Theorem decsubi 9417

Description: Difference between a numeral M and a nonnegative integer N (no underflow). (Contributed by AV, 22-Jul-2021.) (Revised by AV, 6-Sep-2021.)

Hypotheses

Ref	Expression
decaddi.1	|- A e. NN0
decaddi.2	|- B e. NN0
decaddi.3	|- N e. NN0
decaddi.4	|- M = ; A B
decaddci.5	|- ( A + 1 ) = D
decsubi.5	|- ( B - N ) = C

Assertion

Ref	Expression
decsubi	|- ( M - N ) = ; A C

Proof of Theorem decsubi

Step	Hyp	Ref	Expression
1		10nn0 9372	. . . . 5 |- ; 1 0 e. NN0
2		decaddi.1	. . . . 5 |- A e. NN0
3	1, 2	nn0mulcli 9185	. . . 4 |- (; 1 0 x. A ) e. NN0
4	3	nn0cni 9159	. . 3 |- (; 1 0 x. A ) e. CC
5		decaddi.2	. . . 4 |- B e. NN0
6	5	nn0cni 9159	. . 3 |- B e. CC
7		decaddi.3	. . . 4 |- N e. NN0
8	7	nn0cni 9159	. . 3 |- N e. CC
9	4, 6, 8	addsubassi 8222	. 2 |- ( ( (; 1 0 x. A ) + B ) - N ) = ( (; 1 0 x. A ) + ( B - N ) )
10		decaddi.4	. . . 4 |- M = ; A B
11		dfdec10 9358	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
12	10, 11	eqtri 2196	. . 3 |- M = ( (; 1 0 x. A ) + B )
13	12	oveq1i 5875	. 2 |- ( M - N ) = ( ( (; 1 0 x. A ) + B ) - N )
14		dfdec10 9358	. . 3 |- ; A C = ( (; 1 0 x. A ) + C )
15		decsubi.5	. . . . 5 |- ( B - N ) = C
16	15	eqcomi 2179	. . . 4 |- C = ( B - N )
17	16	oveq2i 5876	. . 3 |- ( (; 1 0 x. A ) + C ) = ( (; 1 0 x. A ) + ( B - N ) )
18	14, 17	eqtri 2196	. 2 |- ; A C = ( (; 1 0 x. A ) + ( B - N ) )
19	9, 13, 18	3eqtr4i 2206	1 |- ( M - N ) = ; A C

Syntax hints:   = wceq 1353   e. wcel 2146  (class class class)co 5865   0cc0 7786   1c1 7787   + caddc 7789   x. cmul 7791   - cmin 8102   NN0cn0 9147  ;cdc 9355

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-1cn 7879  ax-1re 7880  ax-icn 7881  ax-addcl 7882  ax-addrcl 7883  ax-mulcl 7884  ax-addcom 7886  ax-mulcom 7887  ax-addass 7888  ax-mulass 7889  ax-distr 7890  ax-i2m1 7891  ax-1rid 7893  ax-0id 7894  ax-rnegex 7895  ax-cnre 7897

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-ral 2458  df-rex 2459  df-reu 2460  df-rab 2462  df-v 2737  df-sbc 2961  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-riota 5821  df-ov 5868  df-oprab 5869  df-mpo 5870  df-sub 8104  df-inn 8891  df-2 8949  df-3 8950  df-4 8951  df-5 8952  df-6 8953  df-7 8954  df-8 8955  df-9 8956  df-n0 9148  df-dec 9356

This theorem is referenced by: (None)