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Mirrors > Home > ILE Home > Th. List > divcanap5 | GIF version |
Description: Cancellation of common factor in a ratio. (Contributed by Jim Kingdon, 25-Feb-2020.) |
Ref | Expression |
---|---|
divcanap5 | ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dividap 7942 | . . . 4 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) → (𝐶 / 𝐶) = 1) | |
2 | 1 | oveq1d 5584 | . . 3 ⊢ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) → ((𝐶 / 𝐶) · (𝐴 / 𝐵)) = (1 · (𝐴 / 𝐵))) |
3 | 2 | 3ad2ant3 962 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 / 𝐶) · (𝐴 / 𝐵)) = (1 · (𝐴 / 𝐵))) |
4 | simp3l 967 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐶 ∈ ℂ) | |
5 | simp1 939 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → 𝐴 ∈ ℂ) | |
6 | simp3 941 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐶 ∈ ℂ ∧ 𝐶 # 0)) | |
7 | simp2 940 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 ∈ ℂ ∧ 𝐵 # 0)) | |
8 | divmuldivap 7953 | . . 3 ⊢ (((𝐶 ∈ ℂ ∧ 𝐴 ∈ ℂ) ∧ ((𝐶 ∈ ℂ ∧ 𝐶 # 0) ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0))) → ((𝐶 / 𝐶) · (𝐴 / 𝐵)) = ((𝐶 · 𝐴) / (𝐶 · 𝐵))) | |
9 | 4, 5, 6, 7, 8 | syl22anc 1171 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 / 𝐶) · (𝐴 / 𝐵)) = ((𝐶 · 𝐴) / (𝐶 · 𝐵))) |
10 | divclap 7919 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) ∈ ℂ) | |
11 | 10 | 3expb 1140 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (𝐴 / 𝐵) ∈ ℂ) |
12 | 11 | mulid2d 7285 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → (1 · (𝐴 / 𝐵)) = (𝐴 / 𝐵)) |
13 | 12 | 3adant3 959 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (1 · (𝐴 / 𝐵)) = (𝐴 / 𝐵)) |
14 | 3, 9, 13 | 3eqtr3d 2123 | 1 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0) ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐶 · 𝐴) / (𝐶 · 𝐵)) = (𝐴 / 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3806 (class class class)co 5569 ℂcc 7127 0cc0 7129 1c1 7130 · cmul 7134 # cap 7834 / cdiv 7913 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3994 ax-un 4218 ax-setind 4310 ax-cnex 7215 ax-resscn 7216 ax-1cn 7217 ax-1re 7218 ax-icn 7219 ax-addcl 7220 ax-addrcl 7221 ax-mulcl 7222 ax-mulrcl 7223 ax-addcom 7224 ax-mulcom 7225 ax-addass 7226 ax-mulass 7227 ax-distr 7228 ax-i2m1 7229 ax-0lt1 7230 ax-1rid 7231 ax-0id 7232 ax-rnegex 7233 ax-precex 7234 ax-cnre 7235 ax-pre-ltirr 7236 ax-pre-ltwlin 7237 ax-pre-lttrn 7238 ax-pre-apti 7239 ax-pre-ltadd 7240 ax-pre-mulgt0 7241 ax-pre-mulext 7242 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rmo 2361 df-rab 2362 df-v 2613 df-sbc 2826 df-dif 2985 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-id 4078 df-po 4081 df-iso 4082 df-xp 4400 df-rel 4401 df-cnv 4402 df-co 4403 df-dm 4404 df-iota 4920 df-fun 4957 df-fv 4963 df-riota 5525 df-ov 5572 df-oprab 5573 df-mpt2 5574 df-pnf 7303 df-mnf 7304 df-xr 7305 df-ltxr 7306 df-le 7307 df-sub 7434 df-neg 7435 df-reap 7828 df-ap 7835 df-div 7914 |
This theorem is referenced by: divcanap7 7962 divadddivap 7968 divcanap5d 8056 8th4div3 8403 flodddiv4 10575 |
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