binom2 - Intuitionistic Logic Explorer

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Theorem binom2 10601

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 7911	. . . 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 7956	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)))
5	2, 3, 2	adddird 7957	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐵 · 𝐴)))
6	3, 2	mulcomd 7953	. . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐴) = (𝐴 · 𝐵))
7	6	oveq2d 5881	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐴) + (𝐵 · 𝐴)) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)))
8	5, 7	eqtrd 2208	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐴) = ((𝐴 · 𝐴) + (𝐴 · 𝐵)))
9	2, 3, 3	adddird 7957	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · 𝐵) = ((𝐴 · 𝐵) + (𝐵 · 𝐵)))
10	8, 9	oveq12d 5883	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + ((𝐴 · 𝐵) + (𝐵 · 𝐵))))
11	2, 2	mulcld 7952	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐴) ∈ ℂ)
12	2, 3	mulcld 7952	. . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)
13	11, 12	addcld 7951	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 · 𝐴) + (𝐴 · 𝐵)) ∈ ℂ)
14	3, 3	mulcld 7952	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 · 𝐵) ∈ ℂ)
15	13, 12, 14	addassd 7954	. . . 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 7954	. . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 · 𝐴) + (𝐴 · 𝐵)) + (𝐴 · 𝐵)) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))))
17	16	oveq1d 5880	. . . 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 2214	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (((𝐴 + 𝐵) · 𝐴) + ((𝐴 + 𝐵) · 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
19	4, 18	eqtrd 2208	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴 + 𝐵) · (𝐴 + 𝐵)) = (((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))) + (𝐵 · 𝐵)))
20		sqval 10548	. . 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 10548	. . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴↑2) = (𝐴 · 𝐴))
23	2, 22	syl 14	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴↑2) = (𝐴 · 𝐴))
24	12	2timesd 9134	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (2 · (𝐴 · 𝐵)) = ((𝐴 · 𝐵) + (𝐴 · 𝐵)))
25	23, 24	oveq12d 5883	. . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐴↑2) + (2 · (𝐴 · 𝐵))) = ((𝐴 · 𝐴) + ((𝐴 · 𝐵) + (𝐴 · 𝐵))))
26		sqval 10548	. . . 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 5883	. 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 2218	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 1353 ∈ wcel 2146 (class class class)co 5865 ℂcc 7784 + caddc 7789 · cmul 7791 2c2 8943 ↑cexp 10489

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 614 ax-in2 615 ax-io 709 ax-5 1445 ax-7 1446 ax-gen 1447 ax-ie1 1491 ax-ie2 1492 ax-8 1502 ax-10 1503 ax-11 1504 ax-i12 1505 ax-bndl 1507 ax-4 1508 ax-17 1524 ax-i9 1528 ax-ial 1532 ax-i5r 1533 ax-13 2148 ax-14 2149 ax-ext 2157 ax-coll 4113 ax-sep 4116 ax-nul 4124 ax-pow 4169 ax-pr 4203 ax-un 4427 ax-setind 4530 ax-iinf 4581 ax-cnex 7877 ax-resscn 7878 ax-1cn 7879 ax-1re 7880 ax-icn 7881 ax-addcl 7882 ax-addrcl 7883 ax-mulcl 7884 ax-mulrcl 7885 ax-addcom 7886 ax-mulcom 7887 ax-addass 7888 ax-mulass 7889 ax-distr 7890 ax-i2m1 7891 ax-0lt1 7892 ax-1rid 7893 ax-0id 7894 ax-rnegex 7895 ax-precex 7896 ax-cnre 7897 ax-pre-ltirr 7898 ax-pre-ltwlin 7899 ax-pre-lttrn 7900 ax-pre-apti 7901 ax-pre-ltadd 7902 ax-pre-mulgt0 7903 ax-pre-mulext 7904

This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1459 df-sb 1761 df-eu 2027 df-mo 2028 df-clab 2162 df-cleq 2168 df-clel 2171 df-nfc 2306 df-ne 2346 df-nel 2441 df-ral 2458 df-rex 2459 df-reu 2460 df-rmo 2461 df-rab 2462 df-v 2737 df-sbc 2961 df-csb 3056 df-dif 3129 df-un 3131 df-in 3133 df-ss 3140 df-nul 3421 df-if 3533 df-pw 3574 df-sn 3595 df-pr 3596 df-op 3598 df-uni 3806 df-int 3841 df-iun 3884 df-br 3999 df-opab 4060 df-mpt 4061 df-tr 4097 df-id 4287 df-po 4290 df-iso 4291 df-iord 4360 df-on 4362 df-ilim 4363 df-suc 4365 df-iom 4584 df-xp 4626 df-rel 4627 df-cnv 4628 df-co 4629 df-dm 4630 df-rn 4631 df-res 4632 df-ima 4633 df-iota 5170 df-fun 5210 df-fn 5211 df-f 5212 df-f1 5213 df-fo 5214 df-f1o 5215 df-fv 5216 df-riota 5821 df-ov 5868 df-oprab 5869 df-mpo 5870 df-1st 6131 df-2nd 6132 df-recs 6296 df-frec 6382 df-pnf 7968 df-mnf 7969 df-xr 7970 df-ltxr 7971 df-le 7972 df-sub 8104 df-neg 8105 df-reap 8506 df-ap 8513 df-div 8603 df-inn 8893 df-2 8951 df-n0 9150 df-z 9227 df-uz 9502 df-seqfrec 10416 df-exp 10490

This theorem is referenced by: binom21 10602 binom2sub 10603 mulbinom2 10606 binom3 10607 nn0opthlem1d 10668 resqrexlemover 10987 resqrexlemcalc1 10991 abstri 11081 amgm2 11095 bdtrilem 11215 pythagtriplem1 12232 pythagtriplem12 12242

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