Theorem divsval 28340

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 4749	. . 3 ⊢ (𝐵 ∈ ( No ∖ { 0s }) ↔ (𝐵 ∈ No ∧ 𝐵 ≠ 0s ))
2		eqeq2 2777	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
3	2	riotabidv 7359	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦) = (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴))
4		oveq1 7407	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
5	4	eqeq1d 2767	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
6	5	riotabidv 7359	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
7		df-divs 28339	. . . 4 ⊢ /su = (𝑦 ∈ No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦))
8		riotaex 7361	. . . 4 ⊢ (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7560	. . 3 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ ( No ∖ { 0s })) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
10	1, 9	sylan2br 606	. 2 ⊢ ((𝐴 ∈ No ∧ (𝐵 ∈ No ∧ 𝐵 ≠ 0s )) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
11	10	3impb 1130	1 ⊢ ((𝐴 ∈ No ∧ 𝐵 ∈ No ∧ 𝐵 ≠ 0s ) → (𝐴 /su 𝐵) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904  {csn 4585  ℩crio 7356  (class class class)co 7400   No csur 27762   0_s c0s 27956   ·_s cmuls 28257   /_su cdivs 28338

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-divs 28339

This theorem is referenced by:  divmulsw  28344  divsclw  28346