Theorem decma2c 9576

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

Step	Hyp	Ref	Expression
1		10nn0 9541	. . 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 9527	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
8	6, 7	eqtri 2227	. . 3 |- M = ( (; 1 0 x. A ) + B )
9		decma.n	. . . 4 |- N = ; C D
10		dfdec10 9527	. . . 4 |- ; C D = ( (; 1 0 x. C ) + D )
11	9, 10	eqtri 2227	. . 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 9527	. . . 4 |- ; G F = ( (; 1 0 x. G ) + F )
18	16, 17	eqtri 2227	. . 3 |- ( ( P x. B ) + D ) = ( (; 1 0 x. G ) + F )
19	1, 2, 3, 4, 5, 8, 11, 12, 13, 14, 15, 18	numma2c 9569	. 2 |- ( ( P x. M ) + N ) = ( (; 1 0 x. E ) + F )
20		dfdec10 9527	. 2 |- ; E F = ( (; 1 0 x. E ) + F )
21	19, 20	eqtr4i 2230	1 |- ( ( P x. M ) + N ) = ; E F

Syntax hints:   = wceq 1373   e. wcel 2177  (class class class)co 5957   0cc0 7945   1c1 7946   + caddc 7948   x. cmul 7950   NN0cn0 9315  ;cdc 9524

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4170  ax-pow 4226  ax-pr 4261  ax-setind 4593  ax-cnex 8036  ax-resscn 8037  ax-1cn 8038  ax-1re 8039  ax-icn 8040  ax-addcl 8041  ax-addrcl 8042  ax-mulcl 8043  ax-addcom 8045  ax-mulcom 8046  ax-addass 8047  ax-mulass 8048  ax-distr 8049  ax-i2m1 8050  ax-1rid 8052  ax-0id 8053  ax-rnegex 8054  ax-cnre 8056

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-br 4052  df-opab 4114  df-id 4348  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-iota 5241  df-fun 5282  df-fv 5288  df-riota 5912  df-ov 5960  df-oprab 5961  df-mpo 5962  df-sub 8265  df-inn 9057  df-2 9115  df-3 9116  df-4 9117  df-5 9118  df-6 9119  df-7 9120  df-8 9121  df-9 9122  df-n0 9316  df-dec 9525

This theorem is referenced by:  2exp16  12835