Theorem decrmanc 9765

Description: Perform a multiply-add of two numerals M and N against a fixed multiplicand P (no carry). (Contributed by AV, 16-Sep-2021.)

Hypotheses

Ref	Expression
decrmanc.a	|- A e. NN0
decrmanc.b	|- B e. NN0
decrmanc.n	|- N e. NN0
decrmanc.m	|- M = ; A B
decrmanc.p	|- P e. NN0
decrmanc.e	|- ( A x. P ) = E
decrmanc.f	|- ( ( B x. P ) + N ) = F

Assertion

Ref	Expression
decrmanc	|- ( ( M x. P ) + N ) = ; E F

Proof of Theorem decrmanc

Step	Hyp	Ref	Expression
1		decrmanc.a	. 2 |- A e. NN0
2		decrmanc.b	. 2 |- B e. NN0
3		0nn0 9511	. 2 |- 0 e. NN0
4		decrmanc.n	. 2 |- N e. NN0
5		decrmanc.m	. 2 |- M = ; A B
6	4	dec0h 9730	. 2 |- N = ; 0 N
7		decrmanc.p	. 2 |- P e. NN0
8	1, 7	nn0mulcli 9534	. . . . 5 |- ( A x. P ) e. NN0
9	8	nn0cni 9508	. . . 4 |- ( A x. P ) e. CC
10	9	addridi 8415	. . 3 |- ( ( A x. P ) + 0 ) = ( A x. P )
11		decrmanc.e	. . 3 |- ( A x. P ) = E
12	10, 11	eqtri 2253	. 2 |- ( ( A x. P ) + 0 ) = E
13		decrmanc.f	. 2 |- ( ( B x. P ) + N ) = F
14	1, 2, 3, 4, 5, 6, 7, 12, 13	decma 9759	1 |- ( ( M x. P ) + N ) = ; E F

Syntax hints:   = wceq 1398   e. wcel 2203  (class class class)co 6050   0cc0 8127   + caddc 8130   x. cmul 8132   NN0cn0 9496  ;cdc 9709

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322  ax-setind 4659  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-cnre 8238

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-iota 5312  df-fun 5354  df-fv 5360  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-sub 8446  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-5 9299  df-6 9300  df-7 9301  df-8 9302  df-9 9303  df-n0 9497  df-dec 9710

This theorem is referenced by: (None)