Theorem subsval 28451

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 7427	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 +s ( -s‘𝑦)) = (𝐴 +s ( -s‘𝑦)))
2		fveq2 6885	. . 3 ⊢ (𝑦 = 𝐵 → ( -s‘𝑦) = ( -s‘𝐵))
3	2	oveq2d 7436	. 2 ⊢ (𝑦 = 𝐵 → (𝐴 +s ( -s‘𝑦)) = (𝐴 +s ( -s‘𝐵)))
4		df-subs 28413	. 2 ⊢ −s = (𝑥 ∈ No, 𝑦 ∈ No ↦ (𝑥 +s ( -s‘𝑦)))
5		ovex 7453	. 2 ⊢ (𝐴 +s ( -s‘𝐵)) ∈ V
6	1, 3, 4, 5	ovmpo 7580	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No) → (𝐴 −s 𝐵) = (𝐴 +s ( -s‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   +_s cadds 28350   -_scnegs 28410   −_s csubs 28411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-subs 28413

This theorem is used by:  subsvald  28452  subscl  28453  subsfo  28456  negsval2  28457  subsid1  28459  subsid  28460  subadds  28461  ltsubs1  28467  z12subscl  28870