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Mirrors > Home > MPE Home > Th. List > sqabsadd | Structured version Visualization version GIF version |
Description: Square of absolute value of sum. Proposition 10-3.7(g) of [Gleason] p. 133. (Contributed by NM, 21-Jan-2007.) |
Ref | Expression |
---|---|
sqabsadd | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cjadd 14495 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 + 𝐵)) = ((∗‘𝐴) + (∗‘𝐵))) | |
2 | 1 | oveq2d 7165 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (∗‘(𝐴 + 𝐵))) = ((𝐴 + 𝐵) · ((∗‘𝐴) + (∗‘𝐵)))) |
3 | cjcl 14459 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (∗‘𝐴) ∈ ℂ) | |
4 | cjcl 14459 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (∗‘𝐵) ∈ ℂ) | |
5 | 3, 4 | anim12i 614 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ)) |
6 | muladd 11065 | . . . 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 685 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · ((∗‘𝐴) + (∗‘𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) + ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
8 | 2, 7 | eqtrd 2855 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (∗‘(𝐴 + 𝐵))) = (((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵)) + ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵)))) |
9 | addcl 10612 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ) | |
10 | absvalsq 14635 | . . 3 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((abs‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) · (∗‘(𝐴 + 𝐵)))) | |
11 | 9, 10 | syl 17 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 + 𝐵))↑2) = ((𝐴 + 𝐵) · (∗‘(𝐴 + 𝐵)))) |
12 | absvalsq 14635 | . . . 4 ⊢ (𝐴 ∈ ℂ → ((abs‘𝐴)↑2) = (𝐴 · (∗‘𝐴))) | |
13 | absvalsq 14635 | . . . . 5 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = (𝐵 · (∗‘𝐵))) | |
14 | mulcom 10616 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) | |
15 | 4, 14 | mpdan 685 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 · (∗‘𝐵)) = ((∗‘𝐵) · 𝐵)) |
16 | 13, 15 | eqtrd 2855 | . . . 4 ⊢ (𝐵 ∈ ℂ → ((abs‘𝐵)↑2) = ((∗‘𝐵) · 𝐵)) |
17 | 12, 16 | oveqan12d 7168 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) = ((𝐴 · (∗‘𝐴)) + ((∗‘𝐵) · 𝐵))) |
18 | mulcl 10614 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) | |
19 | 4, 18 | sylan2 594 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · (∗‘𝐵)) ∈ ℂ) |
20 | 19 | addcjd 14566 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = (2 · (ℜ‘(𝐴 · (∗‘𝐵))))) |
21 | cjmul 14496 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ (∗‘𝐵) ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) | |
22 | 4, 21 | sylan2 594 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · (∗‘(∗‘𝐵)))) |
23 | cjcj 14494 | . . . . . . . 8 ⊢ (𝐵 ∈ ℂ → (∗‘(∗‘𝐵)) = 𝐵) | |
24 | 23 | adantl 484 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(∗‘𝐵)) = 𝐵) |
25 | 24 | oveq2d 7165 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((∗‘𝐴) · (∗‘(∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
26 | 22, 25 | eqtrd 2855 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (∗‘(𝐴 · (∗‘𝐵))) = ((∗‘𝐴) · 𝐵)) |
27 | 26 | oveq2d 7165 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · (∗‘𝐵)) + (∗‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
28 | 20, 27 | eqtr3d 2857 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · (ℜ‘(𝐴 · (∗‘𝐵)))) = ((𝐴 · (∗‘𝐵)) + ((∗‘𝐴) · 𝐵))) |
29 | 17, 28 | oveq12d 7167 | . 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 2865 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘(𝐴 + 𝐵))↑2) = ((((abs‘𝐴)↑2) + ((abs‘𝐵)↑2)) + (2 · (ℜ‘(𝐴 · (∗‘𝐵)))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ‘cfv 6348 (class class class)co 7149 ℂcc 10528 + caddc 10533 · cmul 10535 2c2 11686 ↑cexp 13426 ∗ccj 14450 ℜcre 14451 abscabs 14588 |
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 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 ax-cnex 10586 ax-resscn 10587 ax-1cn 10588 ax-icn 10589 ax-addcl 10590 ax-addrcl 10591 ax-mulcl 10592 ax-mulrcl 10593 ax-mulcom 10594 ax-addass 10595 ax-mulass 10596 ax-distr 10597 ax-i2m1 10598 ax-1ne0 10599 ax-1rid 10600 ax-rnegex 10601 ax-rrecex 10602 ax-cnre 10603 ax-pre-lttri 10604 ax-pre-lttrn 10605 ax-pre-ltadd 10606 ax-pre-mulgt0 10607 ax-pre-sup 10608 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-nel 3123 df-ral 3142 df-rex 3143 df-reu 3144 df-rmo 3145 df-rab 3146 df-v 3493 df-sbc 3769 df-csb 3877 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7107 df-ov 7152 df-oprab 7153 df-mpo 7154 df-om 7574 df-2nd 7683 df-wrecs 7940 df-recs 8001 df-rdg 8039 df-er 8282 df-en 8503 df-dom 8504 df-sdom 8505 df-sup 8899 df-pnf 10670 df-mnf 10671 df-xr 10672 df-ltxr 10673 df-le 10674 df-sub 10865 df-neg 10866 df-div 11291 df-nn 11632 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-seq 13367 df-exp 13427 df-cj 14453 df-re 14454 df-im 14455 df-sqrt 14589 df-abs 14590 |
This theorem is referenced by: abstri 14685 sqabsaddi 14760 cncph 28594 |
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