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Theorem decma2c 9425
Description: Perform a multiply-add of two numerals  M and  N against a fixed multiplier  P (with carry). (Contributed by Mario Carneiro, 18-Feb-2014.) (Revised by AV, 6-Sep-2021.)
Hypotheses
Ref Expression
decma.a  |-  A  e. 
NN0
decma.b  |-  B  e. 
NN0
decma.c  |-  C  e. 
NN0
decma.d  |-  D  e. 
NN0
decma.m  |-  M  = ; A B
decma.n  |-  N  = ; C D
decma2c.p  |-  P  e. 
NN0
decma2c.f  |-  F  e. 
NN0
decma2c.g  |-  G  e. 
NN0
decma2c.e  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
decma2c.2  |-  ( ( P  x.  B )  +  D )  = ; G F
Assertion
Ref Expression
decma2c  |-  ( ( P  x.  M )  +  N )  = ; E F

Proof of Theorem decma2c
StepHypRef Expression
1 10nn0 9390 . . 3  |- ; 1 0  e.  NN0
2 decma.a . . 3  |-  A  e. 
NN0
3 decma.b . . 3  |-  B  e. 
NN0
4 decma.c . . 3  |-  C  e. 
NN0
5 decma.d . . 3  |-  D  e. 
NN0
6 decma.m . . . 4  |-  M  = ; A B
7 dfdec10 9376 . . . 4  |- ; A B  =  ( (; 1 0  x.  A
)  +  B )
86, 7eqtri 2198 . . 3  |-  M  =  ( (; 1 0  x.  A
)  +  B )
9 decma.n . . . 4  |-  N  = ; C D
10 dfdec10 9376 . . . 4  |- ; C D  =  ( (; 1 0  x.  C
)  +  D )
119, 10eqtri 2198 . . 3  |-  N  =  ( (; 1 0  x.  C
)  +  D )
12 decma2c.p . . 3  |-  P  e. 
NN0
13 decma2c.f . . 3  |-  F  e. 
NN0
14 decma2c.g . . 3  |-  G  e. 
NN0
15 decma2c.e . . 3  |-  ( ( P  x.  A )  +  ( C  +  G ) )  =  E
16 decma2c.2 . . . 4  |-  ( ( P  x.  B )  +  D )  = ; G F
17 dfdec10 9376 . . . 4  |- ; G F  =  ( (; 1 0  x.  G
)  +  F )
1816, 17eqtri 2198 . . 3  |-  ( ( P  x.  B )  +  D )  =  ( (; 1 0  x.  G
)  +  F )
191, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18numma2c 9418 . 2  |-  ( ( P  x.  M )  +  N )  =  ( (; 1 0  x.  E
)  +  F )
20 dfdec10 9376 . 2  |- ; E F  =  ( (; 1 0  x.  E
)  +  F )
2119, 20eqtr4i 2201 1  |-  ( ( P  x.  M )  +  N )  = ; E F
Colors of variables: wff set class
Syntax hints:    = wceq 1353    e. wcel 2148  (class class class)co 5869   0cc0 7802   1c1 7803    + caddc 7805    x. cmul 7807   NN0cn0 9165  ;cdc 9373
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-setind 4533  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-cnre 7913
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-iota 5174  df-fun 5214  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-sub 8120  df-inn 8909  df-2 8967  df-3 8968  df-4 8969  df-5 8970  df-6 8971  df-7 8972  df-8 8973  df-9 8974  df-n0 9166  df-dec 9374
This theorem is referenced by: (None)
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