Theorem resubval 41971

Description: Value of real subtraction, which is the (unique) real 𝑥 such that 𝐵 + 𝑥 = 𝐴. (Contributed by Steven Nguyen, 7-Jan-2023.)

Assertion

Ref	Expression
resubval	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem resubval

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

Step	Hyp	Ref	Expression
1		eqeq2 2740	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
2	1	riotabidv 7384	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3		oveq1 7433	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
4	3	eqeq1d 2730	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
5	4	riotabidv 7384	. 2 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6		df-resub 41970	. 2 ⊢ −ℝ = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7		riotaex 7386	. 2 ⊢ (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
8	2, 5, 6, 7	ovmpo 7588	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ℩crio 7381  (class class class)co 7426  ℝcr 11147   + caddc 11151   −_ℝ cresub 41969

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-resub 41970

This theorem is referenced by:  rernegcl  41975  renegadd  41976  rersubcl  41982  resubadd  41983  resubf  41985  resubeqsub  42033