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Mirrors > Home > MPE Home > Th. List > dmgmaddnn0 | Structured version Visualization version GIF version |
Description: If 𝐴 is not a nonpositive integer and 𝑁 is a nonnegative integer, then 𝐴 + 𝑁 is also not a nonpositive integer. (Contributed by Mario Carneiro, 6-Jul-2017.) |
Ref | Expression |
---|---|
dmgmn0.a | ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
dmgmaddnn0.n | ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
Ref | Expression |
---|---|
dmgmaddnn0 | ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dmgmn0.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ))) | |
2 | 1 | eldifad 3552 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | dmgmaddnn0.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
4 | 3 | nn0cnd 11230 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | 2, 4 | addcld 9938 | . 2 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ ℂ) |
6 | eldmgm 24548 | . . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) | |
7 | 1, 6 | sylib 207 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
8 | 7 | simprd 478 | . . 3 ⊢ (𝜑 → ¬ -𝐴 ∈ ℕ0) |
9 | 2, 4 | negdi2d 10285 | . . . . . . 7 ⊢ (𝜑 → -(𝐴 + 𝑁) = (-𝐴 − 𝑁)) |
10 | 9 | oveq1d 6564 | . . . . . 6 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = ((-𝐴 − 𝑁) + 𝑁)) |
11 | 2 | negcld 10258 | . . . . . . 7 ⊢ (𝜑 → -𝐴 ∈ ℂ) |
12 | 11, 4 | npcand 10275 | . . . . . 6 ⊢ (𝜑 → ((-𝐴 − 𝑁) + 𝑁) = -𝐴) |
13 | 10, 12 | eqtrd 2644 | . . . . 5 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
15 | simpr 476 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -(𝐴 + 𝑁) ∈ ℕ0) | |
16 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → 𝑁 ∈ ℕ0) |
17 | 15, 16 | nn0addcld 11232 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) ∈ ℕ0) |
18 | 14, 17 | eqeltrrd 2689 | . . 3 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -𝐴 ∈ ℕ0) |
19 | 8, 18 | mtand 689 | . 2 ⊢ (𝜑 → ¬ -(𝐴 + 𝑁) ∈ ℕ0) |
20 | eldmgm 24548 | . 2 ⊢ ((𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ ((𝐴 + 𝑁) ∈ ℂ ∧ ¬ -(𝐴 + 𝑁) ∈ ℕ0)) | |
21 | 5, 19, 20 | sylanbrc 695 | 1 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∖ cdif 3537 (class class class)co 6549 ℂcc 9813 + caddc 9818 − cmin 10145 -cneg 10146 ℕcn 10897 ℕ0cn0 11169 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 |
This theorem is referenced by: lgamcvg2 24581 gamp1 24584 |
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