Theorem cutsval 28199

Description: The value of the surreal cut operation. (Contributed by Scott Fenton, 8-Dec-2021.)

Assertion

Ref	Expression
cutsval	⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦

Proof of Theorem cutsval

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sltsex1 28182	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ V)
2		sltsss1 28184	. . 3 ⊢ (𝐴 <<s 𝐵 → 𝐴 ⊆ No)
3	1, 2	elpwd 4566	. 2 ⊢ (𝐴 <<s 𝐵 → 𝐴 ∈ 𝒫 No)
4		df-br 5107	. . . 4 ⊢ (𝐴 <<s 𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ <<s )
5	4	biimpi 219	. . 3 ⊢ (𝐴 <<s 𝐵 → ⟨𝐴, 𝐵⟩ ∈ <<s )
6		sltsex2 28183	. . . 4 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ V)
7		elimasng 6100	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
8	1, 6, 7	syl2anc 596	. . 3 ⊢ (𝐴 <<s 𝐵 → (𝐵 ∈ ( <<s “ {𝐴}) ↔ ⟨𝐴, 𝐵⟩ ∈ <<s ))
9	5, 8	mpbird 260	. 2 ⊢ (𝐴 <<s 𝐵 → 𝐵 ∈ ( <<s “ {𝐴}))
10		riotaex 7407	. . 3 ⊢ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ V
11		breq1 5109	. . . . . . 7 ⊢ (𝑎 = 𝐴 → (𝑎 <<s {𝑦} ↔ 𝐴 <<s {𝑦}))
12		breq2 5110	. . . . . . 7 ⊢ (𝑏 = 𝐵 → ({𝑦} <<s 𝑏 ↔ {𝑦} <<s 𝐵))
13	11, 12	bi2anan9 650	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏) ↔ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)))
14	13	rabbidv 3423	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} = {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
15	14	imaeq2d 6065	. . . . . . 7 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
16	15	inteqd 4915	. . . . . 6 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ∩ (bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
17	16	eqeq2d 2774	. . . . 5 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → ((bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)}) ↔ (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
18	14, 17	riotaeqbidv 7406	. . . 4 ⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
19		sneq 4594	. . . . 5 ⊢ (𝑎 = 𝐴 → {𝑎} = {𝐴})
20	19	imaeq2d 6065	. . . 4 ⊢ (𝑎 = 𝐴 → ( <<s “ {𝑎}) = ( <<s “ {𝐴}))
21		df-cuts 28179	. . . 4 ⊢ |s = (𝑎 ∈ 𝒫 No, 𝑏 ∈ ( <<s “ {𝑎}) ↦ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝑎 <<s {𝑦} ∧ {𝑦} <<s 𝑏)})))
22	18, 20, 21	ovmpox 7600	. . 3 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝐵 ∈ ( <<s “ {𝐴}) ∧ (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ V) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
23	10, 22	mp3an3 1480	. 2 ⊢ ((𝐴 ∈ 𝒫 No ∧ 𝐵 ∈ ( <<s “ {𝐴})) → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))
24	3, 9, 23	syl2anc 596	1 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) = (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454  𝒫 cpw 4560  {csn 4588  ⟨cop 4838  ∩ cint 4910   class class class wbr 5106   “ cima 5659  ‘cfv 6561  ℩crio 7402  (class class class)co 7446  Nocsur 28030  bdaycbday 28032   <<s cslts 28176   |s ccuts 28178

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 5253  ax-nul 5264  ax-pr 5396

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-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-slts 28177  df-cuts 28179

This theorem is used by:  cutcuts  28200  cutbday  28203  eqcuts  28204  cutsun12  28209  cutsf  28211  cutbdaylt  28217