Theorem cutcuts 28200

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

Assertion

Ref	Expression
cutcuts	⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))

Proof of Theorem cutcuts

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cutsval 28199	. . 3 ⊢ (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)})))
2		conway 28198	. . . 4 ⊢ (𝐴 <<s 𝐵 → ∃!𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)}))
3		riotacl 7420	. . . 4 ⊢ (∃!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)})) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
4	2, 3	syl 18	. . 3 ⊢ (𝐴 <<s 𝐵 → (℩𝑥 ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} (bday‘𝑥) = ∩ (bday “ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
5	1, 4	eqeltrd 2864	. 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
6		sneq 4594	. . . . . 6 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
7	6	breq2d 5118	. . . . 5 ⊢ (𝑦 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
8	6	breq1d 5116	. . . . 5 ⊢ (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
9	7, 8	anbi12d 644	. . . 4 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
10	9	elrab 3648	. . 3 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
11		3anass 1111	. . 3 ⊢ (((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵) ↔ ((𝐴 |s 𝐵) ∈ No ∧ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
12	10, 11	bitr4i 281	. 2 ⊢ ((𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)} ↔ ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))
13	5, 12	sylib 221	1 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∃!wreu 3367  {crab 3416  {csn 4588  ∩ 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-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  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-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  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-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  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-ord 6386  df-on 6387  df-suc 6389  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7403  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1o 8507  df-2o 8508  df-no 28033  df-lts 28034  df-bday 28035  df-slts 28177  df-cuts 28179

This theorem is used by:  cutscl  28201  cutbday  28203  cutsun12  28209  lesrec  28218  ltsrec  28220  eqcuts3  28223  cofcut2  28341  cofcutr  28343  cofcutrtime  28346  cutmax  28353  cutmin  28354  addsproplem3  28390  addsuniflem  28420  negsproplem3  28449  negsunif  28474  mulsproplem10  28544  sltmuls1  28566  sltmuls2  28567  mulsuniflem  28568  precsexlem11  28636  twocut  28842  pw2cut2  28881