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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 7996 | . . . 4 ⊢ (𝐴 + 𝐵) ∈ ℂ |
4 | 3, 1, 2 | adddii 8002 | . . 3 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) |
5 | 1, 2, 1 | adddiri 8003 | . . . . . 6 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)) |
6 | 2, 1 | mulcomi 7998 | . . . . . . 7 ⊢ (𝐵 · 𝐴) = (𝐴 · 𝐵) |
7 | 6 | oveq2i 5911 | . . . . . 6 ⊢ ((𝐴 · 𝐴) + (𝐵 · 𝐴)) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
8 | 5, 7 | eqtri 2210 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)) |
9 | 1, 2, 2 | adddiri 8003 | . . . . 5 ⊢ ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)) |
10 | 8, 9 | oveq12i 5912 | . . . 4 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
11 | 1, 1 | mulcli 7997 | . . . . . 6 ⊢ (𝐴 · 𝐴) ∈ ℂ |
12 | 1, 2 | mulcli 7997 | . . . . . 6 ⊢ (𝐴 · 𝐵) ∈ ℂ |
13 | 11, 12 | addcli 7996 | . . . . 5 ⊢ ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ |
14 | 2, 2 | mulcli 7997 | . . . . 5 ⊢ (𝐵 · 𝐵) ∈ ℂ |
15 | 13, 12, 14 | addassi 8000 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))) |
16 | 11, 12, 12 | addassi 8000 | . . . . 5 ⊢ (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
17 | 16 | oveq1i 5910 | . . . 4 ⊢ ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
18 | 10, 15, 17 | 3eqtr2i 2216 | . . 3 ⊢ (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
19 | 4, 18 | eqtri 2210 | . 2 ⊢ ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
20 | 3 | sqvali 10640 | . 2 ⊢ ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)) |
21 | 1 | sqvali 10640 | . . . 4 ⊢ (𝐴↑2) = (𝐴 · 𝐴) |
22 | 12 | 2timesi 9084 | . . . 4 ⊢ (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)) |
23 | 21, 22 | oveq12i 5912 | . . 3 ⊢ ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) |
24 | 2 | sqvali 10640 | . . 3 ⊢ (𝐵↑2) = (𝐵 · 𝐵) |
25 | 23, 24 | oveq12i 5912 | . 2 ⊢ (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)) |
26 | 19, 20, 25 | 3eqtr4i 2220 | 1 ⊢ ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) |
Colors of variables: wff set class |
Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5900 ℂcc 7844 + caddc 7849 · cmul 7851 2c2 9005 ↑cexp 10559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-2 9013 df-n0 9212 df-z 9289 df-uz 9564 df-seqfrec 10485 df-exp 10560 |
This theorem is referenced by: 2lgsoddprmlem3d 14944 |
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