Theorem redivvald 42430

Description: Value of real division, which is the (unique) real 𝑥 such that (𝐵 · 𝑥) = 𝐴. (Contributed by SN, 25-Nov-2025.)

Hypotheses

Ref	Expression
redivvald.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
redivvald.b	⊢ (𝜑 → 𝐵 ∈ ℝ)
redivvald.z	⊢ (𝜑 → 𝐵 ≠ 0)

Assertion

Ref	Expression
redivvald	⊢ (𝜑 → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem redivvald

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		redivvald.a	. 2 ⊢ (𝜑 → 𝐴 ∈ ℝ)
2		redivvald.b	. . 3 ⊢ (𝜑 → 𝐵 ∈ ℝ)
3		redivvald.z	. . 3 ⊢ (𝜑 → 𝐵 ≠ 0)
4	2, 3	eldifsnd 4751	. 2 ⊢ (𝜑 → 𝐵 ∈ (ℝ ∖ {0}))
5		eqeq2 2741	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
6	5	riotabidv 7346	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴))
7		oveq1 7394	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
8	7	eqeq1d 2731	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
9	8	riotabidv 7346	. . 3 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
10		df-rediv 42429	. . 3 ⊢ /ℝ = (𝑧 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧))
11		riotaex 7348	. . 3 ⊢ (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴) ∈ V
12	6, 9, 10, 11	ovmpo 7549	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
13	1, 4, 12	syl2anc 584	1 ⊢ (𝜑 → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911  {csn 4589  ℩crio 7343  (class class class)co 7387  ℝcr 11067  0cc0 11068   · cmul 11073   /_ℝ crediv 42428

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-rediv 42429

This theorem is referenced by:  sn-redivcld  42432  redivmuld  42433