Theorem redivvald 42612

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 4740	. 2 ⊢ (𝜑 → 𝐵 ∈ (ℝ ∖ {0}))
5		eqeq2 2745	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
6	5	riotabidv 7314	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴))
7		oveq1 7362	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
8	7	eqeq1d 2735	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
9	8	riotabidv 7314	. . 3 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
10		df-rediv 42611	. . 3 ⊢ /ℝ = (𝑧 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧))
11		riotaex 7316	. . 3 ⊢ (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴) ∈ V
12	6, 9, 10, 11	ovmpo 7515	. 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 1541   ∈ wcel 2113   ≠ wne 2929   ∖ cdif 3895  {csn 4577  ℩crio 7311  (class class class)co 7355  ℝcr 11016  0cc0 11017   · cmul 11022   /_ℝ crediv 42610

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-opab 5158  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-iota 6445  df-fun 6491  df-fv 6497  df-riota 7312  df-ov 7358  df-oprab 7359  df-mpo 7360  df-rediv 42611

This theorem is referenced by:  sn-redivcld  42614  redivmuld  42615