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Theorem muladd11 10206
Description: A simple product of sums expansion. (Contributed by NM, 21-Feb-2005.)
Assertion
Ref Expression
muladd11 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
Proof of Theorem muladd11
StepHypRef Expression
1 ax-1cn 9994 . . . 4 |- 1 e. CC
2 addcl 10018 . . . 4 |- ( ( 1 e. CC /\ A e. CC ) -> ( 1 + A ) e. CC )
31, 2mpan 706 . . 3 |- ( A e. CC -> ( 1 + A ) e. CC )
4 adddi 10025 . . . 4 |- ( ( ( 1 + A ) e. CC /\ 1 e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
51, 4mp3an2 1412 . . 3 |- ( ( ( 1 + A ) e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
63, 5sylan 488 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) )
73mulid1d 10057 . . . 4 |- ( A e. CC -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
87adantr 481 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. 1 ) = ( 1 + A ) )
9 adddir 10031 . . . . 5 |- ( ( 1 e. CC /\ A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
101, 9mp3an1 1411 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( ( 1 x. B ) + ( A x. B ) ) )
11 mulid2 10038 . . . . . 6 |- ( B e. CC -> ( 1 x. B ) = B )
1211adantl 482 . . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( 1 x. B ) = B )
1312oveq1d 6665 . . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. B ) + ( A x. B ) ) = ( B + ( A x. B ) ) )
1410, 13eqtrd 2656 . . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. B ) = ( B + ( A x. B ) ) )
158, 14oveq12d 6668 . 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( 1 + A ) x. 1 ) + ( ( 1 + A ) x. B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
166, 15eqtrd 2656 1 |- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + A ) x. ( 1 + B ) ) = ( ( 1 + A ) + ( B + ( A x. B ) ) ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990  (class class class)co 6650   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-mulcl 9998  ax-mulcom 10000  ax-mulass 10002  ax-distr 10003  ax-1rid 10006  ax-cnre 10009
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653
This theorem is referenced by:  muladd11r  10249  bernneq  12990
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