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Theorem subsval 34336
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 ( -us ‘𝐵)))

Proof of Theorem subsval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7358 . 2 (𝑥 = 𝐴 → (𝑥 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝑦)))
2 fveq2 6839 . . 3 (𝑦 = 𝐵 → ( -us ‘𝑦) = ( -us ‘𝐵))
32oveq2d 7367 . 2 (𝑦 = 𝐵 → (𝐴 +s ( -us ‘𝑦)) = (𝐴 +s ( -us ‘𝐵)))
4 df-subs 34309 . 2 -s = (𝑥 No , 𝑦 No ↦ (𝑥 +s ( -us ‘𝑦)))
5 ovex 7384 . 2 (𝐴 +s ( -us ‘𝐵)) ∈ V
61, 3, 4, 5ovmpo 7509 1 ((𝐴 No 𝐵 No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us ‘𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  wcel 2106  cfv 6493  (class class class)co 7351   No csur 26939   +s cadds 34267   -us cnegs 34306   -s csubs 34307
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-subs 34309
This theorem is referenced by:  subscl  34337  subsid1  34338  subsid  34339  subadds  34340  sltsub1  34344  sltsub2  34345
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