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Theorem xdivval 29755
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 4350 . . 3 (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2 simpl 472 . . . . . 6 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
32eqeq2d 2661 . . . . 5 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
43riotabidva 6667 . . . 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 6705 . . . . . 6 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
76eqeq1d 2653 . . . . 5 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
87riotabidva 6667 . . . 4 (𝑧 = 𝐵 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9 df-xdiv 29754 . . . 4 /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10 riotaex 6655 . . . 4 (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
114, 8, 9, 10ovmpt2 6838 . . 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 1279 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 1054   = wceq 1523  wcel 2030  wne 2823  cdif 3604  {csn 4210  crio 6650  (class class class)co 6690  cr 9973  0cc0 9974  *cxr 10111   ·e cxmu 11983   /𝑒 cxdiv 29753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-xdiv 29754
This theorem is referenced by:  xdivcld  29759  xdivmul  29761  rexdiv  29762  xdivpnfrp  29769
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