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Theorem xdivval 32901
Description: Value of division: the (unique) element 𝑥 such that (𝐵 · 𝑥) = 𝐴. This is meaningful only when 𝐵 is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
Assertion
Ref Expression
xdivval ((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem xdivval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4786 . . 3 (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2 simpl 482 . . . . . 6 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
32eqeq2d 2748 . . . . 5 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
43riotabidva 7407 . . . 4 (𝑦 = 𝐴 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5 simpl 482 . . . . . . 7 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
65oveq1d 7446 . . . . . 6 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
76eqeq1d 2739 . . . . 5 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
87riotabidva 7407 . . . 4 (𝑧 = 𝐵 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9 df-xdiv 32900 . . . 4 /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10 riotaex 7392 . . . 4 (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
114, 8, 9, 10ovmpo 7593 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
121, 11sylan2br 595 . 2 ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13123impb 1115 1 ((𝐴 ∈ ℝ*𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  cdif 3948  {csn 4626  crio 7387  (class class class)co 7431  cr 11154  0cc0 11155  *cxr 11294   ·e cxmu 13153   /𝑒 cxdiv 32899
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-xdiv 32900
This theorem is referenced by:  xdivcld  32905  xdivmul  32907  rexdiv  32908  xdivpnfrp  32915
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