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Theorem negsdi 28082
Description: Distribution of surreal negative over addition. (Contributed by Scott Fenton, 5-Feb-2025.)
Assertion
Ref Expression
negsdi ((𝐴 No 𝐵 No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us𝐴) +s ( -us𝐵)))

Proof of Theorem negsdi
StepHypRef Expression
1 addscl 28014 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 +s 𝐵) ∈ No )
21negsidd 28074 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3 negsid 28073 . . . . 5 (𝐴 No → (𝐴 +s ( -us𝐴)) = 0s )
4 negsid 28073 . . . . 5 (𝐵 No → (𝐵 +s ( -us𝐵)) = 0s )
53, 4oveqan12d 7450 . . . 4 ((𝐴 No 𝐵 No ) → ((𝐴 +s ( -us𝐴)) +s (𝐵 +s ( -us𝐵))) = ( 0s +s 0s ))
6 0sno 27871 . . . . 5 0s No
7 addslid 28001 . . . . 5 ( 0s No → ( 0s +s 0s ) = 0s )
86, 7ax-mp 5 . . . 4 ( 0s +s 0s ) = 0s
95, 8eqtr2di 2794 . . 3 ((𝐴 No 𝐵 No ) → 0s = ((𝐴 +s ( -us𝐴)) +s (𝐵 +s ( -us𝐵))))
10 simpl 482 . . . 4 ((𝐴 No 𝐵 No ) → 𝐴 No )
1110negscld 28069 . . . 4 ((𝐴 No 𝐵 No ) → ( -us𝐴) ∈ No )
12 simpr 484 . . . 4 ((𝐴 No 𝐵 No ) → 𝐵 No )
1312negscld 28069 . . . 4 ((𝐴 No 𝐵 No ) → ( -us𝐵) ∈ No )
1410, 11, 12, 13adds4d 28042 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 +s ( -us𝐴)) +s (𝐵 +s ( -us𝐵))) = ((𝐴 +s 𝐵) +s (( -us𝐴) +s ( -us𝐵))))
152, 9, 143eqtrd 2781 . 2 ((𝐴 No 𝐵 No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us𝐴) +s ( -us𝐵))))
161negscld 28069 . . 3 ((𝐴 No 𝐵 No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17 negscl 28068 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
18 negscl 28068 . . . 4 (𝐵 No → ( -us𝐵) ∈ No )
19 addscl 28014 . . . 4 ((( -us𝐴) ∈ No ∧ ( -us𝐵) ∈ No ) → (( -us𝐴) +s ( -us𝐵)) ∈ No )
2017, 18, 19syl2an 596 . . 3 ((𝐴 No 𝐵 No ) → (( -us𝐴) +s ( -us𝐵)) ∈ No )
2116, 20, 1addscan1d 28033 . 2 ((𝐴 No 𝐵 No ) → (((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us𝐴) +s ( -us𝐵))) ↔ ( -us ‘(𝐴 +s 𝐵)) = (( -us𝐴) +s ( -us𝐵))))
2215, 21mpbid 232 1 ((𝐴 No 𝐵 No ) → ( -us ‘(𝐴 +s 𝐵)) = (( -us𝐴) +s ( -us𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  cfv 6561  (class class class)co 7431   No csur 27684   0s c0s 27867   +s cadds 27992   -us cnegs 28051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-ot 4635  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sle 27790  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec 27971  df-norec2 27982  df-adds 27993  df-negs 28053
This theorem is referenced by:  negsubsdi2d  28110  subsubs4d  28124  zscut  28393  renegscl  28430  readdscl  28431
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