Theorem subsval 28479

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 ( -s‘𝐵)))

Proof of Theorem subsval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq1 7453	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -s‘𝑦)) = (𝐴 +s ( -s‘𝑦)))
2		fveq2 6911	. . 3 ⊢ (𝑦 = 𝐵 → ( -s‘𝑦) = ( -s‘𝐵))
3	2	oveq2d 7462	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -s‘𝑦)) = (𝐴 +s ( -s‘𝐵)))
4		df-subs 28441	. 2 ⊢ −s = (𝑥 ∈ No, 𝑦 ∈ No ↦ (𝑥 +s ( -s‘𝑦)))
5		ovex 7479	. 2 ⊢ (𝐴 +s ( -s‘𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7607	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 −s 𝐵) = (𝐴 +s ( -s‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ‘cfv 6561  (class class class)co 7446  Nocsur 28030   +_s cadds 28378   -_scnegs 28438   −_s csubs 28439

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-iota 6515  df-fun 6563  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-subs 28441

This theorem is used by:  subsvald  28480  subscl  28481  subsfo  28484  negsval2  28485  subsid1  28487  subsid  28488  subadds  28489  ltsubs1  28495  z12subscl  28898