Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 8543 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) |
5 | divmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | apmul1 8541 | . . 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 1220 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ ((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵))) |
8 | 1, 2, 3 | divcanap1d 8544 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
9 | 5, 2 | mulcomd 7780 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐵 · 𝐶)) |
10 | 8, 9 | breq12d 3937 | . 2 ⊢ (𝜑 → (((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵) ↔ 𝐴 # (𝐵 · 𝐶))) |
11 | 2, 5 | mulcld 7779 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ) |
12 | apsym 8361 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝐶) ∈ ℂ) → (𝐴 # (𝐵 · 𝐶) ↔ (𝐵 · 𝐶) # 𝐴)) | |
13 | 1, 11, 12 | syl2anc 408 | . 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 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 0cc0 7613 · cmul 7618 # cap 8336 / cdiv 8425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 |
This theorem is referenced by: tanaddaplem 11434 |
Copyright terms: Public domain | W3C validator |