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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 3948 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | dmgmaddnn0.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
4 | 3 | nn0cnd 11958 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | 2, 4 | addcld 10660 | . 2 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ ℂ) |
6 | eldmgm 25599 | . . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) | |
7 | 1, 6 | sylib 220 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
8 | 7 | simprd 498 | . . 3 ⊢ (𝜑 → ¬ -𝐴 ∈ ℕ0) |
9 | 2, 4 | negdi2d 11011 | . . . . . . 7 ⊢ (𝜑 → -(𝐴 + 𝑁) = (-𝐴 − 𝑁)) |
10 | 9 | oveq1d 7171 | . . . . . 6 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = ((-𝐴 − 𝑁) + 𝑁)) |
11 | 2 | negcld 10984 | . . . . . . 7 ⊢ (𝜑 → -𝐴 ∈ ℂ) |
12 | 11, 4 | npcand 11001 | . . . . . 6 ⊢ (𝜑 → ((-𝐴 − 𝑁) + 𝑁) = -𝐴) |
13 | 10, 12 | eqtrd 2856 | . . . . 5 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
14 | 13 | adantr 483 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
15 | simpr 487 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -(𝐴 + 𝑁) ∈ ℕ0) | |
16 | 3 | adantr 483 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → 𝑁 ∈ ℕ0) |
17 | 15, 16 | nn0addcld 11960 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) ∈ ℕ0) |
18 | 14, 17 | eqeltrrd 2914 | . . 3 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -𝐴 ∈ ℕ0) |
19 | 8, 18 | mtand 814 | . 2 ⊢ (𝜑 → ¬ -(𝐴 + 𝑁) ∈ ℕ0) |
20 | eldmgm 25599 | . 2 ⊢ ((𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ ((𝐴 + 𝑁) ∈ ℂ ∧ ¬ -(𝐴 + 𝑁) ∈ ℕ0)) | |
21 | 5, 19, 20 | sylanbrc 585 | 1 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 (class class class)co 7156 ℂcc 10535 + caddc 10540 − cmin 10870 -cneg 10871 ℕcn 11638 ℕ0cn0 11898 ℤcz 11982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-n0 11899 df-z 11983 |
This theorem is referenced by: lgamcvg2 25632 gamp1 25635 |
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