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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 12874 | . 2 ⊢ (𝐴 ∈ ℚ ↔ ∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦)) | |
2 | zcn 12503 | . . . . . . 7 ⊢ (𝑥 ∈ ℤ → 𝑥 ∈ ℂ) | |
3 | 2 | adantr 481 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑥 ∈ ℂ) |
4 | nncn 12160 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ∈ ℂ) | |
5 | 4 | adantl 482 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑦 ∈ ℂ) |
6 | nnne0 12186 | . . . . . . 7 ⊢ (𝑦 ∈ ℕ → 𝑦 ≠ 0) | |
7 | 6 | adantl 482 | . . . . . 6 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → 𝑦 ≠ 0) |
8 | 3, 5, 7 | divnegd 11943 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → -(𝑥 / 𝑦) = (-𝑥 / 𝑦)) |
9 | znegcl 12537 | . . . . . 6 ⊢ (𝑥 ∈ ℤ → -𝑥 ∈ ℤ) | |
10 | znq 12876 | . . . . . 6 ⊢ ((-𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (-𝑥 / 𝑦) ∈ ℚ) | |
11 | 9, 10 | sylan 580 | . . . . 5 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (-𝑥 / 𝑦) ∈ ℚ) |
12 | 8, 11 | eqeltrd 2838 | . . . 4 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → -(𝑥 / 𝑦) ∈ ℚ) |
13 | negeq 11392 | . . . . 5 ⊢ (𝐴 = (𝑥 / 𝑦) → -𝐴 = -(𝑥 / 𝑦)) | |
14 | 13 | eleq1d 2822 | . . . 4 ⊢ (𝐴 = (𝑥 / 𝑦) → (-𝐴 ∈ ℚ ↔ -(𝑥 / 𝑦) ∈ ℚ)) |
15 | 12, 14 | syl5ibrcom 246 | . . 3 ⊢ ((𝑥 ∈ ℤ ∧ 𝑦 ∈ ℕ) → (𝐴 = (𝑥 / 𝑦) → -𝐴 ∈ ℚ)) |
16 | 15 | rexlimivv 3196 | . 2 ⊢ (∃𝑥 ∈ ℤ ∃𝑦 ∈ ℕ 𝐴 = (𝑥 / 𝑦) → -𝐴 ∈ ℚ) |
17 | 1, 16 | sylbi 216 | 1 ⊢ (𝐴 ∈ ℚ → -𝐴 ∈ ℚ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ≠ wne 2943 ∃wrex 3073 (class class class)co 7356 ℂcc 11048 0cc0 11050 -cneg 11385 / cdiv 11811 ℕcn 12152 ℤcz 12498 ℚcq 12872 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-resscn 11107 ax-1cn 11108 ax-icn 11109 ax-addcl 11110 ax-addrcl 11111 ax-mulcl 11112 ax-mulrcl 11113 ax-mulcom 11114 ax-addass 11115 ax-mulass 11116 ax-distr 11117 ax-i2m1 11118 ax-1ne0 11119 ax-1rid 11120 ax-rnegex 11121 ax-rrecex 11122 ax-cnre 11123 ax-pre-lttri 11124 ax-pre-lttrn 11125 ax-pre-ltadd 11126 ax-pre-mulgt0 11127 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7802 df-1st 7920 df-2nd 7921 df-frecs 8211 df-wrecs 8242 df-recs 8316 df-rdg 8355 df-er 8647 df-en 8883 df-dom 8884 df-sdom 8885 df-pnf 11190 df-mnf 11191 df-xr 11192 df-ltxr 11193 df-le 11194 df-sub 11386 df-neg 11387 df-div 11812 df-nn 12153 df-z 12499 df-q 12873 |
This theorem is referenced by: qsubcl 12892 pcadd2 16761 qsubdrg 20847 vitalilem1 24970 qaa 25681 numdenneg 31657 3cubes 40990 rmxyneg 41221 mpaaeu 41454 |
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