Theorem subsval 28231

Description: The value of surreal subtraction. (Contributed by Scott Fenton, 3-Feb-2025.)

Assertion

Ref	Expression
subsval	⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Proof of Theorem subsval

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

Step	Hyp	Ref	Expression
1		oveq1 7419	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦)))
2		fveq2 6883	. . 3 ⊢ (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵))
3	2	oveq2d 7428	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵)))
4		df-subs 28193	. 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
5		ovex 7445	. 2 ⊢ (𝐴 +s ( -us ‘𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7572	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412   No csur 27782   +_s cadds 28130   -u_s cnegs 28190   -_s csubs 28191

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-ov 7415  df-oprab 7416  df-mpo 7417  df-subs 28193

This theorem is referenced by:  subsvald  28232  subscl  28233  subsfo  28236  negsval2  28237  subsid1  28239  subsid  28240  subadds  28241  ltsubs1  28247  z12subscl  28650