Theorem subsvald 28067

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 28066	. 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 6492  (class class class)co 7360   No csur 27617   +_s cadds 27965   -u_s cnegs 28025   -_s csubs 28026

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-iota 6448  df-fun 6494  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-subs 28028

This theorem is referenced by:  ltsubs2  28083  negsubsdi2d  28086  addsubsassd  28087  addsubsd  28088  ltsubsubsbd  28089  subsubs4d  28100  subsubs2d  28101  subscan1d  28109  subscan2d  28110  zsubscld  28402  elzn0s  28404  zcuts  28413  zseo  28428  recut  28500  renegscl  28504