Theorem subsvald 28141

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 28140	. 2 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
4	1, 2, 3	syl2anc 593	1 ⊢ (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))

Syntax hints:   → wi 4   = wceq 1559   ∈ wcel 2141  ‘cfv 6515  (class class class)co 7390   No csur 27691   +_s cadds 28039   -u_s cnegs 28099   -_s csubs 28100

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-iota 6471  df-fun 6517  df-fv 6523  df-ov 7393  df-oprab 7394  df-mpo 7395  df-subs 28102

This theorem is referenced by:  ltsubs2  28157  negsubsdi2d  28160  addsubsassd  28161  addsubsd  28162  ltsubsubsbd  28163  subsubs4d  28174  subsubs2d  28175  subscan1d  28183  subscan2d  28184  zsubscld  28476  elzn0s  28478  zcuts  28487  zseo  28502  recut  28574  renegscl  28578