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Mirrors > Home > MPE Home > Th. List > zaddcl | Structured version Visualization version 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 | elz2 12080 | . 2 ⊢ (𝑀 ∈ ℤ ↔ ∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦)) | |
2 | elz2 12080 | . 2 ⊢ (𝑁 ∈ ℤ ↔ ∃𝑧 ∈ ℕ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤)) | |
3 | reeanv 3270 | . . 3 ⊢ (∃𝑥 ∈ ℕ ∃𝑧 ∈ ℕ (∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤)) ↔ (∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ ℕ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤))) | |
4 | reeanv 3270 | . . . . 5 ⊢ (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝑀 = (𝑥 − 𝑦) ∧ 𝑁 = (𝑧 − 𝑤)) ↔ (∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤))) | |
5 | nnaddcl 11739 | . . . . . . . . . 10 ⊢ ((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑥 + 𝑧) ∈ ℕ) | |
6 | 5 | adantr 484 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑥 + 𝑧) ∈ ℕ) |
7 | nnaddcl 11739 | . . . . . . . . . 10 ⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 + 𝑤) ∈ ℕ) | |
8 | 7 | adantl 485 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → (𝑦 + 𝑤) ∈ ℕ) |
9 | nncn 11724 | . . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℕ → 𝑥 ∈ ℂ) | |
10 | nncn 11724 | . . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℕ → 𝑧 ∈ ℂ) | |
11 | 9, 10 | anim12i 616 | . . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ)) |
12 | nncn 11724 | . . . . . . . . . . . 12 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
13 | nncn 11724 | . . . . . . . . . . . 12 ⊢ (𝑤 ∈ ℕ → 𝑤 ∈ ℂ) | |
14 | 12, 13 | anim12i 616 | . . . . . . . . . . 11 ⊢ ((𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ) → (𝑦 ∈ ℂ ∧ 𝑤 ∈ ℂ)) |
15 | addsub4 11007 | . . . . . . . . . . 11 ⊢ (((𝑥 ∈ ℂ ∧ 𝑧 ∈ ℂ) ∧ (𝑦 ∈ ℂ ∧ 𝑤 ∈ ℂ)) → ((𝑥 + 𝑧) − (𝑦 + 𝑤)) = ((𝑥 − 𝑦) + (𝑧 − 𝑤))) | |
16 | 11, 14, 15 | syl2an 599 | . . . . . . . . . 10 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 + 𝑧) − (𝑦 + 𝑤)) = ((𝑥 − 𝑦) + (𝑧 − 𝑤))) |
17 | 16 | eqcomd 2744 | . . . . . . . . 9 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 − 𝑦) + (𝑧 − 𝑤)) = ((𝑥 + 𝑧) − (𝑦 + 𝑤))) |
18 | rspceov 7217 | . . . . . . . . 9 ⊢ (((𝑥 + 𝑧) ∈ ℕ ∧ (𝑦 + 𝑤) ∈ ℕ ∧ ((𝑥 − 𝑦) + (𝑧 − 𝑤)) = ((𝑥 + 𝑧) − (𝑦 + 𝑤))) → ∃𝑢 ∈ ℕ ∃𝑣 ∈ ℕ ((𝑥 − 𝑦) + (𝑧 − 𝑤)) = (𝑢 − 𝑣)) | |
19 | 6, 8, 17, 18 | syl3anc 1372 | . . . . . . . 8 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ∃𝑢 ∈ ℕ ∃𝑣 ∈ ℕ ((𝑥 − 𝑦) + (𝑧 − 𝑤)) = (𝑢 − 𝑣)) |
20 | elz2 12080 | . . . . . . . 8 ⊢ (((𝑥 − 𝑦) + (𝑧 − 𝑤)) ∈ ℤ ↔ ∃𝑢 ∈ ℕ ∃𝑣 ∈ ℕ ((𝑥 − 𝑦) + (𝑧 − 𝑤)) = (𝑢 − 𝑣)) | |
21 | 19, 20 | sylibr 237 | . . . . . . 7 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑥 − 𝑦) + (𝑧 − 𝑤)) ∈ ℤ) |
