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Mirrors > Home > MPE Home > Th. List > qnegcl | Structured version Visualization version GIF version |
Description: Closure law for the negative of a rational. (Contributed by NM, 2-Aug-2004.) (Revised by Mario Carneiro, 15-Sep-2014.) |
Ref | Expression |
---|---|
qnegcl | ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elq 11993 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zcn 11584 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
3 | 2 | adantr 466 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ) |
4 | nncn 11230 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | 4 | adantl 467 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
6 | nnne0 11255 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
7 | 6 | adantl 467 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
8 | 3, 5, 7 | divnegd 11016 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → -(𝑥 / 𝑦) = (-𝑥 / 𝑦)) |
9 | znegcl 11614 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
10 | znq 11995 | . . . . . 6 ⊢ ((-𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (-𝑥 / 𝑦) ∈ ℚ) | |
11 | 9, 10 | sylan 569 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (-𝑥 / 𝑦) ∈ ℚ) |
12 | 8, 11 | eqeltrd 2850 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → -(𝑥 / 𝑦) ∈ ℚ) |
13 | negeq 10475 | . . . . 5 ⊢ (𝐴 = (𝑥 / 𝑦) → -𝐴 = -(𝑥 / 𝑦)) | |
14 | 13 | eleq1d 2835 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (-𝐴 ∈ ℚ ↔ -(𝑥 / 𝑦) ∈ ℚ)) |
15 | 12, 14 | syl5ibrcom 237 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → -𝐴 ∈ ℚ)) |
16 | 15 | rexlimivv 3184 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → -𝐴 ∈ ℚ) |
17 | 1, 16 | sylbi 207 | 1 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1631 ∈ wcel 2145 ≠ wne 2943 ∃wrex 3062 (class class class)co 6793 ℂcc 10136 0cc0 10138 -cneg 10469 / cdiv 10886 ℕcn 11222 ℤcz 11579 ℚcq 11991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-resscn 10195 ax-1cn 10196 ax-icn 10197 ax-addcl 10198 ax-addrcl 10199 ax-mulcl 10200 ax-mulrcl 10201 ax-mulcom 10202 ax-addass 10203 ax-mulass 10204 ax-distr 10205 ax-i2m1 10206 ax-1ne0 10207 ax-1rid 10208 ax-rnegex 10209 ax-rrecex 10210 ax-cnre 10211 ax-pre-lttri 10212 ax-pre-lttrn 10213 ax-pre-ltadd 10214 ax-pre-mulgt0 10215 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-nel 3047 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6754 df-ov 6796 df-oprab 6797 df-mpt2 6798 df-om 7213 df-1st 7315 df-2nd 7316 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-er 7896 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10278 df-mnf 10279 df-xr 10280 df-ltxr 10281 df-le 10282 df-sub 10470 df-neg 10471 df-div 10887 df-nn 11223 df-z 11580 df-q 11992 |
This theorem is referenced by: qsubcl 12010 pcadd2 15801 qsubdrg 20013 vitalilem1 23596 qaa 24298 numdenneg 29903 rmxyneg 38011 mpaaeu 38246 |
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