Theorem muladd11r 8054

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 108	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> A e. CC )
2		1cnd 7915	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 1 e. CC )
3	1, 2	addcomd 8049	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( A + 1 ) = ( 1 + A ) )
4		simpr 109	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> B e. CC )
5	4, 2	addcomd 8049	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B + 1 ) = ( 1 + B ) )
6	3, 5	oveq12d 5860	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( 1 + A ) x. ( 1 + B ) ) )
7		muladd11 8031	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
8		mulcl 7880	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
9	4, 8	addcld 7918	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( A x. B ) ) e. CC )
10	2, 1, 9	addassd 7921	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( 1 + ( A + ( B + ( A x. B ) ) ) ) )
11	1, 9	addcld 7918	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) e. CC )
12	2, 11	addcomd 8049	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 1 + ( A + ( B + ( A x. B ) ) ) ) = ( ( A + ( B + ( A x. B ) ) ) + 1 ) )
13	1, 4, 8	addassd 7921	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( A + ( B + ( A x. B ) ) ) )
14		addcl 7878	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
15	14, 8	addcomd 8049	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( ( A x. B ) + ( A + B ) ) )
16	13, 15	eqtr3d 2200	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) = ( ( A x. B ) + ( A + B ) ) )
17	16	oveq1d 5857	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( B + ( A x. B ) ) ) + 1 ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
18	10, 12, 17	3eqtrd 2202	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
19	6, 7, 18	3eqtrd 2202	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136  (class class class)co 5842   CCcc 7751   1c1 7754   + caddc 7756   x. cmul 7758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-resscn 7845  ax-1cn 7846  ax-icn 7848  ax-addcl 7849  ax-mulcl 7851  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-1rid 7860  ax-cnre 7864

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-iota 5153  df-fv 5196  df-ov 5845

This theorem is referenced by: (None)