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Theorem subsvald 27991
Description: The value of surreal subtraction. (Contributed by Scott Fenton, 5-Feb-2025.)
Hypotheses
Ref Expression
subsvald.1 (𝜑𝐴 No )
subsvald.2 (𝜑𝐵 No )
Assertion
Ref Expression
subsvald (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us𝐵)))

Proof of Theorem subsvald
StepHypRef Expression
1 subsvald.1 . 2 (𝜑𝐴 No )
2 subsvald.2 . 2 (𝜑𝐵 No )
3 subsval 27990 . 2 ((𝐴 No 𝐵 No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us𝐵)))
41, 2, 3syl2anc 582 1 (𝜑 → (𝐴 -s 𝐵) = (𝐴 +s ( -us𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cfv 6553  (class class class)co 7426   No csur 27593   +s cadds 27896   -us cnegs 27952   -s csubs 27953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-subs 27955
This theorem is referenced by:  sltsub2  28005  negsubsdi2d  28008  addsubsassd  28009  addsubsd  28010  sltsubsubbd  28011  subsubs4d  28021  subsubs2d  28022  recut  28244  renegscl  28246
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