Theorem divsval 27634

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 4790	. . 3 ⊢ (𝐵 ∈ ( No ∖ { 0s }) ↔ (𝐵 ∈ No ∧ 𝐵 ≠ 0s ))
2		eqeq2 2744	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
3	2	riotabidv 7366	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦) = (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴))
4		oveq1 7415	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
5	4	eqeq1d 2734	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
6	5	riotabidv 7366	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
7		df-divs 27633	. . . 4 ⊢ /su = (𝑦 ∈ No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦))
8		riotaex 7368	. . . 4 ⊢ (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7567	. . 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 1115	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940   ∖ cdif 3945  {csn 4628  ℩crio 7363  (class class class)co 7408   No csur 27140   0_s c0s 27320   ·_s cmuls 27559   /_su cdivs 27632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-divs 27633

This theorem is referenced by:  divsmulw  27637  divsclw  27639