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Mirrors > Home > ILE Home > Th. List > binom2i | 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 7911 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3, 1, 2 | adddii 7917 | . . 3 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) |
5 | 1, 2, 1 | adddiri 7918 | . . . . . 6 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)) |
6 | 2, 1 | mulcomi 7913 | . . . . . . 7 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
7 | 6 | oveq2i 5861 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (𝐵 · 𝐴)) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
8 | 5, 7 | eqtri 2191 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
9 | 1, 2, 2 | adddiri 7918 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)) |
10 | 8, 9 | oveq12i 5862 | . . . 4 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
11 | 1, 1 | mulcli 7912 | . . . . . 6 ⊢ (𝐴 · 𝐴) ∈ ℂ |
12 | 1, 2 | mulcli 7912 | . . . . . 6 ⊢ (𝐴 · 𝐵) ∈ ℂ |
13 | 11, 12 | addcli 7911 | . . . . 5 ⊢ ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ |
14 | 2, 2 | mulcli 7912 | . . . . 5 ⊢ (𝐵 · 𝐵) ∈ ℂ |
15 | 13, 12, 14 | addassi 7915 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
16 | 11, 12, 12 | addassi 7915 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
17 | 16 | oveq1i 5860 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
18 | 10, 15, 17 | 3eqtr2i 2197 | . . 3 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
19 | 4, 18 | eqtri 2191 | . 2 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
20 | 3 | sqvali 10542 | . 2 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
21 | 1 | sqvali 10542 | . . . 4 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
22 | 12 | 2timesi 8995 | . . . 4 ⊢ (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)) |
23 | 21, 22 | oveq12i 5862 | . . 3 ⊢ ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
24 | 2 | sqvali 10542 | . . 3 ⊢ (𝐵↑2) = (𝐵 · 𝐵) |
25 | 23, 24 | oveq12i 5862 | . 2 ⊢ (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
26 | 19, 20, 25 | 3eqtr4i 2201 | 1 ⊢ ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1348 ∈ wcel 2141 (class class class)co 5850 ℂcc 7759 + caddc 7764 · cmul 7766 2c2 8916 ↑cexp 10462 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7852 ax-resscn 7853 ax-1cn 7854 ax-1re 7855 ax-icn 7856 ax-addcl 7857 ax-addrcl 7858 ax-mulcl 7859 ax-mulrcl 7860 ax-addcom 7861 ax-mulcom 7862 ax-addass 7863 ax-mulass 7864 ax-distr 7865 ax-i2m1 7866 ax-0lt1 7867 ax-1rid 7868 ax-0id 7869 ax-rnegex 7870 ax-precex 7871 ax-cnre 7872 ax-pre-ltirr 7873 ax-pre-ltwlin 7874 ax-pre-lttrn 7875 ax-pre-apti 7876 ax-pre-ltadd 7877 ax-pre-mulgt0 7878 ax-pre-mulext 7879 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5806 df-ov 5853 df-oprab 5854 df-mpo 5855 df-1st 6116 df-2nd 6117 df-recs 6281 df-frec 6367 df-pnf 7943 df-mnf 7944 df-xr 7945 df-ltxr 7946 df-le 7947 df-sub 8079 df-neg 8080 df-reap 8481 df-ap 8488 df-div 8577 df-inn 8866 df-2 8924 df-n0 9123 df-z 9200 df-uz 9475 df-seqfrec 10389 df-exp 10463 |
This theorem is referenced by: (None) |
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