Theorem subsvald 28053

Description: The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)

Hypotheses

Ref	Expression
subsvald.1	⊢ (𝜑 → 𝐴 ∈ No )
subsvald.2	⊢ (𝜑 → 𝐵 ∈ No )

Assertion

Ref	Expression
subsvald	⊢ (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Proof of Theorem subsvald

Step	Hyp	Ref	Expression
1		subsvald.1	. 2 ⊢ (𝜑 → 𝐴 ∈ No )
2		subsvald.2	. 2 ⊢ (𝜑 → 𝐵 ∈ No )
3		subsval 28052	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
4	1, 2, 3	syl2anc 585	1 ⊢ (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367   No csur 27603   +_s cadds 27951   -u_s cnegs 28011   -_s csubs 28012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-oprab 7371  df-mpo 7372  df-subs 28014

This theorem is referenced by:  ltsubs2  28069  negsubsdi2d  28072  addsubsassd  28073  addsubsd  28074  ltsubsubsbd  28075  subsubs4d  28086  subsubs2d  28087  subscan1d  28095  subscan2d  28096  zsubscld  28388  elzn0s  28390  zcuts  28399  zseo  28414  recut  28486  renegscl  28490