Theorem decsubi 9566

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 9521	. . . . 5 |- ; 1 0 e. NN0
2		decaddi.1	. . . . 5 |- A e. NN0
3	1, 2	nn0mulcli 9333	. . . 4 |- (; 1 0 x. A ) e. NN0
4	3	nn0cni 9307	. . 3 |- (; 1 0 x. A ) e. CC
5		decaddi.2	. . . 4 |- B e. NN0
6	5	nn0cni 9307	. . 3 |- B e. CC
7		decaddi.3	. . . 4 |- N e. NN0
8	7	nn0cni 9307	. . 3 |- N e. CC
9	4, 6, 8	addsubassi 8363	. 2 |- ( ( (; 1 0 x. A ) + B ) - N ) = ( (; 1 0 x. A ) + ( B - N ) )
10		decaddi.4	. . . 4 |- M = ; A B
11		dfdec10 9507	. . . 4 |- ; A B = ( (; 1 0 x. A ) + B )
12	10, 11	eqtri 2226	. . 3 |- M = ( (; 1 0 x. A ) + B )
13	12	oveq1i 5954	. 2 |- ( M - N ) = ( ( (; 1 0 x. A ) + B ) - N )
14		dfdec10 9507	. . 3 |- ; A C = ( (; 1 0 x. A ) + C )
15		decsubi.5	. . . . 5 |- ( B - N ) = C
16	15	eqcomi 2209	. . . 4 |- C = ( B - N )
17	16	oveq2i 5955	. . 3 |- ( (; 1 0 x. A ) + C ) = ( (; 1 0 x. A ) + ( B - N ) )
18	14, 17	eqtri 2226	. 2 |- ; A C = ( (; 1 0 x. A ) + ( B - N ) )
19	9, 13, 18	3eqtr4i 2236	1 |- ( M - N ) = ; A C

Syntax hints:   = wceq 1373   e. wcel 2176  (class class class)co 5944   0cc0 7925   1c1 7926   + caddc 7928   x. cmul 7930   - cmin 8243   NN0cn0 9295  ;cdc 9504

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-cnre 8036

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rab 2493  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-iota 5232  df-fun 5273  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-sub 8245  df-inn 9037  df-2 9095  df-3 9096  df-4 9097  df-5 9098  df-6 9099  df-7 9100  df-8 9101  df-9 9102  df-n0 9296  df-dec 9505

This theorem is referenced by: (None)