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Theorem decsubi 9716

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 9671	. . . . 5 ⊢ ;10 ∈ ℕ₀
2		decaddi.1	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	nn0mulcli 9483	. . . 4 ⊢ (;10 · 𝐴) ∈ ℕ₀
4	3	nn0cni 9457	. . 3 ⊢ (;10 · 𝐴) ∈ ℂ
5		decaddi.2	. . . 4 ⊢ 𝐵 ∈ ℕ₀
6	5	nn0cni 9457	. . 3 ⊢ 𝐵 ∈ ℂ
7		decaddi.3	. . . 4 ⊢ 𝑁 ∈ ℕ₀
8	7	nn0cni 9457	. . 3 ⊢ 𝑁 ∈ ℂ
9	4, 6, 8	addsubassi 8513	. 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁))
10		decaddi.4	. . . 4 ⊢ 𝑀 = ;𝐴𝐵
11		dfdec10 9657	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
12	10, 11	eqtri 2252	. . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵)
13	12	oveq1i 6038	. 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁)
14		dfdec10 9657	. . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶)
15		decsubi.5	. . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶
16	15	eqcomi 2235	. . . 4 ⊢ 𝐶 = (𝐵 − 𝑁)
17	16	oveq2i 6039	. . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁))
18	14, 17	eqtri 2252	. 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁))
19	9, 13, 18	3eqtr4i 2262	1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Colors of variables: wff set class

Syntax hints: = wceq 1398 ∈ wcel 2202 (class class class)co 6028 0cc0 8075 1c1 8076 + caddc 8078 · cmul 8080 − cmin 8393 ℕ₀cn0 9445 cdc 9654

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-14 2205 ax-ext 2213 ax-sep 4212 ax-pow 4270 ax-pr 4305 ax-setind 4641 ax-cnex 8166 ax-resscn 8167 ax-1cn 8168 ax-1re 8169 ax-icn 8170 ax-addcl 8171 ax-addrcl 8172 ax-mulcl 8173 ax-addcom 8175 ax-mulcom 8176 ax-addass 8177 ax-mulass 8178 ax-distr 8179 ax-i2m1 8180 ax-1rid 8182 ax-0id 8183 ax-rnegex 8184 ax-cnre 8186

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1811 df-eu 2082 df-mo 2083 df-clab 2218 df-cleq 2224 df-clel 2227 df-nfc 2364 df-ne 2404 df-ral 2516 df-rex 2517 df-reu 2518 df-rab 2520 df-v 2805 df-sbc 3033 df-dif 3203 df-un 3205 df-in 3207 df-ss 3214 df-pw 3658 df-sn 3679 df-pr 3680 df-op 3682 df-uni 3899 df-int 3934 df-br 4094 df-opab 4156 df-id 4396 df-xp 4737 df-rel 4738 df-cnv 4739 df-co 4740 df-dm 4741 df-iota 5293 df-fun 5335 df-fv 5341 df-riota 5981 df-ov 6031 df-oprab 6032 df-mpo 6033 df-sub 8395 df-inn 9187 df-2 9245 df-3 9246 df-4 9247 df-5 9248 df-6 9249 df-7 9250 df-8 9251 df-9 9252 df-n0 9446 df-dec 9655

This theorem is referenced by: (None)

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