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Mirrors > Home > ILE Home > Th. List > apdivmuld | GIF version |
Description: Relationship between division and multiplication. (Contributed by Jim Kingdon, 26-Dec-2022.) |
Ref | Expression |
---|---|
divcld.1 | ⊢ (𝜑 → 𝐴 ∈ ℂ) |
divcld.2 | ⊢ (𝜑 → 𝐵 ∈ ℂ) |
divmuld.3 | ⊢ (𝜑 → 𝐶 ∈ ℂ) |
divmulapd.4 | ⊢ (𝜑 → 𝐵 # 0) |
Ref | Expression |
---|---|
apdivmuld | ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | divcld.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) | |
2 | divcld.2 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ ℂ) | |
3 | divmulapd.4 | . . . 4 ⊢ (𝜑 → 𝐵 # 0) | |
4 | 1, 2, 3 | divclapd 8354 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) |
5 | divmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | apmul1 8352 | . . 3 ⊢ (((𝐴 / 𝐵) ∈ ℂ ∧ 𝐶 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 # 0)) → ((𝐴 / 𝐵) # 𝐶 ↔ ((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵))) | |
7 | 4, 5, 2, 3, 6 | syl112anc 1185 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ ((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵))) |
8 | 1, 2, 3 | divcanap1d 8355 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
9 | 5, 2 | mulcomd 7606 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐵 · 𝐶)) |
10 | 8, 9 | breq12d 3880 | . 2 ⊢ (𝜑 → (((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵) ↔ 𝐴 # (𝐵 · 𝐶))) |
11 | 2, 5 | mulcld 7605 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ) |
12 | apsym 8180 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝐶) ∈ ℂ) → (𝐴 # (𝐵 · 𝐶) ↔ (𝐵 · 𝐶) # 𝐴)) | |
13 | 1, 11, 12 | syl2anc 404 | . 2 ⊢ (𝜑 → (𝐴 # (𝐵 · 𝐶) ↔ (𝐵 · 𝐶) # 𝐴)) |
14 | 7, 10, 13 | 3bitrd 213 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∈ wcel 1445 class class class wbr 3867 (class class class)co 5690 ℂcc 7445 0cc0 7447 · cmul 7452 # 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: tanaddaplem 11178 |
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