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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 3975 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
3 | dmgmaddnn0.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0) | |
4 | 3 | nn0cnd 12587 | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℂ) |
5 | 2, 4 | addcld 11278 | . 2 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ ℂ) |
6 | eldmgm 27080 | . . . . 5 ⊢ (𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) | |
7 | 1, 6 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐴 ∈ ℂ ∧ ¬ -𝐴 ∈ ℕ0)) |
8 | 7 | simprd 495 | . . 3 ⊢ (𝜑 → ¬ -𝐴 ∈ ℕ0) |
9 | 2, 4 | negdi2d 11632 | . . . . . . 7 ⊢ (𝜑 → -(𝐴 + 𝑁) = (-𝐴 − 𝑁)) |
10 | 9 | oveq1d 7446 | . . . . . 6 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = ((-𝐴 − 𝑁) + 𝑁)) |
11 | 2 | negcld 11605 | . . . . . . 7 ⊢ (𝜑 → -𝐴 ∈ ℂ) |
12 | 11, 4 | npcand 11622 | . . . . . 6 ⊢ (𝜑 → ((-𝐴 − 𝑁) + 𝑁) = -𝐴) |
13 | 10, 12 | eqtrd 2775 | . . . . 5 ⊢ (𝜑 → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
14 | 13 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) = -𝐴) |
15 | simpr 484 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -(𝐴 + 𝑁) ∈ ℕ0) | |
16 | 3 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → 𝑁 ∈ ℕ0) |
17 | 15, 16 | nn0addcld 12589 | . . . 4 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → (-(𝐴 + 𝑁) + 𝑁) ∈ ℕ0) |
18 | 14, 17 | eqeltrrd 2840 | . . 3 ⊢ ((𝜑 ∧ -(𝐴 + 𝑁) ∈ ℕ0) → -𝐴 ∈ ℕ0) |
19 | 8, 18 | mtand 816 | . 2 ⊢ (𝜑 → ¬ -(𝐴 + 𝑁) ∈ ℕ0) |
20 | eldmgm 27080 | . 2 ⊢ ((𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ)) ↔ ((𝐴 + 𝑁) ∈ ℂ ∧ ¬ -(𝐴 + 𝑁) ∈ ℕ0)) | |
21 | 5, 19, 20 | sylanbrc 583 | 1 ⊢ (𝜑 → (𝐴 + 𝑁) ∈ (ℂ ∖ (ℤ ∖ ℕ))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∖ cdif 3960 (class class class)co 7431 ℂcc 11151 + caddc 11156 − cmin 11490 -cneg 11491 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 |
This theorem is referenced by: lgamcvg2 27113 gamp1 27116 |
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