![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sqabssub | GIF version |
Description: Square of absolute value of difference. (Contributed by NM, 21-Jan-2007.) |
Ref | Expression |
---|---|
sqabssub | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjsub 10900 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) | |
2 | 1 | oveq2d 5890 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵))) = ((𝐴 − 𝐵) · ((∗‘𝐴) − (∗‘𝐵)))) |
3 | cjcl 10856 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | cjcl 10856 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (∗‘𝐵) ∈ ℂ) | |
5 | 3, 4 | anim12i 338 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ)) |
6 | mulsub 8357 | . . . 4 ⊢ (((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) ∧ ((∗‘𝐴) ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ)) → ((𝐴 − 𝐵) · ((∗‘𝐴) − (∗‘𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) | |
7 | 5, 6 | mpdan 421 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · ((∗‘𝐴) − (∗‘𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
8 | 2, 7 | eqtrd 2210 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
9 | subcl 8155 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
10 | absvalsq 11061 | . . 3 ⊢ ((𝐴 − 𝐵) ∈ ℂ → ((abs‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵)))) | |
11 | 9, 10 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 𝐵))↑2) = ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵)))) |
12 | absvalsq 11061 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
13 | absvalsq 11061 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (𝐵 · (∗‘𝐵))) | |
14 | mulcom 7939 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) | |
15 | 4, 14 | mpdan 421 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) |
16 | 13, 15 | eqtrd 2210 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = ((∗‘𝐵) · 𝐵)) |
17 | 12, 16 | oveqan12d 5893 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) = ((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵))) |
18 | mulcl 7937 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) | |
19 | 4, 18 | sylan2 286 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) |
20 | 19 | addcjd 10965 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) |
21 | cjmul 10893 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) | |
22 | 4, 21 | sylan2 286 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) |
23 | cjcj 10891 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (∗‘(∗‘𝐵)) = 𝐵) | |
24 | 23 | adantl 277 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(∗‘𝐵)) = 𝐵) |
25 | 24 | oveq2d 5890 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) · (∗‘(∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
26 | 22, 25 | eqtrd 2210 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
27 | 26 | oveq2d 5890 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
28 | 20, 27 | eqtr3d 2212 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · (ℜ‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
29 | 17, 28 | oveq12d 5892 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
30 | 8, 11, 29 | 3eqtr4d 2220 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 − 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) − (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ‘cfv 5216 (class class class)co 5874 ℂcc 7808 + caddc 7813 · cmul 7815 − cmin 8127 2c2 8969 ↑cexp 10518 ∗ccj 10847 ℜcre 10848 abscabs 11005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 ax-cnex 7901 ax-resscn 7902 ax-1cn 7903 ax-1re 7904 ax-icn 7905 ax-addcl 7906 ax-addrcl 7907 ax-mulcl 7908 ax-mulrcl 7909 ax-addcom 7910 ax-mulcom 7911 ax-addass 7912 ax-mulass 7913 ax-distr 7914 ax-i2m1 7915 ax-0lt1 7916 ax-1rid 7917 ax-0id 7918 ax-rnegex 7919 ax-precex 7920 ax-cnre 7921 ax-pre-ltirr 7922 ax-pre-ltwlin 7923 ax-pre-lttrn 7924 ax-pre-apti 7925 ax-pre-ltadd 7926 ax-pre-mulgt0 7927 ax-pre-mulext 7928 ax-arch 7929 ax-caucvg 7930 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-po 4296 df-iso 4297 df-iord 4366 df-on 4368 df-ilim 4369 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-riota 5830 df-ov 5877 df-oprab 5878 df-mpo 5879 df-1st 6140 df-2nd 6141 df-recs 6305 df-frec 6391 df-pnf 7993 df-mnf 7994 df-xr 7995 df-ltxr 7996 df-le 7997 df-sub 8129 df-neg 8130 df-reap 8531 df-ap 8538 df-div 8629 df-inn 8919 df-2 8977 df-3 8978 df-4 8979 df-n0 9176 df-z 9253 df-uz 9528 df-rp 9653 df-seqfrec 10445 df-exp 10519 df-cj 10850 df-re 10851 df-im 10852 df-rsqrt 11006 df-abs 11007 |
This theorem is referenced by: sqabssubi 11161 |
Copyright terms: Public domain | W3C validator |