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Theorem decsubi 9648

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 9603	. . . . 5 ⊢ ;10 ∈ ℕ₀
2		decaddi.1	. . . . 5 ⊢ 𝐴 ∈ ℕ₀
3	1, 2	nn0mulcli 9415	. . . 4 ⊢ (;10 · 𝐴) ∈ ℕ₀
4	3	nn0cni 9389	. . 3 ⊢ (;10 · 𝐴) ∈ ℂ
5		decaddi.2	. . . 4 ⊢ 𝐵 ∈ ℕ₀
6	5	nn0cni 9389	. . 3 ⊢ 𝐵 ∈ ℂ
7		decaddi.3	. . . 4 ⊢ 𝑁 ∈ ℕ₀
8	7	nn0cni 9389	. . 3 ⊢ 𝑁 ∈ ℂ
9	4, 6, 8	addsubassi 8445	. 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁))
10		decaddi.4	. . . 4 ⊢ 𝑀 = ;𝐴𝐵
11		dfdec10 9589	. . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵)
12	10, 11	eqtri 2250	. . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵)
13	12	oveq1i 6017	. 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁)
14		dfdec10 9589	. . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶)
15		decsubi.5	. . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶
16	15	eqcomi 2233	. . . 4 ⊢ 𝐶 = (𝐵 − 𝑁)
17	16	oveq2i 6018	. . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁))
18	14, 17	eqtri 2250	. 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁))
19	9, 13, 18	3eqtr4i 2260	1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶

Colors of variables: wff set class

Syntax hints: = wceq 1395 ∈ wcel 2200 (class class class)co 6007 0cc0 8007 1c1 8008 + caddc 8010 · cmul 8012 − cmin 8325 ℕ₀cn0 9377 cdc 9586

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-14 2203 ax-ext 2211 ax-sep 4202 ax-pow 4258 ax-pr 4293 ax-setind 4629 ax-cnex 8098 ax-resscn 8099 ax-1cn 8100 ax-1re 8101 ax-icn 8102 ax-addcl 8103 ax-addrcl 8104 ax-mulcl 8105 ax-addcom 8107 ax-mulcom 8108 ax-addass 8109 ax-mulass 8110 ax-distr 8111 ax-i2m1 8112 ax-1rid 8114 ax-0id 8115 ax-rnegex 8116 ax-cnre 8118

This theorem depends on definitions: df-bi 117 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-ral 2513 df-rex 2514 df-reu 2515 df-rab 2517 df-v 2801 df-sbc 3029 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3889 df-int 3924 df-br 4084 df-opab 4146 df-id 4384 df-xp 4725 df-rel 4726 df-cnv 4727 df-co 4728 df-dm 4729 df-iota 5278 df-fun 5320 df-fv 5326 df-riota 5960 df-ov 6010 df-oprab 6011 df-mpo 6012 df-sub 8327 df-inn 9119 df-2 9177 df-3 9178 df-4 9179 df-5 9180 df-6 9181 df-7 9182 df-8 9183 df-9 9184 df-n0 9378 df-dec 9587

This theorem is referenced by: (None)

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