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Mirrors > Home > ILE Home > Th. List > divvalap | GIF version |
Description: Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is apart from zero. (Contributed by Jim Kingdon, 21-Feb-2020.) |
Ref | Expression |
---|---|
divvalap | ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 939 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐴 ∈ ℂ) | |
2 | simp2 940 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ ℂ) | |
3 | 0cn 7162 | . . . . . 6 ⊢ 0 ∈ ℂ | |
4 | apne 7779 | . . . . . 6 ⊢ ((𝐵 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐵 # 0 → 𝐵 ≠ 0)) | |
5 | 3, 4 | mpan2 416 | . . . . 5 ⊢ (𝐵 ∈ ℂ → (𝐵 # 0 → 𝐵 ≠ 0)) |
6 | 5 | adantl 271 | . . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 # 0 → 𝐵 ≠ 0)) |
7 | 6 | 3impia 1136 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ≠ 0) |
8 | eldifsn 3519 | . . 3 ⊢ (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0)) | |
9 | 2, 7, 8 | sylanbrc 408 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ (ℂ ∖ {0})) |
10 | receuap 7815 | . . 3 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) | |
11 | riotacl 5507 | . . 3 ⊢ (∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴 → (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) | |
12 | 10, 11 | syl 14 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) |
13 | eqeq2 2091 | . . . 4 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴)) | |
14 | 13 | riotabidv 5495 | . . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴)) |
15 | oveq1 5544 | . . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥)) | |
16 | 15 | eqeq1d 2090 | . . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴)) |
17 | 16 | riotabidv 5495 | . . 3 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
18 | df-div 7817 | . . 3 ⊢ / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧)) | |
19 | 14, 17, 18 | ovmpt2g 5660 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0}) ∧ (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
20 | 1, 9, 12, 19 | syl3anc 1170 | 1 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (℩𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 920 = wceq 1285 ∈ wcel 1434 ≠ wne 2246 ∃!wreu 2351 ∖ cdif 2971 {csn 3400 class class class wbr 3787 ℩crio 5492 (class class class)co 5537 ℂcc 7030 0cc0 7032 · cmul 7037 # cap 7737 / cdiv 7816 |
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 2064 ax-sep 3898 ax-pow 3950 ax-pr 3966 ax-un 4190 ax-setind 4282 ax-cnex 7118 ax-resscn 7119 ax-1cn 7120 ax-1re 7121 ax-icn 7122 ax-addcl 7123 ax-addrcl 7124 ax-mulcl 7125 ax-mulrcl 7126 ax-addcom 7127 ax-mulcom 7128 ax-addass 7129 ax-mulass 7130 ax-distr 7131 ax-i2m1 7132 ax-0lt1 7133 ax-1rid 7134 ax-0id 7135 ax-rnegex 7136 ax-precex 7137 ax-cnre 7138 ax-pre-ltirr 7139 ax-pre-ltwlin 7140 ax-pre-lttrn 7141 ax-pre-apti 7142 ax-pre-ltadd 7143 ax-pre-mulgt0 7144 ax-pre-mulext 7145 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1687 df-eu 1945 df-mo 1946 df-clab 2069 df-cleq 2075 df-clel 2078 df-nfc 2209 df-ne 2247 df-nel 2341 df-ral 2354 df-rex 2355 df-reu 2356 df-rmo 2357 df-rab 2358 df-v 2604 df-sbc 2817 df-dif 2976 df-un 2978 df-in 2980 df-ss 2987 df-pw 3386 df-sn 3406 df-pr 3407 df-op 3409 df-uni 3604 df-br 3788 df-opab 3842 df-id 4050 df-po 4053 df-iso 4054 df-xp 4371 df-rel 4372 df-cnv 4373 df-co 4374 df-dm 4375 df-iota 4891 df-fun 4928 df-fv 4934 df-riota 5493 df-ov 5540 df-oprab 5541 df-mpt2 5542 df-pnf 7206 df-mnf 7207 df-xr 7208 df-ltxr 7209 df-le 7210 df-sub 7337 df-neg 7338 df-reap 7731 df-ap 7738 df-div 7817 |
This theorem is referenced by: divmulap 7819 divclap 7822 |
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