Theorem decsubi 9636

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 9591	. . . . 5 |- ; 1 0 e. NN0
2		decaddi.1	. . . . 5 |- A e. NN0
3	1, 2	nn0mulcli 9403	. . . 4 |- (; 1 0 x. A ) e. NN0
4	3	nn0cni 9377	. . 3 |- (; 1 0 x. A ) e. CC
5		decaddi.2	. . . 4 |- B e. NN0
6	5	nn0cni 9377	. . 3 |- B e. CC
7		decaddi.3	. . . 4 |- N e. NN0
8	7	nn0cni 9377	. . 3 |- N e. CC
9	4, 6, 8	addsubassi 8433	. 2 |- ( ( (; 1 0 x. A ) + B ) - N ) = ( (; 1 0 x. A ) + ( B - N ) )
10		decaddi.4	. . . 4 |- M = ; A B
11		dfdec10 9577	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
12	10, 11	eqtri 2250	. . 3 |- M = ( (; 1 0 x. A ) + B )
13	12	oveq1i 6010	. 2 |- ( M - N ) = ( ( (; 1 0 x. A ) + B ) - N )
14		dfdec10 9577	. . 3 |- ; A C = ( (; 1 0 x. A ) + C )
15		decsubi.5	. . . . 5 |- ( B - N ) = C
16	15	eqcomi 2233	. . . 4 |- C = ( B - N )
17	16	oveq2i 6011	. . 3 |- ( (; 1 0 x. A ) + C ) = ( (; 1 0 x. A ) + ( B - N ) )
18	14, 17	eqtri 2250	. 2 |- ; A C = ( (; 1 0 x. A ) + ( B - N ) )
19	9, 13, 18	3eqtr4i 2260	1 |- ( M - N ) = ; A C

Syntax hints:   = wceq 1395   e. wcel 2200  (class class class)co 6000   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   - cmin 8313   NN0cn0 9365  ;cdc 9574

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-cnre 8106

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-iota 5277  df-fun 5319  df-fv 5325  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-sub 8315  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-5 9168  df-6 9169  df-7 9170  df-8 9171  df-9 9172  df-n0 9366  df-dec 9575

This theorem is referenced by: (None)