Theorem divsval 28092

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 2741	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
3	2	riotabidv 7346	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦) = (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴))
4		oveq1 7394	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
5	4	eqeq1d 2731	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
6	5	riotabidv 7346	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
7		df-divs 28091	. . . 4 ⊢ /su = (𝑦 ∈ No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦))
8		riotaex 7348	. . . 4 ⊢ (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7549	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ( No ∖ { 0s })) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
10	1, 9	sylan2br 595	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐵 ∈ No ∧ 𝐵 ≠ 0s )) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
11	10	3impb 1114	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911  {csn 4589  ℩crio 7343  (class class class)co 7387   No csur 27551   0_s c0s 27734   ·_s cmuls 28009   /_su cdivs 28090

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-divs 28091

This theorem is referenced by:  divsmulw  28096  divsclw  28098