MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  negsdi Structured version   Visualization version   GIF version

Theorem negsdi 28097
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 28029 . . . 4 ((𝐴 No 𝐵 No ) → (𝐴 +s 𝐵) ∈ No )
21negsidd 28089 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = 0s )
3 negsid 28088 . . . . 5 (𝐴 No → (𝐴 +s ( -us𝐴)) = 0s )
4 negsid 28088 . . . . 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 27886 . . . . 5 0s No
7 addslid 28016 . . . . 5 ( 0s No → ( 0s +s 0s ) = 0s )
86, 7ax-mp 5 . . . 4 ( 0s +s 0s ) = 0s
95, 8eqtr2di 2792 . . 3 ((𝐴 No 𝐵 No ) → 0s = ((𝐴 +s ( -us𝐴)) +s (𝐵 +s ( -us𝐵))))
10 simpl 482 . . . 4 ((𝐴 No 𝐵 No ) → 𝐴 No )
1110negscld 28084 . . . 4 ((𝐴 No 𝐵 No ) → ( -us𝐴) ∈ No )
12 simpr 484 . . . 4 ((𝐴 No 𝐵 No ) → 𝐵 No )
1312negscld 28084 . . . 4 ((𝐴 No 𝐵 No ) → ( -us𝐵) ∈ No )
1410, 11, 12, 13adds4d 28057 . . 3 ((𝐴 No 𝐵 No ) → ((𝐴 +s ( -us𝐴)) +s (𝐵 +s ( -us𝐵))) = ((𝐴 +s 𝐵) +s (( -us𝐴) +s ( -us𝐵))))
152, 9, 143eqtrd 2779 . 2 ((𝐴 No 𝐵 No ) → ((𝐴 +s 𝐵) +s ( -us ‘(𝐴 +s 𝐵))) = ((𝐴 +s 𝐵) +s (( -us𝐴) +s ( -us𝐵))))
161negscld 28084 . . 3 ((𝐴 No 𝐵 No ) → ( -us ‘(𝐴 +s 𝐵)) ∈ No )
17 negscl 28083 . . . 4 (𝐴 No → ( -us𝐴) ∈ No )
18 negscl 28083 . . . 4 (𝐵 No → ( -us𝐵) ∈ No )
19 addscl 28029 . . . 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 28048 . 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 1537  wcel 2106  cfv 6563  (class class class)co 7431   No csur 27699   0s c0s 27882   +s cadds 28007   -us cnegs 28066
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-ot 4640  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sle 27805  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec 27986  df-norec2 27997  df-adds 28008  df-negs 28068
This theorem is referenced by:  negsubsdi2d  28125  subsubs4d  28139  zscut  28408  renegscl  28445  readdscl  28446
  Copyright terms: Public domain W3C validator