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Mirrors > Home > ILE Home > Th. List > zaddcllempos | GIF version |
Description: Lemma for zaddcl 8285. Special case in which 𝑁 is a positive integer. (Contributed by Jim Kingdon, 14-Mar-2020.) |
Ref | Expression |
---|---|
zaddcllempos | ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = 1 → (𝑀 + 𝑥) = (𝑀 + 1)) | |
2 | 1 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 1 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 1) ∈ ℤ)) |
3 | 2 | imbi2d 219 | . . 3 ⊢ (𝑥 = 1 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ))) |
4 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑀 + 𝑥) = (𝑀 + 𝑦)) | |
5 | 4 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 𝑦) ∈ ℤ)) |
6 | 5 | imbi2d 219 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 𝑦) ∈ ℤ))) |
7 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑀 + 𝑥) = (𝑀 + (𝑦 + 1))) | |
8 | 7 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
9 | 8 | imbi2d 219 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
10 | oveq2 5520 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑀 + 𝑥) = (𝑀 + 𝑁)) | |
11 | 10 | eleq1d 2106 | . . . 4 ⊢ (𝑥 = 𝑁 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 𝑁) ∈ ℤ)) |
12 | 11 | imbi2d 219 | . . 3 ⊢ (𝑥 = 𝑁 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 𝑁) ∈ ℤ))) |
13 | peano2z 8281 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ) | |
14 | peano2z 8281 | . . . . . 6 ⊢ ((𝑀 + 𝑦) ∈ ℤ → ((𝑀 + 𝑦) + 1) ∈ ℤ) | |
15 | zcn 8250 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
16 | 15 | adantl 262 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
17 | nncn 7922 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | 17 | adantr 261 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 𝑦 ∈ ℂ) |
19 | 1cnd 7043 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 1 ∈ ℂ) | |
20 | 16, 18, 19 | addassd 7049 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → ((𝑀 + 𝑦) + 1) = (𝑀 + (𝑦 + 1))) |
21 | 20 | eleq1d 2106 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (((𝑀 + 𝑦) + 1) ∈ ℤ ↔ (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
22 | 14, 21 | syl5ib 143 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → ((𝑀 + 𝑦) ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
23 | 22 | ex 108 | . . . 4 ⊢ (𝑦 ∈ ℕ → (𝑀 ∈ ℤ → ((𝑀 + 𝑦) ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
24 | 23 | a2d 23 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝑀 ∈ ℤ → (𝑀 + 𝑦) ∈ ℤ) → (𝑀 ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
25 | 3, 6, 9, 12, 13, 24 | nnind 7930 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℤ → (𝑀 + 𝑁) ∈ ℤ)) |
26 | 25 | impcom 116 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 = wceq 1243 ∈ wcel 1393 (class class class)co 5512 ℂcc 6887 1c1 6890 + caddc 6892 ℕcn 7914 ℤcz 8245 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 ax-cnex 6975 ax-resscn 6976 ax-1cn 6977 ax-1re 6978 ax-icn 6979 ax-addcl 6980 ax-addrcl 6981 ax-mulcl 6982 ax-addcom 6984 ax-addass 6986 ax-distr 6988 ax-i2m1 6989 ax-0id 6992 ax-rnegex 6993 ax-cnre 6995 |
This theorem depends on definitions: df-bi 110 df-3or 886 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-br 3765 df-opab 3819 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-iota 4867 df-fun 4904 df-fv 4910 df-riota 5468 df-ov 5515 df-oprab 5516 df-mpt2 5517 df-sub 7184 df-neg 7185 df-inn 7915 df-n0 8182 df-z 8246 |
This theorem is referenced by: zaddcl 8285 |
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