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Theorem divsval 28230
Description: The value of surreal division. (Contributed by Scott Fenton, 12-Mar-2025.)
Assertion
Ref Expression
divsval ((𝐴 No 𝐵 No 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem divsval
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eldifsn 4791 . . 3 (𝐵 ∈ ( No ∖ { 0s }) ↔ (𝐵 No 𝐵 ≠ 0s ))
2 eqeq2 2747 . . . . 5 (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
32riotabidv 7390 . . . 4 (𝑦 = 𝐴 → (𝑥 No (𝑧 ·s 𝑥) = 𝑦) = (𝑥 No (𝑧 ·s 𝑥) = 𝐴))
4 oveq1 7438 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
54eqeq1d 2737 . . . . 5 (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
65riotabidv 7390 . . . 4 (𝑧 = 𝐵 → (𝑥 No (𝑧 ·s 𝑥) = 𝐴) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
7 df-divs 28229 . . . 4 /su = (𝑦 No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (𝑥 No (𝑧 ·s 𝑥) = 𝑦))
8 riotaex 7392 . . . 4 (𝑥 No (𝐵 ·s 𝑥) = 𝐴) ∈ V
93, 6, 7, 8ovmpo 7593 . . 3 ((𝐴 No 𝐵 ∈ ( No ∖ { 0s })) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
101, 9sylan2br 595 . 2 ((𝐴 No ∧ (𝐵 No 𝐵 ≠ 0s )) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
11103impb 1114 1 ((𝐴 No 𝐵 No 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  cdif 3960  {csn 4631  crio 7387  (class class class)co 7431   No csur 27699   0s c0s 27882   ·s cmuls 28147   /su cdivs 28228
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-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  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-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-divs 28229
This theorem is referenced by:  divsmulw  28233  divsclw  28235
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