22 | oveq12 7179 | . . . . . . . 8 ⊢ ((𝑀 = (𝑥 − 𝑦) ∧ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) = ((𝑥 − 𝑦) + (𝑧 − 𝑤))) | |
23 | 22 | eleq1d 2817 | . . . . . . 7 ⊢ ((𝑀 = (𝑥 − 𝑦) ∧ 𝑁 = (𝑧 − 𝑤)) → ((𝑀 + 𝑁) ∈ ℤ ↔ ((𝑥 − 𝑦) + (𝑧 − 𝑤)) ∈ ℤ)) |
24 | 21, 23 | syl5ibrcom 250 | . . . . . 6 ⊢ (((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) ∧ (𝑦 ∈ ℕ ∧ 𝑤 ∈ ℕ)) → ((𝑀 = (𝑥 − 𝑦) ∧ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) ∈ ℤ)) |
25 | 24 | rexlimdvva 3204 | . . . . 5 ⊢ ((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) → (∃𝑦 ∈ ℕ ∃𝑤 ∈ ℕ (𝑀 = (𝑥 − 𝑦) ∧ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) ∈ ℤ)) |
26 | 4, 25 | syl5bir 246 | . . . 4 ⊢ ((𝑥 ∈ ℕ ∧ 𝑧 ∈ ℕ) → ((∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) ∈ ℤ)) |
27 | 26 | rexlimivv 3202 | . . 3 ⊢ (∃𝑥 ∈ ℕ ∃𝑧 ∈ ℕ (∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) ∈ ℤ) |
28 | 3, 27 | sylbir 238 | . 2 ⊢ ((∃𝑥 ∈ ℕ ∃𝑦 ∈ ℕ 𝑀 = (𝑥 − 𝑦) ∧ ∃𝑧 ∈ ℕ ∃𝑤 ∈ ℕ 𝑁 = (𝑧 − 𝑤)) → (𝑀 + 𝑁) ∈ ℤ) |
29 | 1, 2, 28 | syl2anb 601 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 + 𝑁) ∈ ℤ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∃wrex 3054 (class class class)co 7170 ℂcc 10613 + caddc 10618 − cmin 10948 ℕcn 11716 ℤcz 12062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-n0 11977 df-z 12063 |
This theorem is referenced by: peano2z 12104 zsubcl 12105 zrevaddcl 12108 zdivadd 12134 zaddcld 12172 eluzaddi 12353 eluzsubi 12354 nn0pzuz 12387 fzen 13015 fzaddel 13032 fzadd2 13033 fzrev3 13064 fzrevral3 13085 elfzmlbp 13109 fzoun 13165 fzoaddel 13181 zpnn0elfzo 13201 elfzomelpfzo 13232 fzoshftral 13245 modsumfzodifsn 13403 ccatsymb 14025 ccatval21sw 14028 lswccatn0lsw 14034 swrdccatin2 14180 revccat 14217 2cshw 14264 cshweqrep 14272 2cshwcshw 14276 cshwcsh2id 14279 cshco 14287 climshftlem 15021 isershft 15113 iseraltlem2 15132 fsumzcl 15185 zrisefaccl 15466 summodnegmod 15732 dvds2ln 15734 dvds2add 15735 dvdsadd 15747 dvdsadd2b 15751 addmodlteqALT 15770 3dvdsdec 15777 3dvds2dec 15778 opoe 15808 opeo 15810 divalglem2 15840 ndvdsadd 15855 gcdaddmlem 15967 pythagtriplem9 16261 difsqpwdvds 16323 gzaddcl 16373 mod2xnegi 16507 cshwshashlem2 16533 cycsubgcl 18467 efgredleme 18987 zaddablx 19111 pgpfac1lem2 19316 zsubrg 20270 zringmulg 20297 expghm 20316 mulgghm2 20317 cygznlem3 20388 iaa 25073 dchrisumlem1 26225 axlowdimlem16 26903 crctcshwlkn0lem4 27751 crctcshwlkn0 27759 clwwlkccatlem 27926 clwwisshclwwslemlem 27950 ballotlemsima 32052 mzpclall 40121 mzpindd 40140 rmxyadd 40315 jm2.18 40382 inductionexd 41311 dvdsn1add 43022 stoweidlem34 43117 fourierswlem 43313 2elfz2melfz 44344 opoeALTV 44669 opeoALTV 44670 even3prm2 44705 mogoldbblem 44706 gbowgt5 44748 gboge9 44750 sbgoldbst 44764 2zrngamgm 45031 |
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