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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > numdenneg | Structured version Visualization version GIF version |
Description: Numerator and denominator of the negative. (Contributed by Thierry Arnoux, 27-Oct-2017.) |
Ref | Expression |
---|---|
numdenneg | ⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | qnegcl 12983 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 ∈ ℚ) | |
2 | qnumcl 16715 | . . 3 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℤ) | |
3 | 2 | znegcld 12701 | . 2 ⊢ (𝑄 ∈ ℚ → -(numer‘𝑄) ∈ ℤ) |
4 | qdencl 16716 | . 2 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℕ) | |
5 | 4 | nnzd 12618 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℤ) |
6 | neggcd 16501 | . . . 4 ⊢ (((numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℤ) → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) | |
7 | 2, 5, 6 | syl2anc 582 | . . 3 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = ((numer‘𝑄) gcd (denom‘𝑄))) |
8 | qnumdencoprm 16720 | . . 3 ⊢ (𝑄 ∈ ℚ → ((numer‘𝑄) gcd (denom‘𝑄)) = 1) | |
9 | 7, 8 | eqtrd 2765 | . 2 ⊢ (𝑄 ∈ ℚ → (-(numer‘𝑄) gcd (denom‘𝑄)) = 1) |
10 | qeqnumdivden 16721 | . . . 4 ⊢ (𝑄 ∈ ℚ → 𝑄 = ((numer‘𝑄) / (denom‘𝑄))) | |
11 | 10 | negeqd 11486 | . . 3 ⊢ (𝑄 ∈ ℚ → -𝑄 = -((numer‘𝑄) / (denom‘𝑄))) |
12 | 2 | zcnd 12700 | . . . 4 ⊢ (𝑄 ∈ ℚ → (numer‘𝑄) ∈ ℂ) |
13 | 4 | nncnd 12261 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ∈ ℂ) |
14 | 4 | nnne0d 12295 | . . . 4 ⊢ (𝑄 ∈ ℚ → (denom‘𝑄) ≠ 0) |
15 | 12, 13, 14 | divnegd 12036 | . . 3 ⊢ (𝑄 ∈ ℚ → -((numer‘𝑄) / (denom‘𝑄)) = (-(numer‘𝑄) / (denom‘𝑄))) |
16 | 11, 15 | eqtrd 2765 | . 2 ⊢ (𝑄 ∈ ℚ → -𝑄 = (-(numer‘𝑄) / (denom‘𝑄))) |
17 | qnumdenbi 16719 | . . 3 ⊢ ((-𝑄 ∈ ℚ ∧ -(numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℕ) → (((-(numer‘𝑄) gcd (denom‘𝑄)) = 1 ∧ -𝑄 = (-(numer‘𝑄) / (denom‘𝑄))) ↔ ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄)))) | |
18 | 17 | biimpa 475 | . 2 ⊢ (((-𝑄 ∈ ℚ ∧ -(numer‘𝑄) ∈ ℤ ∧ (denom‘𝑄) ∈ ℕ) ∧ ((-(numer‘𝑄) gcd (denom‘𝑄)) = 1 ∧ -𝑄 = (-(numer‘𝑄) / (denom‘𝑄)))) → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
19 | 1, 3, 4, 9, 16, 18 | syl32anc 1375 | 1 ⊢ (𝑄 ∈ ℚ → ((numer‘-𝑄) = -(numer‘𝑄) ∧ (denom‘-𝑄) = (denom‘𝑄))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 1c1 11141 -cneg 11477 / cdiv 11903 ℕcn 12245 ℤcz 12591 ℚcq 12965 gcd cgcd 16472 numercnumer 16708 denomcdenom 16709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-n0 12506 df-z 12592 df-uz 12856 df-q 12966 df-rp 13010 df-fl 13793 df-mod 13871 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-dvds 16235 df-gcd 16473 df-numer 16710 df-denom 16711 |
This theorem is referenced by: divnumden2 32664 |
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