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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 8736 | . . 3 ⊢ (𝜑 → (𝐴 / 𝐵) ∈ ℂ) |
5 | divmuld.3 | . . 3 ⊢ (𝜑 → 𝐶 ∈ ℂ) | |
6 | apmul1 8734 | . . 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 1242 | . 2 ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ ((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵))) |
8 | 1, 2, 3 | divcanap1d 8737 | . . 3 ⊢ (𝜑 → ((𝐴 / 𝐵) · 𝐵) = 𝐴) |
9 | 5, 2 | mulcomd 7969 | . . 3 ⊢ (𝜑 → (𝐶 · 𝐵) = (𝐵 · 𝐶)) |
10 | 8, 9 | breq12d 4013 | . 2 ⊢ (𝜑 → (((𝐴 / 𝐵) · 𝐵) # (𝐶 · 𝐵) ↔ 𝐴 # (𝐵 · 𝐶))) |
11 | 2, 5 | mulcld 7968 | . . 3 ⊢ (𝜑 → (𝐵 · 𝐶) ∈ ℂ) |
12 | apsym 8553 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 · 𝐶) ∈ ℂ) → (𝐴 # (𝐵 · 𝐶) ↔ (𝐵 · 𝐶) # 𝐴)) | |
13 | 1, 11, 12 | syl2anc 411 | . 2 ⊢ (𝜑 → (𝐴 # (𝐵 · 𝐶) ↔ (𝐵 · 𝐶) # 𝐴)) |
14 | 7, 10, 13 | 3bitrd 214 | 1 ⊢ (𝜑 → ((𝐴 / 𝐵) # 𝐶 ↔ (𝐵 · 𝐶) # 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∈ wcel 2148 class class class wbr 4000 (class class class)co 5869 ℂcc 7800 0cc0 7802 · cmul 7807 # cap 8528 / cdiv 8618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4206 ax-un 4430 ax-setind 4533 ax-cnex 7893 ax-resscn 7894 ax-1cn 7895 ax-1re 7896 ax-icn 7897 ax-addcl 7898 ax-addrcl 7899 ax-mulcl 7900 ax-mulrcl 7901 ax-addcom 7902 ax-mulcom 7903 ax-addass 7904 ax-mulass 7905 ax-distr 7906 ax-i2m1 7907 ax-0lt1 7908 ax-1rid 7909 ax-0id 7910 ax-rnegex 7911 ax-precex 7912 ax-cnre 7913 ax-pre-ltirr 7914 ax-pre-ltwlin 7915 ax-pre-lttrn 7916 ax-pre-apti 7917 ax-pre-ltadd 7918 ax-pre-mulgt0 7919 ax-pre-mulext 7920 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-br 4001 df-opab 4062 df-id 4290 df-po 4293 df-iso 4294 df-xp 4629 df-rel 4630 df-cnv 4631 df-co 4632 df-dm 4633 df-iota 5174 df-fun 5214 df-fv 5220 df-riota 5825 df-ov 5872 df-oprab 5873 df-mpo 5874 df-pnf 7984 df-mnf 7985 df-xr 7986 df-ltxr 7987 df-le 7988 df-sub 8120 df-neg 8121 df-reap 8522 df-ap 8529 df-div 8619 |
This theorem is referenced by: tanaddaplem 11730 |
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