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Mirrors > Home > MPE Home > Th. List > binom2i | Structured version Visualization version GIF version |
Description: The square of a binomial. (Contributed by NM, 11-Aug-1999.) |
Ref | Expression |
---|---|
binom2.1 | ⊢ 𝐴 ∈ ℂ |
binom2.2 | ⊢ 𝐵 ∈ ℂ |
Ref | Expression |
---|---|
binom2i | ⊢ ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | binom2.1 | . . . . 5 ⊢ 𝐴 ∈ ℂ | |
2 | binom2.2 | . . . . 5 ⊢ 𝐵 ∈ ℂ | |
3 | 1, 2 | addcli 10646 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3, 1, 2 | adddii 10652 | . . 3 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) |
5 | 1, 2, 1 | adddiri 10653 | . . . . . 6 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)) |
6 | 2, 1 | mulcomi 10648 | . . . . . . 7 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
7 | 6 | oveq2i 7166 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (𝐵 · 𝐴)) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
8 | 5, 7 | eqtri 2844 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
9 | 1, 2, 2 | adddiri 10653 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)) |
10 | 8, 9 | oveq12i 7167 | . . . 4 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
11 | 1, 1 | mulcli 10647 | . . . . . 6 ⊢ (𝐴 · 𝐴) ∈ ℂ |
12 | 1, 2 | mulcli 10647 | . . . . . 6 ⊢ (𝐴 · 𝐵) ∈ ℂ |
13 | 11, 12 | addcli 10646 | . . . . 5 ⊢ ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ |
14 | 2, 2 | mulcli 10647 | . . . . 5 ⊢ (𝐵 · 𝐵) ∈ ℂ |
15 | 13, 12, 14 | addassi 10650 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
16 | 11, 12, 12 | addassi 10650 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
17 | 16 | oveq1i 7165 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
18 | 10, 15, 17 | 3eqtr2i 2850 | . . 3 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
19 | 4, 18 | eqtri 2844 | . 2 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
20 | 3 | sqvali 13542 | . 2 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
21 | 1 | sqvali 13542 | . . . 4 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
22 | 12 | 2timesi 11774 | . . . 4 ⊢ (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)) |
23 | 21, 22 | oveq12i 7167 | . . 3 ⊢ ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
24 | 2 | sqvali 13542 | . . 3 ⊢ (𝐵↑2) = (𝐵 · 𝐵) |
25 | 23, 24 | oveq12i 7167 | . 2 ⊢ (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
26 | 19, 20, 25 | 3eqtr4i 2854 | 1 ⊢ ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 (class class class)co 7155 ℂcc 10534 + caddc 10539 · cmul 10541 2c2 11691 ↑cexp 13428 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 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-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-nn 11638 df-2 11699 df-n0 11897 df-z 11981 df-uz 12243 df-seq 13369 df-exp 13429 |
This theorem is referenced by: binom2 13578 nn0opthlem1 13627 2lgsoddprmlem3d 25988 ax5seglem7 26720 norm-ii-i 28913 quad3 32913 |
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