Theorem resubval 43109

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 2775	. . 3 ⊢ (𝑦 = 𝐴 → ((𝑧 + 𝑥) = 𝑦 ↔ (𝑧 + 𝑥) = 𝐴))
2	1	riotabidv 7371	. 2 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦) = (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴))
3		oveq1 7419	. . . 4 ⊢ (𝑧 = 𝐵 → (𝑧 + 𝑥) = (𝐵 + 𝑥))
4	3	eqeq1d 2765	. . 3 ⊢ (𝑧 = 𝐵 → ((𝑧 + 𝑥) = 𝐴 ↔ (𝐵 + 𝑥) = 𝐴))
5	4	riotabidv 7371	. 2 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝐴) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))
6		df-resub 43108	. 2 ⊢ −ℝ = (𝑦 ∈ ℝ, 𝑧 ∈ ℝ ↦ (℩𝑥 ∈ ℝ (𝑧 + 𝑥) = 𝑦))
7		riotaex 7373	. 2 ⊢ (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴) ∈ V
8	2, 5, 6, 7	ovmpo 7572	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 −ℝ 𝐵) = (℩𝑥 ∈ ℝ (𝐵 + 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ℩crio 7368  (class class class)co 7412  ℝcr 11100   + caddc 11104   −_ℝ cresub 43107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6494  df-fun 6540  df-fv 6546  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-resub 43108

This theorem is referenced by:  rernegcl  43113  renegadd  43114  rersubcl  43120  resubadd  43121  resubf  43123  resubeqsub  43172