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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 10641 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3, 1, 2 | adddii 10647 | . . 3 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) |
5 | 1, 2, 1 | adddiri 10648 | . . . . . 6 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)) |
6 | 2, 1 | mulcomi 10643 | . . . . . . 7 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
7 | 6 | oveq2i 7161 | . . . . . 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 10648 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)) |
10 | 8, 9 | oveq12i 7162 | . . . 4 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
11 | 1, 1 | mulcli 10642 | . . . . . 6 ⊢ (𝐴 · 𝐴) ∈ ℂ |
12 | 1, 2 | mulcli 10642 | . . . . . 6 ⊢ (𝐴 · 𝐵) ∈ ℂ |
13 | 11, 12 | addcli 10641 | . . . . 5 ⊢ ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ |
14 | 2, 2 | mulcli 10642 | . . . . 5 ⊢ (𝐵 · 𝐵) ∈ ℂ |
15 | 13, 12, 14 | addassi 10645 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
16 | 11, 12, 12 | addassi 10645 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
17 | 16 | oveq1i 7160 | . . . 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 13537 | . 2 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
21 | 1 | sqvali 13537 | . . . 4 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
22 | 12 | 2timesi 11769 | . . . 4 ⊢ (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)) |
23 | 21, 22 | oveq12i 7162 | . . 3 ⊢ ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
24 | 2 | sqvali 13537 | . . 3 ⊢ (𝐵↑2) = (𝐵 · 𝐵) |
25 | 23, 24 | oveq12i 7162 | . 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 7150 ℂcc 10529 + caddc 10534 · cmul 10536 2c2 11686 ↑cexp 13423 |
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 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 |
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 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-nn 11633 df-2 11694 df-n0 11892 df-z 11976 df-uz 12238 df-seq 13364 df-exp 13424 |
This theorem is referenced by: binom2 13573 nn0opthlem1 13622 2lgsoddprmlem3d 25983 ax5seglem7 26715 norm-ii-i 28908 quad3 32908 |
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