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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 7779 | . . . . 5 ⊢ (𝐵 ∈ ℂ → -𝐵 ∈ ℂ) | |
2 | 1 | 3ad2ant2 968 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -𝐵 ∈ ℂ) |
3 | simp1 946 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐴 ∈ ℂ) | |
4 | simp2 947 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ ℂ) | |
5 | simp3 948 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 # 0) | |
6 | div12ap 8258 | . . . 4 ⊢ ((-𝐵 ∈ ℂ ∧ 𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (-𝐵 · (𝐴 / 𝐵)) = (𝐴 · (-𝐵 / 𝐵))) | |
7 | 2, 3, 4, 5, 6 | syl112anc 1185 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐵 · (𝐴 / 𝐵)) = (𝐴 · (-𝐵 / 𝐵))) |
8 | divnegap 8270 | . . . . . . 7 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐵 / 𝐵) = (-𝐵 / 𝐵)) | |
9 | 4, 8 | syld3an1 1227 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐵 / 𝐵) = (-𝐵 / 𝐵)) |
10 | dividap 8265 | . . . . . . . 8 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 / 𝐵) = 1) | |
11 | 10 | 3adant1 964 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐵 / 𝐵) = 1) |
12 | 11 | negeqd 7774 | . . . . . 6 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -(𝐵 / 𝐵) = -1) |
13 | 9, 12 | eqtr3d 2129 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐵 / 𝐵) = -1) |
14 | 13 | oveq2d 5706 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 · (-𝐵 / 𝐵)) = (𝐴 · -1)) |
15 | ax-1cn 7535 | . . . . . . . 8 ⊢ 1 ∈ ℂ | |
16 | 15 | negcli 7847 | . . . . . . 7 ⊢ -1 ∈ ℂ |
17 | mulcom 7568 | . . . . . . 7 ⊢ ((𝐴 ∈ ℂ ∧ -1 ∈ ℂ) → (𝐴 · -1) = (-1 · 𝐴)) | |
18 | 16, 17 | mpan2 417 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (𝐴 · -1) = (-1 · 𝐴)) |
19 | mulm1 7975 | . . . . . 6 ⊢ (𝐴 ∈ ℂ → (-1 · 𝐴) = -𝐴) | |
20 | 18, 19 | eqtrd 2127 | . . . . 5 ⊢ (𝐴 ∈ ℂ → (𝐴 · -1) = -𝐴) |
21 | 20 | 3ad2ant1 967 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 · -1) = -𝐴) |
22 | 14, 21 | eqtrd 2127 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 · (-𝐵 / 𝐵)) = -𝐴) |
23 | 7, 22 | eqtrd 2127 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐵 · (𝐴 / 𝐵)) = -𝐴) |
24 | negcl 7779 | . . . 4 ⊢ (𝐴 ∈ ℂ → -𝐴 ∈ ℂ) | |
25 | 24 | 3ad2ant1 967 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -𝐴 ∈ ℂ) |
26 | divclap 8242 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) | |
27 | negap0 8203 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 # 0 ↔ -𝐵 # 0)) | |
28 | 27 | biimpa 291 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ 𝐵 # 0) → -𝐵 # 0) |
29 | 28 | 3adant1 964 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → -𝐵 # 0) |
30 | divmulap 8239 | . . 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 1185 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ((-𝐴 / -𝐵) = (𝐴 / 𝐵) ↔ (-𝐵 · (𝐴 / 𝐵)) = -𝐴)) |
32 | 23, 31 | mpbird 166 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (-𝐴 / -𝐵) = (𝐴 / 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∧ w3a 927 = wceq 1296 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 ℂcc 7445 0cc0 7447 1c1 7448 · cmul 7452 -cneg 7751 # cap 8155 / cdiv 8236 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-cnex 7533 ax-resscn 7534 ax-1cn 7535 ax-1re 7536 ax-icn 7537 ax-addcl 7538 ax-addrcl 7539 ax-mulcl 7540 ax-mulrcl 7541 ax-addcom 7542 ax-mulcom 7543 ax-addass 7544 ax-mulass 7545 ax-distr 7546 ax-i2m1 7547 ax-0lt1 7548 ax-1rid 7549 ax-0id 7550 ax-rnegex 7551 ax-precex 7552 ax-cnre 7553 ax-pre-ltirr 7554 ax-pre-ltwlin 7555 ax-pre-lttrn 7556 ax-pre-apti 7557 ax-pre-ltadd 7558 ax-pre-mulgt0 7559 ax-pre-mulext 7560 |
This theorem depends on definitions: df-bi 116 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-nel 2358 df-ral 2375 df-rex 2376 df-reu 2377 df-rmo 2378 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-br 3868 df-opab 3922 df-id 4144 df-po 4147 df-iso 4148 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-iota 5014 df-fun 5051 df-fv 5057 df-riota 5646 df-ov 5693 df-oprab 5694 df-mpt2 5695 df-pnf 7621 df-mnf 7622 df-xr 7623 df-ltxr 7624 df-le 7625 df-sub 7752 df-neg 7753 df-reap 8149 df-ap 8156 df-div 8237 |
This theorem is referenced by: divneg2ap 8300 div2negapd 8369 div2subap 8399 |
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