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.

Step	Hyp	Ref	Expression
1		eldifsn 4786	. . 3 ⊢ (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2		simpl 482	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
3	2	eqeq2d 2748	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
4	3	riotabidva 7407	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5		simpl 482	. . . . . . 7 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
6	5	oveq1d 7446	. . . . . 6 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
7	6	eqeq1d 2739	. . . . 5 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
8	7	riotabidva 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
11	4, 8, 9, 10	ovmpo 7593	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
12	1, 11	sylan2br 595	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13	12	3impb 1115	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))

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  ℝcr 11154  0cc0 11155  ℝ^*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