Theorem muladd11r 8475

Description: A simple product of sums expansion. (Contributed by AV, 30-Jul-2021.)

Assertion

Ref	Expression
muladd11r	|- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )

Proof of Theorem muladd11r

Step	Hyp	Ref	Expression
1		simpl 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		1cnd 8335	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 1 e. CC )
3	1, 2	addcomd 8470	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + 1 ) = ( 1 + A ) )
4		simpr 110	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
5	4, 2	addcomd 8470	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B + 1 ) = ( 1 + B ) )
6	3, 5	oveq12d 6096	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( 1 + A ) x. ( 1 + B ) ) )
7		muladd11 8452	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
8		mulcl 8299	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
9	4, 8	addcld 8338	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( A x. B ) ) e. CC )
10	2, 1, 9	addassd 8341	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( 1 + ( A + ( B + ( A x. B ) ) ) ) )
11	1, 9	addcld 8338	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) e. CC )
12	2, 11	addcomd 8470	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 1 + ( A + ( B + ( A x. B ) ) ) ) = ( ( A + ( B + ( A x. B ) ) ) + 1 ) )
13	1, 4, 8	addassd 8341	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( A + ( B + ( A x. B ) ) ) )
14		addcl 8297	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
15	14, 8	addcomd 8470	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( ( A x. B ) + ( A + B ) ) )
16	13, 15	eqtr3d 2273	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) = ( ( A x. B ) + ( A + B ) ) )
17	16	oveq1d 6093	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( B + ( A x. B ) ) ) + 1 ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
18	10, 12, 17	3eqtrd 2275	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
19	6, 7, 18	3eqtrd 2275	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209  (class class class)co 6078   CCcc 8170   1c1 8173   + caddc 8175   x. cmul 8177

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-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-ext 2220  ax-resscn 8264  ax-1cn 8265  ax-icn 8267  ax-addcl 8268  ax-mulcl 8270  ax-addcom 8272  ax-mulcom 8273  ax-addass 8274  ax-mulass 8275  ax-distr 8276  ax-1rid 8279  ax-cnre 8283

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-br 4129  df-iota 5335  df-fv 5383  df-ov 6081

This theorem is referenced by: (None)