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Mirrors > Home > ILE Home > Th. List > zaddcllempos | GIF version |
Description: Lemma for zaddcl 9118. 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 5790 | . . . . 5 ⊢ (𝑥 = 1 → (𝑀 + 𝑥) = (𝑀 + 1)) | |
2 | 1 | eleq1d 2209 | . . . 4 ⊢ (𝑥 = 1 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 1) ∈ ℤ)) |
3 | 2 | imbi2d 229 | . . 3 ⊢ (𝑥 = 1 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ))) |
4 | oveq2 5790 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑀 + 𝑥) = (𝑀 + 𝑦)) | |
5 | 4 | eleq1d 2209 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 𝑦) ∈ ℤ)) |
6 | 5 | imbi2d 229 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 𝑦) ∈ ℤ))) |
7 | oveq2 5790 | . . . . 5 ⊢ (𝑥 = (𝑦 + 1) → (𝑀 + 𝑥) = (𝑀 + (𝑦 + 1))) | |
8 | 7 | eleq1d 2209 | . . . 4 ⊢ (𝑥 = (𝑦 + 1) → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
9 | 8 | imbi2d 229 | . . 3 ⊢ (𝑥 = (𝑦 + 1) → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
10 | oveq2 5790 | . . . . 5 ⊢ (𝑥 = 𝑁 → (𝑀 + 𝑥) = (𝑀 + 𝑁)) | |
11 | 10 | eleq1d 2209 | . . . 4 ⊢ (𝑥 = 𝑁 → ((𝑀 + 𝑥) ∈ ℤ ↔ (𝑀 + 𝑁) ∈ ℤ)) |
12 | 11 | imbi2d 229 | . . 3 ⊢ (𝑥 = 𝑁 → ((𝑀 ∈ ℤ → (𝑀 + 𝑥) ∈ ℤ) ↔ (𝑀 ∈ ℤ → (𝑀 + 𝑁) ∈ ℤ))) |
13 | peano2z 9114 | . . 3 ⊢ (𝑀 ∈ ℤ → (𝑀 + 1) ∈ ℤ) | |
14 | peano2z 9114 | . . . . . 6 ⊢ ((𝑀 + 𝑦) ∈ ℤ → ((𝑀 + 𝑦) + 1) ∈ ℤ) | |
15 | zcn 9083 | . . . . . . . . 9 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℂ) | |
16 | 15 | adantl 275 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
17 | nncn 8752 | . . . . . . . . 9 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
18 | 17 | adantr 274 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 𝑦 ∈ ℂ) |
19 | 1cnd 7806 | . . . . . . . 8 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → 1 ∈ ℂ) | |
20 | 16, 18, 19 | addassd 7812 | . . . . . . 7 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → ((𝑀 + 𝑦) + 1) = (𝑀 + (𝑦 + 1))) |
21 | 20 | eleq1d 2209 | . . . . . 6 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → (((𝑀 + 𝑦) + 1) ∈ ℤ ↔ (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
22 | 14, 21 | syl5ib 153 | . . . . 5 ⊢ ((𝑦 ∈ ℕ ∧ 𝑀 ∈ ℤ) → ((𝑀 + 𝑦) ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ)) |
23 | 22 | ex 114 | . . . 4 ⊢ (𝑦 ∈ ℕ → (𝑀 ∈ ℤ → ((𝑀 + 𝑦) ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
24 | 23 | a2d 26 | . . 3 ⊢ (𝑦 ∈ ℕ → ((𝑀 ∈ ℤ → (𝑀 + 𝑦) ∈ ℤ) → (𝑀 ∈ ℤ → (𝑀 + (𝑦 + 1)) ∈ ℤ))) |
25 | 3, 6, 9, 12, 13, 24 | nnind 8760 | . 2 ⊢ (𝑁 ∈ ℕ → (𝑀 ∈ ℤ → (𝑀 + 𝑁) ∈ ℤ)) |
26 | 25 | impcom 124 | 1 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℕ) → (𝑀 + 𝑁) ∈ ℤ) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1332 ∈ wcel 1481 (class class class)co 5782 ℂcc 7642 1c1 7645 + caddc 7647 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-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-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-0id 7752 ax-rnegex 7753 ax-cnre 7755 |
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-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-sub 7959 df-neg 7960 df-inn 8745 df-n0 9002 df-z 9079 |
This theorem is referenced by: zaddcl 9118 |
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