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Theorem divvalap 8634
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.)
Assertion
Ref Expression
divvalap ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem divvalap
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 997 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐴 ∈ ℂ)
2 simp2 998 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ ℂ)
3 0cn 7952 . . . . . 6 0 ∈ ℂ
4 apne 8583 . . . . . 6 ((𝐵 ∈ ℂ ∧ 0 ∈ ℂ) → (𝐵 # 0 → 𝐵 ≠ 0))
53, 4mpan2 425 . . . . 5 (𝐵 ∈ ℂ → (𝐵 # 0 → 𝐵 ≠ 0))
65adantl 277 . . . 4 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐵 # 0 → 𝐵 ≠ 0))
763impia 1200 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ≠ 0)
8 eldifsn 3721 . . 3 (𝐵 ∈ (ℂ ∖ {0}) ↔ (𝐵 ∈ ℂ ∧ 𝐵 ≠ 0))
92, 7, 8sylanbrc 417 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → 𝐵 ∈ (ℂ ∖ {0}))
10 receuap 8629 . . 3 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → ∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴)
11 riotacl 5848 . . 3 (∃!𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴 → (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ)
1210, 11syl 14 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ∧ 𝐵 # 0) → (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ)
13 eqeq2 2187 . . . 4 (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
1413riotabidv 5836 . . 3 (𝑧 = 𝐴 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧) = (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴))
15 oveq1 5885 . . . . 5 (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
1615eqeq1d 2186 . . . 4 (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
1716riotabidv 5836 . . 3 (𝑦 = 𝐵 → (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝐴) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
18 df-div 8633 . . 3 / = (𝑧 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑥 ∈ ℂ (𝑦 · 𝑥) = 𝑧))
1914, 17, 18ovmpog 6012 . 2 ((𝐴 ∈ ℂ ∧ 𝐵 ∈ (ℂ ∖ {0}) ∧ (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴) ∈ ℂ) → (𝐴 / 𝐵) = (𝑥 ∈ ℂ (𝐵 · 𝑥) = 𝐴))
201, 9, 12, 19syl3anc 1238 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 978   = wceq 1353  wcel 2148  wne 2347  ∃!wreu 2457  cdif 3128  {csn 3594   class class class wbr 4005  crio 5833  (class class class)co 5878  cc 7812  0cc0 7814   · cmul 7819   # cap 8541   / cdiv 8632
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 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932
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 2741  df-sbc 2965  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-po 4298  df-iso 4299  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-iota 5180  df-fun 5220  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633
This theorem is referenced by:  divmulap  8635  divclap  8638
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