Theorem muladd11r 8313

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 8173	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 1 e. CC )
3	1, 2	addcomd 8308	. . 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 8308	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B + 1 ) = ( 1 + B ) )
6	3, 5	oveq12d 6025	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( 1 + A ) x. ( 1 + B ) ) )
7		muladd11 8290	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
8		mulcl 8137	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
9	4, 8	addcld 8177	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( A x. B ) ) e. CC )
10	2, 1, 9	addassd 8180	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( 1 + ( A + ( B + ( A x. B ) ) ) ) )
11	1, 9	addcld 8177	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) e. CC )
12	2, 11	addcomd 8308	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 1 + ( A + ( B + ( A x. B ) ) ) ) = ( ( A + ( B + ( A x. B ) ) ) + 1 ) )
13	1, 4, 8	addassd 8180	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( A + ( B + ( A x. B ) ) ) )
14		addcl 8135	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
15	14, 8	addcomd 8308	. . . . 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 6022	. . 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 6007   CCcc 8008   1c1 8011   + caddc 8013   x. cmul 8015

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 8102  ax-1cn 8103  ax-icn 8105  ax-addcl 8106  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-1rid 8117  ax-cnre 8121

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 2801  df-un 3201  df-in 3203  df-ss 3210  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-iota 5278  df-fv 5326  df-ov 6010

This theorem is referenced by: (None)