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Mirrors > Home > ILE Home > Th. List > decsubi | GIF version |
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.) |
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 |
Ref | Expression |
---|---|
decsubi | ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 10nn0 8892 | . . . . 5 ⊢ ;10 ∈ ℕ0 | |
2 | decaddi.1 | . . . . 5 ⊢ 𝐴 ∈ ℕ0 | |
3 | 1, 2 | nn0mulcli 8709 | . . . 4 ⊢ (;10 · 𝐴) ∈ ℕ0 |
4 | 3 | nn0cni 8683 | . . 3 ⊢ (;10 · 𝐴) ∈ ℂ |
5 | decaddi.2 | . . . 4 ⊢ 𝐵 ∈ ℕ0 | |
6 | 5 | nn0cni 8683 | . . 3 ⊢ 𝐵 ∈ ℂ |
7 | decaddi.3 | . . . 4 ⊢ 𝑁 ∈ ℕ0 | |
8 | 7 | nn0cni 8683 | . . 3 ⊢ 𝑁 ∈ ℂ |
9 | 4, 6, 8 | addsubassi 7771 | . 2 ⊢ (((;10 · 𝐴) + 𝐵) − 𝑁) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
10 | decaddi.4 | . . . 4 ⊢ 𝑀 = ;𝐴𝐵 | |
11 | dfdec10 8878 | . . . 4 ⊢ ;𝐴𝐵 = ((;10 · 𝐴) + 𝐵) | |
12 | 10, 11 | eqtri 2108 | . . 3 ⊢ 𝑀 = ((;10 · 𝐴) + 𝐵) |
13 | 12 | oveq1i 5662 | . 2 ⊢ (𝑀 − 𝑁) = (((;10 · 𝐴) + 𝐵) − 𝑁) |
14 | dfdec10 8878 | . . 3 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + 𝐶) | |
15 | decsubi.5 | . . . . 5 ⊢ (𝐵 − 𝑁) = 𝐶 | |
16 | 15 | eqcomi 2092 | . . . 4 ⊢ 𝐶 = (𝐵 − 𝑁) |
17 | 16 | oveq2i 5663 | . . 3 ⊢ ((;10 · 𝐴) + 𝐶) = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
18 | 14, 17 | eqtri 2108 | . 2 ⊢ ;𝐴𝐶 = ((;10 · 𝐴) + (𝐵 − 𝑁)) |
19 | 9, 13, 18 | 3eqtr4i 2118 | 1 ⊢ (𝑀 − 𝑁) = ;𝐴𝐶 |
Colors of variables: wff set class |
Syntax hints: = wceq 1289 ∈ wcel 1438 (class class class)co 5652 0cc0 7348 1c1 7349 + caddc 7351 · cmul 7353 − cmin 7651 ℕ0cn0 8671 ;cdc 8875 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 ax-setind 4353 ax-cnex 7434 ax-resscn 7435 ax-1cn 7436 ax-1re 7437 ax-icn 7438 ax-addcl 7439 ax-addrcl 7440 ax-mulcl 7441 ax-addcom 7443 ax-mulcom 7444 ax-addass 7445 ax-mulass 7446 ax-distr 7447 ax-i2m1 7448 ax-1rid 7450 ax-0id 7451 ax-rnegex 7452 ax-cnre 7454 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-fal 1295 df-nf 1395 df-sb 1693 df-eu 1951 df-mo 1952 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ne 2256 df-ral 2364 df-rex 2365 df-reu 2366 df-rab 2368 df-v 2621 df-sbc 2841 df-dif 3001 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-uni 3654 df-int 3689 df-br 3846 df-opab 3900 df-id 4120 df-xp 4444 df-rel 4445 df-cnv 4446 df-co 4447 df-dm 4448 df-iota 4980 df-fun 5017 df-fv 5023 df-riota 5608 df-ov 5655 df-oprab 5656 df-mpt2 5657 df-sub 7653 df-inn 8421 df-2 8479 df-3 8480 df-4 8481 df-5 8482 df-6 8483 df-7 8484 df-8 8485 df-9 8486 df-n0 8672 df-dec 8876 |
This theorem is referenced by: (None) |
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