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Theorem subsval 27921
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 ( -us𝐵)))

Proof of Theorem subsval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7411 . 2 (𝑥 = 𝐴 → (𝑥 +s ( -us𝑦)) = (𝐴 +s ( -us𝑦)))
2 fveq2 6884 . . 3 (𝑦 = 𝐵 → ( -us𝑦) = ( -us𝐵))
32oveq2d 7420 . 2 (𝑦 = 𝐵 → (𝐴 +s ( -us𝑦)) = (𝐴 +s ( -us𝐵)))
4 df-subs 27886 . 2 -s = (𝑥 No , 𝑦 No ↦ (𝑥 +s ( -us𝑦)))
5 ovex 7437 . 2 (𝐴 +s ( -us𝐵)) ∈ V
61, 3, 4, 5ovmpo 7563 1 ((𝐴 No 𝐵 No ) → (𝐴 -s 𝐵) = (𝐴 +s ( -us𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  cfv 6536  (class class class)co 7404   No csur 27524   +s cadds 27827   -us cnegs 27883   -s csubs 27884
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-subs 27886
This theorem is referenced by:  subsvald  27922  subscl  27923  negsval2  27925  subsid1  27927  subsid  27928  subadds  27929  sltsub1  27935
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