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Theorem subsfo 28095
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 28094 . 2 -s :( No × No )⟶ No
2 0sno 27871 . . . . 5 0s No
3 opelxpi 5722 . . . . 5 ((𝑥 No ∧ 0s No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
42, 3mpan2 691 . . . 4 (𝑥 No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5 subsval 28090 . . . . . 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 28058 . . . . . . 7 ( -us ‘ 0s ) = 0s
87oveq2i 7442 . . . . . 6 (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9 addsrid 27997 . . . . . 6 (𝑥 No → (𝑥 +s 0s ) = 𝑥)
108, 9eqtrid 2789 . . . . 5 (𝑥 No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
116, 10eqtr2d 2778 . . . 4 (𝑥 No 𝑥 = (𝑥 -s 0s ))
12 fveq2 6906 . . . . . 6 (𝑦 = ⟨𝑥, 0s ⟩ → ( -s𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13 df-ov 7434 . . . . . 6 (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
1412, 13eqtr4di 2795 . . . . 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 3063 . 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 1540  wcel 2108  wral 3061  wrex 3070  cop 4632   × cxp 5683  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431   No csur 27684   0s c0s 27867   +s cadds 27992   -us cnegs 28051   -s csubs 28052
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-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-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  df-subs 28054
This theorem is referenced by:  zssno  28367
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