Theorem muladd11r 8075

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 7936	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> 1 e. CC )
3	1, 2	addcomd 8070	. . 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 8070	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( B + 1 ) = ( 1 + B ) )
6	3, 5	oveq12d 5871	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + 1 ) x. ( B + 1 ) ) = ( ( 1 + A ) x. ( 1 + B ) ) )
7		muladd11 8052	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
8		mulcl 7901	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC )
9	4, 8	addcld 7939	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( B + ( A x. B ) ) e. CC )
10	2, 1, 9	addassd 7942	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( 1 + ( A + ( B + ( A x. B ) ) ) ) )
11	1, 9	addcld 7939	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) e. CC )
12	2, 11	addcomd 8070	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( 1 + ( A + ( B + ( A x. B ) ) ) ) = ( ( A + ( B + ( A x. B ) ) ) + 1 ) )
13	1, 4, 8	addassd 7942	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( A + ( B + ( A x. B ) ) ) )
14		addcl 7899	. . . . . 6 |- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC )
15	14, 8	addcomd 8070	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) + ( A x. B ) ) = ( ( A x. B ) + ( A + B ) ) )
16	13, 15	eqtr3d 2205	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( A + ( B + ( A x. B ) ) ) = ( ( A x. B ) + ( A + B ) ) )
17	16	oveq1d 5868	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( A + ( B + ( A x. B ) ) ) + 1 ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
18	10, 12, 17	3eqtrd 2207	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) + ( B + ( A x. B ) ) ) = ( ( ( A x. B ) + ( A + B ) ) + 1 ) )
19	6, 7, 18	3eqtrd 2207	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 1348   e. wcel 2141  (class class class)co 5853   CCcc 7772   1c1 7775   + caddc 7777   x. cmul 7779

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-resscn 7866  ax-1cn 7867  ax-icn 7869  ax-addcl 7870  ax-mulcl 7872  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-1rid 7881  ax-cnre 7885

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-iota 5160  df-fv 5206  df-ov 5856

This theorem is referenced by: (None)