Theorem muladd11r 8325

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 8185	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 1 e. CC )
3	1, 2	addcomd 8320	. . 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 8320	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B + 1 ) = ( 1 + B ) )
6	3, 5	oveq12d 6031	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( 1 + A ) x. ( 1 + B ) ) )
7		muladd11 8302	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
8		mulcl 8149	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
9	4, 8	addcld 8189	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( A x. B ) ) e. CC )
10	2, 1, 9	addassd 8192	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( 1 + ( A + ( B + ( A x. B ) ) ) ) )
11	1, 9	addcld 8189	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) e. CC )
12	2, 11	addcomd 8320	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 1 + ( A + ( B + ( A x. B ) ) ) ) = ( ( A + ( B + ( A x. B ) ) ) + 1 ) )
13	1, 4, 8	addassd 8192	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( A + ( B + ( A x. B ) ) ) )
14		addcl 8147	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
15	14, 8	addcomd 8320	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( ( A x. B ) + ( A + B ) ) )
16	13, 15	eqtr3d 2264	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) = ( ( A x. B ) + ( A + B ) ) )
17	16	oveq1d 6028	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( B + ( A x. B ) ) ) + 1 ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
18	10, 12, 17	3eqtrd 2266	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
19	6, 7, 18	3eqtrd 2266	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 1395   e. wcel 2200  (class class class)co 6013   CCcc 8020   1c1 8023   + caddc 8025   x. cmul 8027

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-resscn 8114  ax-1cn 8115  ax-icn 8117  ax-addcl 8118  ax-mulcl 8120  ax-addcom 8122  ax-mulcom 8123  ax-addass 8124  ax-mulass 8125  ax-distr 8126  ax-1rid 8129  ax-cnre 8133

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-br 4087  df-iota 5284  df-fv 5332  df-ov 6016

This theorem is referenced by: (None)