Theorem xdivval 33467

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 4748	. . 3 ⊢ (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2		simpl 488	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
3	2	eqeq2d 2772	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
4	3	riotabidva 7388	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5		simpl 488	. . . . . . 7 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
6	5	oveq1d 7427	. . . . . 6 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
7	6	eqeq1d 2763	. . . . 5 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
8	7	riotabidva 7388	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9		df-xdiv 33466	. . . 4 ⊢ /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10		riotaex 7373	. . . 4 ⊢ (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
11	4, 8, 9, 10	ovmpo 7572	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
12	1, 11	sylan2br 607	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13	12	3impb 1132	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∖ cdif 3896  {csn 4584  ℩crio 7368  (class class class)co 7412  ℝcr 11180  0cc0 11181  ℝ^*cxr 11323   ·_e cxmu 13221   /_𝑒 cxdiv 33465

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6487  df-fun 6533  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-xdiv 33466

This theorem is used by:  xdivcld  33471  xdivmul  33473  rexdiv  33474  xdivpnfrp  33481