Theorem xdivval 33004

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 4726	. . 3 ⊢ (𝐵 ∈ (ℝ ∖ {0}) ↔ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0))
2		simpl 483	. . . . . 6 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → 𝑦 = 𝐴)
3	2	eqeq2d 2751	. . . . 5 ⊢ ((𝑦 = 𝐴 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝑦 ↔ (𝑧 ·e 𝑥) = 𝐴))
4	3	riotabidva 7339	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴))
5		simpl 483	. . . . . . 7 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → 𝑧 = 𝐵)
6	5	oveq1d 7378	. . . . . 6 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → (𝑧 ·e 𝑥) = (𝐵 ·e 𝑥))
7	6	eqeq1d 2742	. . . . 5 ⊢ ((𝑧 = 𝐵 ∧ 𝑥 ∈ ℝ*) → ((𝑧 ·e 𝑥) = 𝐴 ↔ (𝐵 ·e 𝑥) = 𝐴))
8	7	riotabidva 7339	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
9		df-xdiv 33003	. . . 4 ⊢ /𝑒 = (𝑦 ∈ ℝ*, 𝑧 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ* (𝑧 ·e 𝑥) = 𝑦))
10		riotaex 7324	. . . 4 ⊢ (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴) ∈ V
11	4, 8, 9, 10	ovmpo 7523	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
12	1, 11	sylan2br 601	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ (𝐵 ∈ ℝ ∧ 𝐵 ≠ 0)) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))
13	12	3impb 1120	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ ∧ 𝐵 ≠ 0) → (𝐴 /𝑒 𝐵) = (℩𝑥 ∈ ℝ* (𝐵 ·e 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ≠ wne 2935   ∖ cdif 3887  {csn 4562  ℩crio 7319  (class class class)co 7363  ℝcr 11035  0cc0 11036  ℝ^*cxr 11176   ·_e cxmu 13060   /_𝑒 cxdiv 33002

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-sbc 3731  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6448  df-fun 6494  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-xdiv 33003

This theorem is referenced by:  xdivcld  33008  xdivmul  33010  rexdiv  33011  xdivpnfrp  33018