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Mirrors > Home > ILE Home > Th. List > divmulap3 | GIF version |
Description: Relationship between division and multiplication. (Contributed by Jim Kingdon, 22-Feb-2020.) |
Ref | Expression |
---|---|
divmulap3 | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divmulap2 8040 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐶 · 𝐵))) | |
2 | mulcom 7373 | . . . . 5 ⊢ ((𝐵 ∈ ℂ ∧ 𝐶 ∈ ℂ) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) | |
3 | 2 | adantrr 463 | . . . 4 ⊢ ((𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
4 | 3 | 3adant1 957 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐵 · 𝐶) = (𝐶 · 𝐵)) |
5 | 4 | eqeq2d 2094 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → (𝐴 = (𝐵 · 𝐶) ↔ 𝐴 = (𝐶 · 𝐵))) |
6 | 1, 5 | bitr4d 189 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ (𝐶 ∈ ℂ ∧ 𝐶 # 0)) → ((𝐴 / 𝐶) = 𝐵 ↔ 𝐴 = (𝐵 · 𝐶))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 (class class class)co 5590 ℂcc 7250 0cc0 7252 · cmul 7257 # cap 7957 / cdiv 8036 |
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 3922 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-cnex 7338 ax-resscn 7339 ax-1cn 7340 ax-1re 7341 ax-icn 7342 ax-addcl 7343 ax-addrcl 7344 ax-mulcl 7345 ax-mulrcl 7346 ax-addcom 7347 ax-mulcom 7348 ax-addass 7349 ax-mulass 7350 ax-distr 7351 ax-i2m1 7352 ax-0lt1 7353 ax-1rid 7354 ax-0id 7355 ax-rnegex 7356 ax-precex 7357 ax-cnre 7358 ax-pre-ltirr 7359 ax-pre-ltwlin 7360 ax-pre-lttrn 7361 ax-pre-apti 7362 ax-pre-ltadd 7363 ax-pre-mulgt0 7364 ax-pre-mulext 7365 |
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 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-br 3812 df-opab 3866 df-id 4083 df-po 4086 df-iso 4087 df-xp 4406 df-rel 4407 df-cnv 4408 df-co 4409 df-dm 4410 df-iota 4933 df-fun 4970 df-fv 4976 df-riota 5546 df-ov 5593 df-oprab 5594 df-mpt2 5595 df-pnf 7426 df-mnf 7427 df-xr 7428 df-ltxr 7429 df-le 7430 df-sub 7557 df-neg 7558 df-reap 7951 df-ap 7958 df-div 8037 |
This theorem is referenced by: rec11rap 8075 divmulap3d 8187 3lcm2e6woprm 10847 6lcm4e12 10848 divgcdcoprm0 10862 |
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