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Mirrors > Home > ILE Home > Th. List > div2negap | GIF version |
Description: Quotient of two negatives. (Contributed by Jim Kingdon, 27-Feb-2020.) |
Ref | Expression |
---|---|
div2negap | ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-A / -B) = (A / B)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | negcl 7008 | . . . . 5 ⊢ (B ∈ ℂ → -B ∈ ℂ) | |
2 | 1 | 3ad2ant2 925 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -B ∈ ℂ) |
3 | simp1 903 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → A ∈ ℂ) | |
4 | simp2 904 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → B ∈ ℂ) | |
5 | simp3 905 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → B # 0) | |
6 | div12ap 7455 | . . . 4 ⊢ ((-B ∈ ℂ ∧ A ∈ ℂ ∧ (B ∈ ℂ ∧ B # 0)) → (-B · (A / B)) = (A · (-B / B))) | |
7 | 2, 3, 4, 5, 6 | syl112anc 1138 | . . 3 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-B · (A / B)) = (A · (-B / B))) |
8 | divnegap 7465 | . . . . . . 7 ⊢ ((B ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -(B / B) = (-B / B)) | |
9 | 4, 8 | syld3an1 1180 | . . . . . 6 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -(B / B) = (-B / B)) |
10 | dividap 7460 | . . . . . . . 8 ⊢ ((B ∈ ℂ ∧ B # 0) → (B / B) = 1) | |
11 | 10 | 3adant1 921 | . . . . . . 7 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (B / B) = 1) |
12 | 11 | negeqd 7003 | . . . . . 6 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -(B / B) = -1) |
13 | 9, 12 | eqtr3d 2071 | . . . . 5 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-B / B) = -1) |
14 | 13 | oveq2d 5471 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A · (-B / B)) = (A · -1)) |
15 | ax-1cn 6776 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
16 | 15 | negcli 7075 | . . . . . . 7 ⊢ -1 ∈ ℂ |
17 | mulcom 6808 | . . . . . . 7 ⊢ ((A ∈ ℂ ∧ -1 ∈ ℂ) → (A · -1) = (-1 · A)) | |
18 | 16, 17 | mpan2 401 | . . . . . 6 ⊢ (A ∈ ℂ → (A · -1) = (-1 · A)) |
19 | mulm1 7193 | . . . . . 6 ⊢ (A ∈ ℂ → (-1 · A) = -A) | |
20 | 18, 19 | eqtrd 2069 | . . . . 5 ⊢ (A ∈ ℂ → (A · -1) = -A) |
21 | 20 | 3ad2ant1 924 | . . . 4 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A · -1) = -A) |
22 | 14, 21 | eqtrd 2069 | . . 3 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A · (-B / B)) = -A) |
23 | 7, 22 | eqtrd 2069 | . 2 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-B · (A / B)) = -A) |
24 | negcl 7008 | . . . 4 ⊢ (A ∈ ℂ → -A ∈ ℂ) | |
25 | 24 | 3ad2ant1 924 | . . 3 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -A ∈ ℂ) |
26 | divclap 7439 | . . 3 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (A / B) ∈ ℂ) | |
27 | negap0 7412 | . . . . 5 ⊢ (B ∈ ℂ → (B # 0 ↔ -B # 0)) | |
28 | 27 | biimpa 280 | . . . 4 ⊢ ((B ∈ ℂ ∧ B # 0) → -B # 0) |
29 | 28 | 3adant1 921 | . . 3 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → -B # 0) |
30 | divmulap 7436 | . . 3 ⊢ ((-A ∈ ℂ ∧ (A / B) ∈ ℂ ∧ (-B ∈ ℂ ∧ -B # 0)) → ((-A / -B) = (A / B) ↔ (-B · (A / B)) = -A)) | |
31 | 25, 26, 2, 29, 30 | syl112anc 1138 | . 2 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → ((-A / -B) = (A / B) ↔ (-B · (A / B)) = -A)) |
32 | 23, 31 | mpbird 156 | 1 ⊢ ((A ∈ ℂ ∧ B ∈ ℂ ∧ B # 0) → (-A / -B) = (A / B)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∧ w3a 884 = wceq 1242 ∈ wcel 1390 class class class wbr 3755 (class class class)co 5455 ℂcc 6709 0cc0 6711 1c1 6712 · cmul 6716 -cneg 6980 # cap 7365 / cdiv 7433 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-nul 3874 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-setind 4220 ax-iinf 4254 ax-cnex 6774 ax-resscn 6775 ax-1cn 6776 ax-1re 6777 ax-icn 6778 ax-addcl 6779 ax-addrcl 6780 ax-mulcl 6781 ax-mulrcl 6782 ax-addcom 6783 ax-mulcom 6784 ax-addass 6785 ax-mulass 6786 ax-distr 6787 ax-i2m1 6788 ax-1rid 6790 ax-0id 6791 ax-rnegex 6792 ax-precex 6793 ax-cnre 6794 ax-pre-ltirr 6795 ax-pre-ltwlin 6796 ax-pre-lttrn 6797 ax-pre-apti 6798 ax-pre-ltadd 6799 ax-pre-mulgt0 6800 ax-pre-mulext 6801 |
This theorem depends on definitions: df-bi 110 df-dc 742 df-3or 885 df-3an 886 df-tru 1245 df-fal 1248 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ne 2203 df-nel 2204 df-ral 2305 df-rex 2306 df-reu 2307 df-rmo 2308 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-dif 2914 df-un 2916 df-in 2918 df-ss 2925 df-nul 3219 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-tr 3846 df-eprel 4017 df-id 4021 df-po 4024 df-iso 4025 df-iord 4069 df-on 4071 df-suc 4074 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 df-riota 5411 df-ov 5458 df-oprab 5459 df-mpt2 5460 df-1st 5709 df-2nd 5710 df-recs 5861 df-irdg 5897 df-1o 5940 df-2o 5941 df-oadd 5944 df-omul 5945 df-er 6042 df-ec 6044 df-qs 6048 df-ni 6288 df-pli 6289 df-mi 6290 df-lti 6291 df-plpq 6328 df-mpq 6329 df-enq 6331 df-nqqs 6332 df-plqqs 6333 df-mqqs 6334 df-1nqqs 6335 df-rq 6336 df-ltnqqs 6337 df-enq0 6407 df-nq0 6408 df-0nq0 6409 df-plq0 6410 df-mq0 6411 df-inp 6449 df-i1p 6450 df-iplp 6451 df-iltp 6453 df-enr 6654 df-nr 6655 df-ltr 6658 df-0r 6659 df-1r 6660 df-0 6718 df-1 6719 df-r 6721 df-lt 6724 df-pnf 6859 df-mnf 6860 df-xr 6861 df-ltxr 6862 df-le 6863 df-sub 6981 df-neg 6982 df-reap 7359 df-ap 7366 df-div 7434 |
This theorem is referenced by: divneg2ap 7494 div2negapd 7562 |
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