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Theorem decsubi 9475

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 ∈ ℕ₀
decaddi.2	⊢ 𝐵 ∈ ℕ₀
decaddi.3	⊢ 𝑁 ∈ ℕ₀
decaddi.4	⊢ 𝑀 = ;𝐴𝐵
decaddci.5	⊢ (𝐴 + 1) = 𝐷
decsubi.5	⊢ (𝐵 − 𝑁) = 𝐶

**Assertion**
Ref	Expression
decsubi	⊢ (𝑀 − 𝑁) = ;𝐴𝐶

**Proof of Theorem decsubi**
Step	Hyp	Ref	Expression
1		10nn0 9430	. . . . 5 ⊢ ;10 ∈ ℕ₀
2		decaddi.1	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	nn0mulcli 9243	. . . 4 ⊢ (;10 · 𝐴) ∈ ℕ₀
4	3	nn0cni 9217	. . 3 ⊢ (;10 · 𝐴) ∈ ℂ
5		decaddi.2	. . . 4 ⊢ 𝐵 ∈ ℕ₀
6	5	nn0cni 9217	. . 3 ⊢ 𝐵 ∈ ℂ
7		decaddi.3	. . . 4 ⊢ 𝑁 ∈ ℕ₀
8	7	nn0cni 9217	. . 3 ⊢ 𝑁 ∈ ℂ
9	4, 6, 8	addsubassi 8277	. 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁))
10		decaddi.4	. . . 4 ⊢ 𝑀 = ;𝐴𝐵
11		dfdec10 9416	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
12	10, 11	eqtri 2210	. . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵)
13	12	oveq1i 5905	. 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁)
14		dfdec10 9416	. . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶)
15		decsubi.5	. . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶
16	15	eqcomi 2193	. . . 4 ⊢ 𝐶 = (𝐵 − 𝑁)
17	16	oveq2i 5906	. . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁))
18	14, 17	eqtri 2210	. 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁))
19	9, 13, 18	3eqtr4i 2220	1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Colors of variables: wff set class

Syntax hints: = wceq 1364 ∈ wcel 2160 (class class class)co 5895 0cc0 7840 1c1 7841 + caddc 7843 · cmul 7845 − cmin 8157 ℕ₀cn0 9205 cdc 9413

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 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-setind 4554 ax-cnex 7931 ax-resscn 7932 ax-1cn 7933 ax-1re 7934 ax-icn 7935 ax-addcl 7936 ax-addrcl 7937 ax-mulcl 7938 ax-addcom 7940 ax-mulcom 7941 ax-addass 7942 ax-mulass 7943 ax-distr 7944 ax-i2m1 7945 ax-1rid 7947 ax-0id 7948 ax-rnegex 7949 ax-cnre 7951

This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-reu 2475 df-rab 2477 df-v 2754 df-sbc 2978 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-br 4019 df-opab 4080 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-iota 5196 df-fun 5237 df-fv 5243 df-riota 5851 df-ov 5898 df-oprab 5899 df-mpo 5900 df-sub 8159 df-inn 8949 df-2 9007 df-3 9008 df-4 9009 df-5 9010 df-6 9011 df-7 9012 df-8 9013 df-9 9014 df-n0 9206 df-dec 9414

This theorem is referenced by: (None)

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