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Theorem subsfo 28110
Description: Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.)
Assertion
Ref Expression
subsfo -s :( No × No )–onto No

Proof of Theorem subsfo
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 subsf 28109 . 2 -s :( No × No )⟶ No
2 0sno 27886 . . . . 5 0s No
3 opelxpi 5726 . . . . 5 ((𝑥 No ∧ 0s No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
42, 3mpan2 691 . . . 4 (𝑥 No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5 subsval 28105 . . . . . 6 ((𝑥 No ∧ 0s No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
62, 5mpan2 691 . . . . 5 (𝑥 No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
7 negs0s 28073 . . . . . . 7 ( -us ‘ 0s ) = 0s
87oveq2i 7442 . . . . . 6 (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9 addsrid 28012 . . . . . 6 (𝑥 No → (𝑥 +s 0s ) = 𝑥)
108, 9eqtrid 2787 . . . . 5 (𝑥 No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
116, 10eqtr2d 2776 . . . 4 (𝑥 No 𝑥 = (𝑥 -s 0s ))
12 fveq2 6907 . . . . . 6 (𝑦 = ⟨𝑥, 0s ⟩ → ( -s𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13 df-ov 7434 . . . . . 6 (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
1412, 13eqtr4di 2793 . . . . 5 (𝑦 = ⟨𝑥, 0s ⟩ → ( -s𝑦) = (𝑥 -s 0s ))
1514rspceeqv 3645 . . . 4 ((⟨𝑥, 0s ⟩ ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s𝑦))
164, 11, 15syl2anc 584 . . 3 (𝑥 No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s𝑦))
1716rgen 3061 . 2 𝑥 No 𝑦 ∈ ( No × No )𝑥 = ( -s𝑦)
18 dffo3 7122 . 2 ( -s :( No × No )–onto No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 No 𝑦 ∈ ( No × No )𝑥 = ( -s𝑦)))
191, 17, 18mpbir2an 711 1 -s :( No × No )–onto No
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  wrex 3068  cop 4637   × cxp 5687  wf 6559  ontowfo 6561  cfv 6563  (class class class)co 7431   No csur 27699   0s c0s 27882   +s cadds 28007   -us cnegs 28066   -s csubs 28067
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-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-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  df-subs 28069
This theorem is referenced by:  zssno  28382
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