Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > divval | Structured version Visualization version GIF version |
Description: Value of division: if 𝐴 and 𝐵 are complex numbers with 𝐵 nonzero, then (𝐴 / 𝐵) is the (unique) complex number such that (𝐵 · 𝑥) = 𝐴. (Contributed by NM, 8-May-1999.) (Revised by Mario Carneiro, 17-Feb-2014.) |
Ref | Expression |
---|---|
divval | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldifsn 4317 | . . 3 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
2 | eqeq2 2633 | . . . . 5 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴)) | |
3 | 2 | riotabidv 6613 | . . . 4 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴)) |
4 | oveq1 6657 | . . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥)) | |
5 | 4 | eqeq1d 2624 | . . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴)) |
6 | 5 | riotabidv 6613 | . . . 4 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
7 | df-div 10685 | . . . 4 ⊢ / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧)) | |
8 | riotaex 6615 | . . . 4 ⊢ (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ V | |
9 | 3, 6, 7, 8 | ovmpt2 6796 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0})) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
10 | 1, 9 | sylan2br 493 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
11 | 10 | 3impb 1260 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 ≠ 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∖ cdif 3571 {csn 4177 ℩crio 6610 (class class class)co 6650 ℂcc 9934 0cc0 9936 · cmul 9941 / cdiv 10684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-iota 5851 df-fun 5890 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-div 10685 |
This theorem is referenced by: divmul 10688 divcl 10691 cnflddiv 19776 divcn 22671 rexdiv 29634 |
Copyright terms: Public domain | W3C validator |