Theorem divsval 28129

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 4737	. . 3 ⊢ (𝐵 ∈ ( No ∖ { 0s }) ↔ (𝐵 ∈ No ∧ 𝐵 ≠ 0s ))
2		eqeq2 2745	. . . . 5 ⊢ (𝑦 = 𝐴 → ((𝑧 ·s 𝑥) = 𝑦 ↔ (𝑧 ·s 𝑥) = 𝐴))
3	2	riotabidv 7311	. . . 4 ⊢ (𝑦 = 𝐴 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦) = (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴))
4		oveq1 7359	. . . . . 6 ⊢ (𝑧 = 𝐵 → (𝑧 ·s 𝑥) = (𝐵 ·s 𝑥))
5	4	eqeq1d 2735	. . . . 5 ⊢ (𝑧 = 𝐵 → ((𝑧 ·s 𝑥) = 𝐴 ↔ (𝐵 ·s 𝑥) = 𝐴))
6	5	riotabidv 7311	. . . 4 ⊢ (𝑧 = 𝐵 → (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝐴) = (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴))
7		df-divs 28128	. . . 4 ⊢ /su = (𝑦 ∈ No , 𝑧 ∈ ( No ∖ { 0s }) ↦ (℩𝑥 ∈ No (𝑧 ·s 𝑥) = 𝑦))
8		riotaex 7313	. . . 4 ⊢ (℩𝑥 ∈ No (𝐵 ·s 𝑥) = 𝐴) ∈ V
9	3, 6, 7, 8	ovmpo 7512	. . 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 1541   ∈ wcel 2113   ≠ wne 2929   ∖ cdif 3895  {csn 4575  ℩crio 7308  (class class class)co 7352   No csur 27579   0_s c0s 27767   ·_s cmuls 28046   /_su cdivs 28127

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6442  df-fun 6488  df-fv 6494  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-divs 28128

This theorem is referenced by:  divsmulw  28133  divsclw  28135