Theorem divsval 28632

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.

Step	Hyp	Ref	Expression
1		eldifsn 4750	. . 3 ⊢ (𝐵 ∈ (No ∖ { 0s }) ↔ (𝐵 ∈ No ∧ 𝐵 ≠ 0s ))
2		eqeq2 2775	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
3	2	riotabidv 7410	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦) = (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴))
4		oveq1 7458	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
5	4	eqeq1d 2765	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
6	5	riotabidv 7410	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
7		df-divs 28631	. . . 4 ⊢ /su = (𝑦 ∈ No, 𝑧 ∈ (No ∖ { 0s }) ↦ (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦))
8		riotaex 7412	. . . 4 ⊢ (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7612	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ (No ∖ { 0s })) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
10	1, 9	sylan2br 607	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐵 ∈ No ∧ 𝐵 ≠ 0s )) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
11	10	3impb 1132	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ∖ cdif 4048  {csn 4588  ℩crio 7407  (class class class)co 7451  Nocsur 28054   0_s c0s 28248   ·_s cmuls 28549   /_su cdivs 28630

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-iota 6520  df-fun 6568  df-fv 6574  df-riota 7408  df-ov 7454  df-oprab 7455  df-mpo 7456  df-divs 28631

This theorem is used by:  divmulsw  28636  divsclw  28638