Theorem decsubi 9855

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 9809	. . . . 5 |- ; 1 0 e. NN0
2		decaddi.1	. . . . 5 |- A e. NN0
3	1, 2	nn0mulcli 9612	. . . 4 |- (; 1 0 x. A ) e. NN0
4	3	nn0cni 9586	. . 3 |- (; 1 0 x. A ) e. CC
5		decaddi.2	. . . 4 |- B e. NN0
6	5	nn0cni 9586	. . 3 |- B e. CC
7		decaddi.3	. . . 4 |- N e. NN0
8	7	nn0cni 9586	. . 3 |- N e. CC
9	4, 6, 8	addsubassi 8624	. 2 |- ( ( (; 1 0 x. A ) + B ) - N ) = ( (; 1 0 x. A ) + ( B - N ) )
10		decaddi.4	. . . 4 |- M = ; A B
11		dfdec10 9791	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
12	10, 11	eqtri 2259	. . 3 |- M = ( (; 1 0 x. A ) + B )
13	12	oveq1i 6098	. 2 |- ( M - N ) = ( ( (; 1 0 x. A ) + B ) - N )
14		dfdec10 9791	. . 3 |- ; A C = ( (; 1 0 x. A ) + C )
15		decsubi.5	. . . . 5 |- ( B - N ) = C
16	15	eqcomi 2242	. . . 4 |- C = ( B - N )
17	16	oveq2i 6099	. . 3 |- ( (; 1 0 x. A ) + C ) = ( (; 1 0 x. A ) + ( B - N ) )
18	14, 17	eqtri 2259	. 2 |- ; A C = ( (; 1 0 x. A ) + ( B - N ) )
19	9, 13, 18	3eqtr4i 2269	1 |- ( M - N ) = ; A C

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209  (class class class)co 6088   0cc0 8185   1c1 8186   + caddc 8188   x. cmul 8190   - cmin 8504   NN0cn0 9574  ;cdc 9788

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4251  ax-pow 4313  ax-pr 4348  ax-setind 4686  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-mulcom 8286  ax-addass 8287  ax-mulass 8288  ax-distr 8289  ax-i2m1 8290  ax-1rid 8292  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-iota 5340  df-fun 5382  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-sub 8506  df-inn 9314  df-2 9372  df-3 9373  df-4 9374  df-5 9375  df-6 9376  df-7 9377  df-8 9378  df-9 9379  df-n0 9575  df-dec 9789

This theorem is used by: (None)