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Theorem xdivval 30142
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 4507 . . 3 (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2 simpl 475 . . . . . 6 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
32eqeq2d 2810 . . . . 5 ((𝑦 = 𝐴𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
43riotabidva 6856 . . . 4 (𝑦 = 𝐴 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5 simpl 475 . . . . . . 7 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
65oveq1d 6894 . . . . . 6 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
76eqeq1d 2802 . . . . 5 ((𝑧 = 𝐵𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
87riotabidva 6856 . . . 4 (𝑧 = 𝐵 → (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9 df-xdiv 30141 . . . 4 /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10 riotaex 6844 . . . 4 (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
114, 8, 9, 10ovmpt2 7031 . . 3 ((𝐴 ∈ ℝ*𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
121, 11sylan2br 589 . 2 ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13123impb 1144 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 385  w3a 1108   = wceq 1653  wcel 2157  wne 2972  cdif 3767  {csn 4369  crio 6839  (class class class)co 6879  cr 10224  0cc0 10225  *cxr 10363   ·e cxmu 12191   /𝑒 cxdiv 30140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2378  ax-ext 2778  ax-sep 4976  ax-nul 4984  ax-pr 5098
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2592  df-eu 2610  df-clab 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3388  df-sbc 3635  df-dif 3773  df-un 3775  df-in 3777  df-ss 3784  df-nul 4117  df-if 4279  df-sn 4370  df-pr 4372  df-op 4376  df-uni 4630  df-br 4845  df-opab 4907  df-id 5221  df-xp 5319  df-rel 5320  df-cnv 5321  df-co 5322  df-dm 5323  df-iota 6065  df-fun 6104  df-fv 6110  df-riota 6840  df-ov 6882  df-oprab 6883  df-mpt2 6884  df-xdiv 30141
This theorem is referenced by:  xdivcld  30146  xdivmul  30148  rexdiv  30149  xdivpnfrp  30156
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