Theorem xdivval 30597

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 4721	. . 3 ⊢ (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2		simpl 485	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
3	2	eqeq2d 2834	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
4	3	riotabidva 7135	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5		simpl 485	. . . . . . 7 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
6	5	oveq1d 7173	. . . . . 6 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
7	6	eqeq1d 2825	. . . . 5 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
8	7	riotabidva 7135	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9		df-xdiv 30596	. . . 4 ⊢ /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10		riotaex 7120	. . . 4 ⊢ (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
11	4, 8, 9, 10	ovmpo 7312	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
12	1, 11	sylan2br 596	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13	12	3impb 1111	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ≠ wne 3018   ∖ cdif 3935  {csn 4569  ℩crio 7115  (class class class)co 7158  ℝcr 10538  0cc0 10539  ℝ^*cxr 10676   ·_e cxmu 12509   /_𝑒 cxdiv 30595

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-iota 6316  df-fun 6359  df-fv 6365  df-riota 7116  df-ov 7161  df-oprab 7162  df-mpo 7163  df-xdiv 30596

This theorem is referenced by:  xdivcld  30601  xdivmul  30603  rexdiv  30604  xdivpnfrp  30611