Theorem subsval 27988

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 7431	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦)))
2		fveq2 6900	. . 3 ⊢ (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵))
3	2	oveq2d 7440	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵)))
4		df-subs 27953	. 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
5		ovex 7457	. 2 ⊢ (𝐴 +s ( -us ‘𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7585	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098  ‘cfv 6551  (class class class)co 7424   No csur 27591   +_s cadds 27894   -u_s cnegs 27950   -_s csubs 27951

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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-subs 27953

This theorem is referenced by:  subsvald  27989  subscl  27990  negsval2  27992  subsid1  27994  subsid  27995  subadds  27996  sltsub1  28002