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Mirrors > Home > MPE Home > Th. List > absabv | Structured version Visualization version GIF version |
Description: The regular absolute value function on the complex numbers is in fact an absolute value under our definition. (Contributed by Mario Carneiro, 4-Dec-2014.) |
Ref | Expression |
---|---|
absabv | ⊢ abs ∈ (AbsVal‘ℂfld) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2824 | . . 3 ⊢ (⊤ → (AbsVal‘ℂfld) = (AbsVal‘ℂfld)) | |
2 | cnfldbas 20551 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → ℂ = (Base‘ℂfld)) |
4 | cnfldadd 20552 | . . . 4 ⊢ + = (+g‘ℂfld) | |
5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → + = (+g‘ℂfld)) |
6 | cnfldmul 20553 | . . . 4 ⊢ · = (.r‘ℂfld) | |
7 | 6 | a1i 11 | . . 3 ⊢ (⊤ → · = (.r‘ℂfld)) |
8 | cnfld0 20571 | . . . 4 ⊢ 0 = (0g‘ℂfld) | |
9 | 8 | a1i 11 | . . 3 ⊢ (⊤ → 0 = (0g‘ℂfld)) |
10 | cnring 20569 | . . . 4 ⊢ ℂfld ∈ Ring | |
11 | 10 | a1i 11 | . . 3 ⊢ (⊤ → ℂfld ∈ Ring) |
12 | absf 14699 | . . . 4 ⊢ abs:ℂ⟶ℝ | |
13 | 12 | a1i 11 | . . 3 ⊢ (⊤ → abs:ℂ⟶ℝ) |
14 | abs0 14647 | . . . 4 ⊢ (abs‘0) = 0 | |
15 | 14 | a1i 11 | . . 3 ⊢ (⊤ → (abs‘0) = 0) |
16 | absgt0 14686 | . . . . 5 ⊢ (𝑥 ∈ ℂ → (𝑥 ≠ 0 ↔ 0 < (abs‘𝑥))) | |
17 | 16 | biimpa 479 | . . . 4 ⊢ ((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
18 | 17 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ 𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) → 0 < (abs‘𝑥)) |
19 | absmul 14656 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) | |
20 | 19 | ad2ant2r 745 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
21 | 20 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 · 𝑦)) = ((abs‘𝑥) · (abs‘𝑦))) |
22 | abstri 14692 | . . . . 5 ⊢ ((𝑥 ∈ ℂ ∧ 𝑦 ∈ ℂ) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) | |
23 | 22 | ad2ant2r 745 | . . . 4 ⊢ (((𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) |
24 | 23 | 3adant1 1126 | . . 3 ⊢ ((⊤ ∧ (𝑥 ∈ ℂ ∧ 𝑥 ≠ 0) ∧ (𝑦 ∈ ℂ ∧ 𝑦 ≠ 0)) → (abs‘(𝑥 + 𝑦)) ≤ ((abs‘𝑥) + (abs‘𝑦))) |
25 | 1, 3, 5, 7, 9, 11, 13, 15, 18, 21, 24 | isabvd 19593 | . 2 ⊢ (⊤ → abs ∈ (AbsVal‘ℂfld)) |
26 | 25 | mptru 1544 | 1 ⊢ abs ∈ (AbsVal‘ℂfld) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ⊤wtru 1538 ∈ wcel 2114 ≠ wne 3018 class class class wbr 5068 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 ℂcc 10537 ℝcr 10538 0cc0 10539 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 abscabs 14595 Basecbs 16485 +gcplusg 16567 .rcmulr 16568 0gc0g 16715 Ringcrg 19299 AbsValcabv 19589 ℂfldccnfld 20547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-ico 12747 df-fz 12896 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-plusg 16580 df-mulr 16581 df-starv 16582 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-grp 18108 df-minusg 18109 df-cmn 18910 df-mgp 19242 df-ring 19301 df-cring 19302 df-abv 19590 df-cnfld 20548 |
This theorem is referenced by: cnnrg 23391 cnindmet 23768 qabsabv 26207 |
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