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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 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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