![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > zaddcl | GIF version |
Description: Closure of addition of integers. (Contributed by NM, 9-May-2004.) (Proof shortened by Mario Carneiro, 16-May-2014.) |
Ref | Expression |
---|---|
zaddcl | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elz 9080 | . . . 4 ⊢ (𝑁 ∈ ℤ ↔ (𝑁 ∈ ℝ ∧ (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ))) | |
2 | 1 | simprbi 273 | . . 3 ⊢ (𝑁 ∈ ℤ → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
3 | 2 | adantl 275 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ)) |
4 | zcn 9083 | . . . . . . 7 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
5 | 4 | adantr 274 | . . . . . 6 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℂ) |
6 | 5 | addid1d 7935 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 0) = 𝑀) |
7 | simpl 108 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → 𝑀 ∈ ℤ) | |
8 | 6, 7 | eqeltrd 2217 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 0) ∈ ℤ) |
9 | oveq2 5790 | . . . . 5 ⊢ (𝑁 = 0 → (𝑀 + 𝑁) = (𝑀 + 0)) | |
10 | 9 | eleq1d 2209 | . . . 4 ⊢ (𝑁 = 0 → ((𝑀 + 𝑁) ∈ ℤ ↔ (𝑀 + 0) ∈ ℤ)) |
11 | 8, 10 | syl5ibrcom 156 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 = 0 → (𝑀 + 𝑁) ∈ ℤ)) |
12 | zaddcllempos 9115 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | |
13 | 12 | ex 114 | . . . 4 ⊢ (𝑀 ∈ ℤ → (𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
14 | 13 | adantr 274 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
15 | zre 9082 | . . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ) | |
16 | zaddcllemneg 9117 | . . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ ∧ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) | |
17 | 16 | 3expia 1184 | . . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℝ) → (-𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
18 | 15, 17 | sylan2 284 | . . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (-𝑁 ∈ ℕ → (𝑀 + 𝑁) ∈ ℤ)) |
19 | 11, 14, 18 | 3jaod 1283 | . 2 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ -𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ)) |
20 | 3, 19 | mpd 13 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∨ w3o 962 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 ℝcr 7643 0cc0 7644 + caddc 7647 -cneg 7958 ℕcn 8744 ℤcz 9078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-1cn 7737 ax-1re 7738 ax-icn 7739 ax-addcl 7740 ax-addrcl 7741 ax-mulcl 7742 ax-addcom 7744 ax-addass 7746 ax-distr 7748 ax-i2m1 7749 ax-0lt1 7750 ax-0id 7752 ax-rnegex 7753 ax-cnre 7755 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 ax-pre-ltadd 7760 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-reu 2424 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-id 4223 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-riota 5738 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: zsubcl 9119 zrevaddcl 9128 zdivadd 9164 zaddcld 9201 eluzaddi 9376 eluzsubi 9377 eluzadd 9378 nn0pzuz 9409 fzen 9854 fzaddel 9870 fzrev3 9898 fzrevral3 9918 elfzmlbp 9940 fzoaddel 10000 zpnn0elfzo 10015 elfzomelpfzo 10039 fzoshftral 10046 climshftlemg 11103 fsumzcl 11203 summodnegmod 11560 dvds2ln 11562 dvds2add 11563 dvdsadd 11572 dvdsadd2b 11576 addmodlteqALT 11593 3dvdsdec 11598 3dvds2dec 11599 opoe 11628 opeo 11630 ndvdsadd 11664 |
Copyright terms: Public domain | W3C validator |