Theorem mulbinom2 11114

Description: The square of a binomial with factor. (Contributed by AV, 19-Jul-2021.)

Assertion

Ref	Expression
mulbinom2	|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( C x. A ) + B ) ^ 2 ) = ( ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

Proof of Theorem mulbinom2

Step	Hyp	Ref	Expression
1		mulcl 8312	. . . . 5 |- ( ( C e. CC /\ A e. CC ) -> ( C x. A ) e. CC )
2	1	ancoms 268	. . . 4 |- ( ( A e. CC /\ C e. CC ) -> ( C x. A ) e. CC )
3	2	3adant2 1047	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. A ) e. CC )
4		simp2 1029	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
5		binom2 11109	. . 3 |- ( ( ( C x. A ) e. CC /\ B e. CC ) -> ( ( ( C x. A ) + B ) ^ 2 ) = ( ( ( ( C x. A ) ^ 2 ) + ( 2 x. ( ( C x. A ) x. B ) ) ) + ( B ^ 2 ) ) )
6	3, 4, 5	syl2anc 415	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( C x. A ) + B ) ^ 2 ) = ( ( ( ( C x. A ) ^ 2 ) + ( 2 x. ( ( C x. A ) x. B ) ) ) + ( B ^ 2 ) ) )
7		mulass 8316	. . . . . . 7 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( ( C x. A ) x. B ) = ( C x. ( A x. B ) ) )
8	7	3coml 1241	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C x. A ) x. B ) = ( C x. ( A x. B ) ) )
9	8	oveq2d 6104	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( 2 x. ( ( C x. A ) x. B ) ) = ( 2 x. ( C x. ( A x. B ) ) ) )
10		2cnd 9386	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> 2 e. CC )
11		simp3 1030	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
12		mulcl 8312	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	12	3adant3 1048	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
14	10, 11, 13	mulassd 8355	. . . . 5 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( 2 x. C ) x. ( A x. B ) ) = ( 2 x. ( C x. ( A x. B ) ) ) )
15	9, 14	eqtr4d 2274	. . . 4 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( 2 x. ( ( C x. A ) x. B ) ) = ( ( 2 x. C ) x. ( A x. B ) ) )
16	15	oveq2d 6104	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( C x. A ) ^ 2 ) + ( 2 x. ( ( C x. A ) x. B ) ) ) = ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) ) )
17	16	oveq1d 6103	. 2 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( ( C x. A ) ^ 2 ) + ( 2 x. ( ( C x. A ) x. B ) ) ) + ( B ^ 2 ) ) = ( ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) ) + ( B ^ 2 ) ) )
18	6, 17	eqtrd 2271	1 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( C x. A ) + B ) ^ 2 ) = ( ( ( ( C x. A ) ^ 2 ) + ( ( 2 x. C ) x. ( A x. B ) ) ) + ( B ^ 2 ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6088   CCcc 8183   + caddc 8188   x. cmul 8190   2c2 9364   ^cexp 10996

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-mulrcl 8284  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-precex 8295  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301  ax-pre-mulgt0 8302  ax-pre-mulext 8303

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-reap 8911  df-ap 8918  df-div 9011  df-inn 9314  df-2 9372  df-n0 9575  df-z 9656  df-uz 9938  df-seqfrec 10906  df-exp 10997

This theorem is used by:  mulsubdivbinom2ap  11171