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Theorem xdivval 32642
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 4791 . . 3 (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2 simpl 482 . . . . . 6 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
32eqeq2d 2739 . . . . 5 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
43riotabidva 7396 . . . 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 7435 . . . . . 6 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
76eqeq1d 2730 . . . . 5 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
87riotabidva 7396 . . . 4 (𝑧 = 𝐵 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9 df-xdiv 32641 . . . 4 /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10 riotaex 7380 . . . 4 (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
114, 8, 9, 10ovmpo 7581 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
121, 11sylan2br 594 . 2 ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13123impb 1113 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 1085   = wceq 1534  wcel 2099  wne 2937  cdif 3944  {csn 4629  crio 7375  (class class class)co 7420  cr 11137  0cc0 11138  *cxr 11277   ·e cxmu 13123   /𝑒 cxdiv 32640
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-xdiv 32641
This theorem is referenced by:  xdivcld  32646  xdivmul  32648  rexdiv  32649  xdivpnfrp  32656
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