Theorem cutcuts 27851

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 27850	. . 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 27849	. . . 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 7366	. . . 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 17	. . 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 2861	. 2 ⊢ (𝐴 <<s 𝐵 → (𝐴 |s 𝐵) ∈ {𝑦 ∈ No ∣ (𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵)})
6		sneq 4591	. . . . . 6 ⊢ (𝑦 = (𝐴 |s 𝐵) → {𝑦} = {(𝐴 |s 𝐵)})
7	6	breq2d 5111	. . . . 5 ⊢ (𝑦 = (𝐴 |s 𝐵) → (𝐴 <<s {𝑦} ↔ 𝐴 <<s {(𝐴 |s 𝐵)}))
8	6	breq1d 5109	. . . . 5 ⊢ (𝑦 = (𝐴 |s 𝐵) → ({𝑦} <<s 𝐵 ↔ {(𝐴 |s 𝐵)} <<s 𝐵))
9	7, 8	anbi12d 641	. . . 4 ⊢ (𝑦 = (𝐴 |s 𝐵) → ((𝐴 <<s {𝑦} ∧ {𝑦} <<s 𝐵) ↔ (𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵)))
10	9	elrab 3650	. . 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 1105	. . 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 280	. 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 220	1 ⊢ (𝐴 <<s 𝐵 → ((𝐴 |s 𝐵) ∈ No ∧ 𝐴 <<s {(𝐴 |s 𝐵)} ∧ {(𝐴 |s 𝐵)} <<s 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1097   = wceq 1559   ∈ wcel 2141  ∃!wreu 3364  {crab 3413  {csn 4581  ∩ cint 4904   class class class wbr 5099   “ cima 5648  ‘cfv 6517  ℩crio 7348  (class class class)co 7392   No csur 27681   bday cbday 27683   <<s cslts 27827   |s ccuts 27829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5321  ax-pr 5389  ax-un 7714

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3745  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4905  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5540  df-eprel 5545  df-po 5553  df-so 5554  df-fr 5598  df-we 5600  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-rn 5656  df-res 5657  df-ima 5658  df-ord 6345  df-on 6346  df-suc 6348  df-iota 6473  df-fun 6519  df-fn 6520  df-f 6521  df-f1 6522  df-fo 6523  df-f1o 6524  df-fv 6525  df-riota 7349  df-ov 7395  df-oprab 7396  df-mpo 7397  df-1o 8432  df-2o 8433  df-no 27684  df-lts 27685  df-bday 27686  df-slts 27828  df-cuts 27830

This theorem is referenced by:  cutscl  27852  cutbday  27854  cutsun12  27860  lesrec  27869  ltsrec  27871  eqcuts3  27874  cofcut2  27992  cofcutr  27994  cofcutrtime  27997  cutmax  28004  cutmin  28005  addsproplem3  28041  addsuniflem  28071  negsproplem3  28100  negsunif  28125  mulsproplem10  28195  sltmuls1  28217  sltmuls2  28218  mulsuniflem  28219  precsexlem11  28287  twocut  28493  pw2cut2  28532