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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 10151 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 − 𝐵)) = ((∗‘𝐴) − (∗‘𝐵))) | |
2 | 1 | oveq2d 5605 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵))) = ((𝐴 − 𝐵) · ((∗‘𝐴) − (∗‘𝐵)))) |
3 | cjcl 10107 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | cjcl 10107 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (∗‘𝐵) ∈ ℂ) | |
5 | 3, 4 | anim12i 331 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ)) |
6 | mulsub 7780 | . . . 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 412 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · ((∗‘𝐴) − (∗‘𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
8 | 2, 7 | eqtrd 2115 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 − 𝐵) · (∗‘(𝐴 − 𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) − ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
9 | subcl 7582 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 − 𝐵) ∈ ℂ) | |
10 | absvalsq 10311 | . . 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 10311 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
13 | absvalsq 10311 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (𝐵 · (∗‘𝐵))) | |
14 | mulcom 7372 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) | |
15 | 4, 14 | mpdan 412 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) |
16 | 13, 15 | eqtrd 2115 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = ((∗‘𝐵) · 𝐵)) |
17 | 12, 16 | oveqan12d 5608 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) = ((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵))) |
18 | mulcl 7370 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) | |
19 | 4, 18 | sylan2 280 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) |
20 | 19 | addcjd 10216 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) |
21 | cjmul 10144 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) | |
22 | 4, 21 | sylan2 280 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) |
23 | cjcj 10142 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (∗‘(∗‘𝐵)) = 𝐵) | |
24 | 23 | adantl 271 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(∗‘𝐵)) = 𝐵) |
25 | 24 | oveq2d 5605 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) · (∗‘(∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
26 | 22, 25 | eqtrd 2115 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
27 | 26 | oveq2d 5605 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
28 | 20, 27 | eqtr3d 2117 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · (ℜ‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
29 | 17, 28 | oveq12d 5607 | . 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 2125 | 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 102 = wceq 1285 ∈ wcel 1434 ‘cfv 4967 (class class class)co 5589 ℂcc 7249 + caddc 7254 · cmul 7256 − cmin 7554 2c2 8364 ↑cexp 9789 ∗ccj 10098 ℜcre 10099 abscabs 10255 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-mulrcl 7345 ax-addcom 7346 ax-mulcom 7347 ax-addass 7348 ax-mulass 7349 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-1rid 7353 ax-0id 7354 ax-rnegex 7355 ax-precex 7356 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 ax-pre-mulgt0 7363 ax-pre-mulext 7364 ax-arch 7365 ax-caucvg 7366 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-if 3374 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-po 4086 df-iso 4087 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-reap 7950 df-ap 7957 df-div 8036 df-inn 8315 df-2 8373 df-3 8374 df-4 8375 df-n0 8564 df-z 8645 df-uz 8913 df-rp 9028 df-iseq 9739 df-iexp 9790 df-cj 10101 df-re 10102 df-im 10103 df-rsqrt 10256 df-abs 10257 |
This theorem is referenced by: sqabssubi 10411 |
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