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Theorem binom2 10873

Description: The square of a binomial. (Contributed by FL, 10-Dec-2006.)

**Assertion**
Ref	Expression
binom2	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))

**Proof of Theorem binom2**
Step	Hyp	Ref	Expression
1		addcl 8124	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 + 𝐵) ∈ ℂ)
2		simpl 109	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐴 ∈ ℂ)
3		simpr 110	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → 𝐵 ∈ ℂ)
4	1, 2, 3	adddid 8171	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)))
5	2, 3, 2	adddird 8172	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)))
6	3, 2	mulcomd 8168	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐴) = (𝐴 · 𝐵))
7	6	oveq2d 6017	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐴) + (𝐵 · 𝐴)) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)))
8	5, 7	eqtrd 2262	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)))
9	2, 3, 3	adddird 8172	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)))
10	8, 9	oveq12d 6019	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))))
11	2, 2	mulcld 8167	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐴) ∈ ℂ)
12	2, 3	mulcld 8167	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
13	11, 12	addcld 8166	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ)
14	3, 3	mulcld 8167	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐵) ∈ ℂ)
15	13, 12, 14	addassd 8169	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))))
16	11, 12, 12	addassd 8169	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))))
17	16	oveq1d 6016	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) + (𝐵 · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
18	10, 15, 17	3eqtr2d 2268	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
19	4, 18	eqtrd 2262	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
20		sqval 10819	. . 3 ⊢ ((𝐴 + 𝐵) ∈ ℂ → ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)))
21	1, 20	syl 14	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑2) = ((𝐴 + 𝐵) · (𝐴 + 𝐵)))
22		sqval 10819	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
23	2, 22	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑2) = (𝐴 · 𝐴))
24	12	2timesd 9354	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)))
25	23, 24	oveq12d 6019	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))))
26		sqval 10819	. . . 4 ⊢ (𝐵 ∈ ℂ → (𝐵↑2) = (𝐵 · 𝐵))
27	3, 26	syl 14	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵↑2) = (𝐵 · 𝐵))
28	25, 27	oveq12d 6019	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
29	19, 21, 28	3eqtr4d 2272	1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵)↑2) = (((𝐴↑2) + (2 · (𝐴 · 𝐵))) + (𝐵↑2)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 = wceq 1395 ∈ wcel 2200 (class class class)co 6001 ℂcc 7997 + caddc 8002 · cmul 8004 2c2 9161 ↑cexp 10760

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4199 ax-sep 4202 ax-nul 4210 ax-pow 4258 ax-pr 4293 ax-un 4524 ax-setind 4629 ax-iinf 4680 ax-cnex 8090 ax-resscn 8091 ax-1cn 8092 ax-1re 8093 ax-icn 8094 ax-addcl 8095 ax-addrcl 8096 ax-mulcl 8097 ax-mulrcl 8098 ax-addcom 8099 ax-mulcom 8100 ax-addass 8101 ax-mulass 8102 ax-distr 8103 ax-i2m1 8104 ax-0lt1 8105 ax-1rid 8106 ax-0id 8107 ax-rnegex 8108 ax-precex 8109 ax-cnre 8110 ax-pre-ltirr 8111 ax-pre-ltwlin 8112 ax-pre-lttrn 8113 ax-pre-apti 8114 ax-pre-ltadd 8115 ax-pre-mulgt0 8116 ax-pre-mulext 8117

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-iun 3967 df-br 4084 df-opab 4146 df-mpt 4147 df-tr 4183 df-id 4384 df-po 4387 df-iso 4388 df-iord 4457 df-on 4459 df-ilim 4460 df-suc 4462 df-iom 4683 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-rn 4730 df-res 4731 df-ima 4732 df-iota 5278 df-fun 5320 df-fn 5321 df-f 5322 df-f1 5323 df-fo 5324 df-f1o 5325 df-fv 5326 df-riota 5954 df-ov 6004 df-oprab 6005 df-mpo 6006 df-1st 6286 df-2nd 6287 df-recs 6451 df-frec 6537 df-pnf 8183 df-mnf 8184 df-xr 8185 df-ltxr 8186 df-le 8187 df-sub 8319 df-neg 8320 df-reap 8722 df-ap 8729 df-div 8820 df-inn 9111 df-2 9169 df-n0 9370 df-z 9447 df-uz 9723 df-seqfrec 10670 df-exp 10761

This theorem is referenced by: binom21 10874 binom2sub 10875 mulbinom2 10878 binom3 10879 nn0opthlem1d 10942 resqrexlemover 11521 resqrexlemcalc1 11525 abstri 11615 amgm2 11629 bdtrilem 11750 pythagtriplem1 12788 pythagtriplem12 12798

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