Theorem resubval 41815

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 2738	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
2	1	riotabidv 7363	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3		oveq1 7412	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
4	3	eqeq1d 2728	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
5	4	riotabidv 7363	. 2 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6		df-resub 41814	. 2 ⊢ −ℝ = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7		riotaex 7365	. 2 ⊢ (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
8	2, 5, 6, 7	ovmpo 7564	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ℩crio 7360  (class class class)co 7405  ℝcr 11111   + caddc 11115   −_ℝ cresub 41813

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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6489  df-fun 6539  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-resub 41814

This theorem is referenced by:  rernegcl  41819  renegadd  41820  rersubcl  41826  resubadd  41827  resubf  41829  resubeqsub  41877