Theorem resubval 42374

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 2747	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
2	1	riotabidv 7390	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3		oveq1 7438	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
4	3	eqeq1d 2737	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
5	4	riotabidv 7390	. 2 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6		df-resub 42373	. 2 ⊢ −ℝ = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7		riotaex 7392	. 2 ⊢ (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
8	2, 5, 6, 7	ovmpo 7593	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ℩crio 7387  (class class class)co 7431  ℝcr 11152   + caddc 11156   −_ℝ cresub 42372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-resub 42373

This theorem is referenced by:  rernegcl  42378  renegadd  42379  rersubcl  42385  resubadd  42386  resubf  42388  resubeqsub  42436