Theorem mulbinom2 10833

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 8082	. . . . 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 1019	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( C x. A ) e. CC )
4		simp2 1001	. . 3 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> B e. CC )
5		binom2 10828	. . 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 411	. 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 8086	. . . . . . 7 |- ( ( C e. CC /\ A e. CC /\ B e. CC ) -> ( ( C x. A ) x. B ) = ( C x. ( A x. B ) ) )
8	7	3coml 1213	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( C x. A ) x. B ) = ( C x. ( A x. B ) ) )
9	8	oveq2d 5978	. . . . 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 9139	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> 2 e. CC )
11		simp3 1002	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> C e. CC )
12		mulcl 8082	. . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
13	12	3adant3 1020	. . . . . 6 |- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A x. B ) e. CC )
14	10, 11, 13	mulassd 8126	. . . . 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 2242	. . . 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 5978	. . 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 5977	. 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 2239	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 ) ) )

Syntax hints:   -> wi 4   /\ w3a 981   = wceq 1373   e. wcel 2177  (class class class)co 5962   CCcc 7953   + caddc 7958   x. cmul 7960   2c2 9117   ^cexp 10715

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4170  ax-sep 4173  ax-nul 4181  ax-pow 4229  ax-pr 4264  ax-un 4493  ax-setind 4598  ax-iinf 4649  ax-cnex 8046  ax-resscn 8047  ax-1cn 8048  ax-1re 8049  ax-icn 8050  ax-addcl 8051  ax-addrcl 8052  ax-mulcl 8053  ax-mulrcl 8054  ax-addcom 8055  ax-mulcom 8056  ax-addass 8057  ax-mulass 8058  ax-distr 8059  ax-i2m1 8060  ax-0lt1 8061  ax-1rid 8062  ax-0id 8063  ax-rnegex 8064  ax-precex 8065  ax-cnre 8066  ax-pre-ltirr 8067  ax-pre-ltwlin 8068  ax-pre-lttrn 8069  ax-pre-apti 8070  ax-pre-ltadd 8071  ax-pre-mulgt0 8072  ax-pre-mulext 8073

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-nel 2473  df-ral 2490  df-rex 2491  df-reu 2492  df-rmo 2493  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-if 3576  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3860  df-int 3895  df-iun 3938  df-br 4055  df-opab 4117  df-mpt 4118  df-tr 4154  df-id 4353  df-po 4356  df-iso 4357  df-iord 4426  df-on 4428  df-ilim 4429  df-suc 4431  df-iom 4652  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-iota 5246  df-fun 5287  df-fn 5288  df-f 5289  df-f1 5290  df-fo 5291  df-f1o 5292  df-fv 5293  df-riota 5917  df-ov 5965  df-oprab 5966  df-mpo 5967  df-1st 6244  df-2nd 6245  df-recs 6409  df-frec 6495  df-pnf 8139  df-mnf 8140  df-xr 8141  df-ltxr 8142  df-le 8143  df-sub 8275  df-neg 8276  df-reap 8678  df-ap 8685  df-div 8776  df-inn 9067  df-2 9125  df-n0 9326  df-z 9403  df-uz 9679  df-seqfrec 10625  df-exp 10716

This theorem is referenced by:  mulsubdivbinom2ap  10888