Theorem redivvald 43317

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 4750	. 2 ⊢ (𝜑 → 𝐵 ∈ (ℝ ∖ {0}))
5		eqeq2 2772	. . . 4 ⊢ (𝑧 = 𝐴 → ((𝑦 · 𝑥) = 𝑧 ↔ (𝑦 · 𝑥) = 𝐴))
6	5	riotabidv 7372	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧) = (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴))
7		oveq1 7420	. . . . 5 ⊢ (𝑦 = 𝐵 → (𝑦 · 𝑥) = (𝐵 · 𝑥))
8	7	eqeq1d 2762	. . . 4 ⊢ (𝑦 = 𝐵 → ((𝑦 · 𝑥) = 𝐴 ↔ (𝐵 · 𝑥) = 𝐴))
9	8	riotabidv 7372	. . 3 ⊢ (𝑦 = 𝐵 → (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
10		df-rediv 43316	. . 3 ⊢ /ℝ = (𝑧 ∈ ℝ, 𝑦 ∈ (ℝ ∖ {0}) ↦ (℩𝑥 ∈ ℝ (𝑦 · 𝑥) = 𝑧))
11		riotaex 7374	. . 3 ⊢ (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴) ∈ V
12	6, 9, 10, 11	ovmpo 7573	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ (ℝ ∖ {0})) → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))
13	1, 4, 12	syl2anc 596	1 ⊢ (𝜑 → (𝐴 /ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 · 𝑥) = 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ≠ wne 2955   ∖ cdif 3896  {csn 4584  ℩crio 7369  (class class class)co 7413  ℝcr 11123  0cc0 11124   · cmul 11129   /_ℝ crediv 43315

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 2732  ax-sep 5251  ax-nul 5263  ax-pr 5398

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 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  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 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-iota 6489  df-fun 6535  df-fv 6541  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-rediv 43316

This theorem is used by:  sn-redivcld  43319  redivmuld  43320