Theorem subsval 28333

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 7424	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦)))
2		fveq2 6882	. . 3 ⊢ (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵))
3	2	oveq2d 7433	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵)))
4		df-subs 28295	. 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
5		ovex 7450	. 2 ⊢ (𝐴 +s ( -us ‘𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7577	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ‘cfv 6537  (class class class)co 7417   No csur 27884   +_s cadds 28232   -u_s cnegs 28292   -_s csubs 28293

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-subs 28295

This theorem is used by:  subsvald  28334  subscl  28335  subsfo  28338  negsval2  28339  subsid1  28341  subsid  28342  subadds  28343  ltsubs1  28349  z12subscl  28752