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Theorem decsubi 9841

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 9796	. . . . 5 ⊢ ;10 ∈ ℕ₀
2		decaddi.1	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	nn0mulcli 9603	. . . 4 ⊢ (;10 · 𝐴) ∈ ℕ₀
4	3	nn0cni 9577	. . 3 ⊢ (;10 · 𝐴) ∈ ℂ
5		decaddi.2	. . . 4 ⊢ 𝐵 ∈ ℕ₀
6	5	nn0cni 9577	. . 3 ⊢ 𝐵 ∈ ℂ
7		decaddi.3	. . . 4 ⊢ 𝑁 ∈ ℕ₀
8	7	nn0cni 9577	. . 3 ⊢ 𝑁 ∈ ℂ
9	4, 6, 8	addsubassi 8617	. 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁))
10		decaddi.4	. . . 4 ⊢ 𝑀 = ;𝐴𝐵
11		dfdec10 9782	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
12	10, 11	eqtri 2259	. . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵)
13	12	oveq1i 6095	. 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁)
14		dfdec10 9782	. . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶)
15		decsubi.5	. . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶
16	15	eqcomi 2242	. . . 4 ⊢ 𝐶 = (𝐵 − 𝑁)
17	16	oveq2i 6096	. . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁))
18	14, 17	eqtri 2259	. 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁))
19	9, 13, 18	3eqtr4i 2269	1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Colors of variables: wff set class

This proof depends on syntax axioms: = wceq 1402 ∈ wcel 2209 (class class class)co 6085 0cc0 8179 1c1 8180 + caddc 8182 · cmul 8184 − cmin 8497 ℕ₀cn0 9565 cdc 9779

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 4249 ax-pow 4311 ax-pr 4346 ax-setind 4684 ax-cnex 8270 ax-resscn 8271 ax-1cn 8272 ax-1re 8273 ax-icn 8274 ax-addcl 8275 ax-addrcl 8276 ax-mulcl 8277 ax-addcom 8279 ax-mulcom 8280 ax-addass 8281 ax-mulass 8282 ax-distr 8283 ax-i2m1 8284 ax-1rid 8286 ax-0id 8287 ax-rnegex 8288 ax-cnre 8290

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 4131 df-opab 4193 df-id 4438 df-xp 4780 df-rel 4781 df-cnv 4782 df-co 4783 df-dm 4784 df-iota 5337 df-fun 5379 df-fv 5385 df-riota 6038 df-ov 6088 df-oprab 6089 df-mpo 6090 df-sub 8499 df-inn 9306 df-2 9364 df-3 9365 df-4 9366 df-5 9367 df-6 9368 df-7 9369 df-8 9370 df-9 9371 df-n0 9566 df-dec 9780

This theorem is used by: (None)

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