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

Theorem divsval 28076
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 4786 . . 3 (𝐵 ∈ ( No ∖ { 0s }) ↔ (𝐵 No 𝐵 ≠ 0s ))
2 eqeq2 2739 . . . . 5 (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
32riotabidv 7372 . . . 4 (𝑦 = 𝐴 → (𝑥 No (𝑧 ·s 𝑥) = 𝑦) = (𝑥 No (𝑧 ·s 𝑥) = 𝐴))
4 oveq1 7421 . . . . . 6 (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
54eqeq1d 2729 . . . . 5 (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
65riotabidv 7372 . . . 4 (𝑧 = 𝐵 → (𝑥 No (𝑧 ·s 𝑥) = 𝐴) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
7 df-divs 28075 . . . 4 /su = (𝑦 No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (𝑥 No (𝑧 ·s 𝑥) = 𝑦))
8 riotaex 7374 . . . 4 (𝑥 No (𝐵 ·s 𝑥) = 𝐴) ∈ V
93, 6, 7, 8ovmpo 7575 . . 3 ((𝐴 No 𝐵 ∈ ( No ∖ { 0s })) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
101, 9sylan2br 594 . 2 ((𝐴 No ∧ (𝐵 No 𝐵 ≠ 0s )) → (𝐴 /su 𝐵) = (𝑥 No (𝐵 ·s 𝑥) = 𝐴))
11103impb 1113 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 1085   = wceq 1534  wcel 2099  wne 2935  cdif 3941  {csn 4624  crio 7369  (class class class)co 7414   No csur 27560   0s c0s 27742   ·s cmuls 27993   /su cdivs 28074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-sbc 3775  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-divs 28075
This theorem is referenced by:  divsmulw  28079  divsclw  28081
  Copyright terms: Public domain W3C validator