1 | absge0i 14759 | . 2 ⊢ 0 ≤ (abs‘𝐴) |
3 | abs2sqlti.2 | . . 3 ⊢ 𝐵 ∈ ℂ | |
4 | 3 | absge0i 14759 | . 2 ⊢ 0 ≤ (abs‘𝐵) |
5 | 1 | abscli 14758 | . . 3 ⊢ (abs‘𝐴) ∈ ℝ |
6 | 3 | abscli 14758 | . . 3 ⊢ (abs‘𝐵) ∈ ℝ |
7 | 5, 6 | lt2sqi 13555 | . 2 ⊢ ((0 ≤ (abs‘𝐴) ∧ 0 ≤ (abs‘𝐵)) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))) |
8 | 2, 4, 7 | mp2an 690 | 1 ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 < clt 10678 ≤ cle 10679 2c2 11695 ↑cexp 13432 abscabs 14596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-2nd 7693 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-sup 8909 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-rp 12393 df-seq 13373 df-exp 13433 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 |
This theorem is referenced by: abs2sqlt 32928 |
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