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Theorem xdivval 28958
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 4260 . . 3 (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2 simpl 472 . . . . . 6 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
32eqeq2d 2620 . . . . 5 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
43riotabidva 6527 . . . 4 (𝑦 = 𝐴 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5 simpl 472 . . . . . . 7 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
65oveq1d 6564 . . . . . 6 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
76eqeq1d 2612 . . . . 5 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
87riotabidva 6527 . . . 4 (𝑧 = 𝐵 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9 df-xdiv 28957 . . . 4 /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10 riotaex 6515 . . . 4 (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
114, 8, 9, 10ovmpt2 6694 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
121, 11sylan2br 492 . 2 ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13123impb 1252 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 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  cdif 3537  {csn 4125  crio 6510  (class class class)co 6549  cr 9814  0cc0 9815  *cxr 9952   ·e cxmu 11821   /𝑒 cxdiv 28956
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-xdiv 28957
This theorem is referenced by:  xdivcld  28962  xdivmul  28964  rexdiv  28965  xdivpnfrp  28972
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