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Mirrors > Home > MPE Home > Th. List > abslem2 | Structured version Visualization version GIF version |
Description: Lemma involving absolute values. (Contributed by NM, 11-Oct-1999.) (Revised by Mario Carneiro, 29-May-2016.) |
Ref | Expression |
---|---|
abslem2 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((∗‘(𝐴 / (abs‘𝐴))) · 𝐴) + ((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = (2 · (abs‘𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | absvalsq 14640 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
2 | 1 | adantr 483 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) |
3 | abscl 14638 | . . . . . . . . . . 11 ⊢ (𝐴 ∈ ℂ → (abs‘𝐴) ∈ ℝ) | |
4 | 3 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℝ) |
5 | 4 | recnd 10669 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ∈ ℂ) |
6 | 5 | sqvald 13508 | . . . . . . . 8 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((abs‘𝐴)↑2) = ((abs‘𝐴) · (abs‘𝐴))) |
7 | 2, 6 | eqtr3d 2858 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 · (∗‘𝐴)) = ((abs‘𝐴) · (abs‘𝐴))) |
8 | 7 | oveq1d 7171 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / (abs‘𝐴)) = (((abs‘𝐴) · (abs‘𝐴)) / (abs‘𝐴))) |
9 | simpl 485 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → 𝐴 ∈ ℂ) | |
10 | 9 | cjcld 14555 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘𝐴) ∈ ℂ) |
11 | abs00 14649 | . . . . . . . . 9 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) = 0 ↔ 𝐴 = 0)) | |
12 | 11 | necon3bid 3060 | . . . . . . . 8 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴) ≠ 0 ↔ 𝐴 ≠ 0)) |
13 | 12 | biimpar 480 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (abs‘𝐴) ≠ 0) |
14 | 9, 10, 5, 13 | div23d 11453 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴 · (∗‘𝐴)) / (abs‘𝐴)) = ((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) |
15 | 5, 5, 13 | divcan3d 11421 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((abs‘𝐴) · (abs‘𝐴)) / (abs‘𝐴)) = (abs‘𝐴)) |
16 | 8, 14, 15 | 3eqtr3d 2864 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((𝐴 / (abs‘𝐴)) · (∗‘𝐴)) = (abs‘𝐴)) |
17 | 16 | fveq2d 6674 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = (∗‘(abs‘𝐴))) |
18 | 9, 5, 13 | divcld 11416 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (𝐴 / (abs‘𝐴)) ∈ ℂ) |
19 | 18, 10 | cjmuld 14580 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = ((∗‘(𝐴 / (abs‘𝐴))) · (∗‘(∗‘𝐴)))) |
20 | 9 | cjcjd 14558 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘(∗‘𝐴)) = 𝐴) |
21 | 20 | oveq2d 7172 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((∗‘(𝐴 / (abs‘𝐴))) · (∗‘(∗‘𝐴))) = ((∗‘(𝐴 / (abs‘𝐴))) · 𝐴)) |
22 | 19, 21 | eqtrd 2856 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = ((∗‘(𝐴 / (abs‘𝐴))) · 𝐴)) |
23 | 4 | cjred 14585 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (∗‘(abs‘𝐴)) = (abs‘𝐴)) |
24 | 17, 22, 23 | 3eqtr3d 2864 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → ((∗‘(𝐴 / (abs‘𝐴))) · 𝐴) = (abs‘𝐴)) |
25 | 24, 16 | oveq12d 7174 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((∗‘(𝐴 / (abs‘𝐴))) · 𝐴) + ((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = ((abs‘𝐴) + (abs‘𝐴))) |
26 | 5 | 2timesd 11881 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (2 · (abs‘𝐴)) = ((abs‘𝐴) + (abs‘𝐴))) |
27 | 25, 26 | eqtr4d 2859 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0) → (((∗‘(𝐴 / (abs‘𝐴))) · 𝐴) + ((𝐴 / (abs‘𝐴)) · (∗‘𝐴))) = (2 · (abs‘𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 + caddc 10540 · cmul 10542 / cdiv 11297 2c2 11693 ↑cexp 13430 ∗ccj 14455 abscabs 14593 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-sup 8906 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-n0 11899 df-z 11983 df-uz 12245 df-rp 12391 df-seq 13371 df-exp 13431 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 |
This theorem is referenced by: bcsiALT 28956 |
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