Theorem sltrec 31924

Description: A comparison law for surreals considered as cuts of sets of surreals. (Contributed by Scott Fenton, 11-Dec-2021.)

Assertion

Ref	Expression
sltrec	|- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( X <s Y <-> ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) ) )

Distinct variable groups:   A, b, c   B, b, c   C, b, c   D, b, c   X, b, c   Y, b, c

Proof of Theorem sltrec

Step	Hyp	Ref	Expression
1		simplr 792	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> C < <s D )
2		simpll 790	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> A < <s B )
3		simprr 796	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> Y = ( C |s D ) )
4		simprl 794	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> X = ( A |s B ) )
5		slerec 31923	. . . . 5 |- ( ( ( C < <s D /\ A < <s B ) /\ ( Y = ( C |s D ) /\ X = ( A |s B ) ) ) -> ( Y ≤s X <-> ( A. b e. B Y <s b /\ A. c e. C c <s X ) ) )
6	1, 2, 3, 4, 5	syl22anc 1327	. . . 4 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( Y ≤s X <-> ( A. b e. B Y <s b /\ A. c e. C c <s X ) ) )
7		ancom 466	. . . 4 |- ( ( A. b e. B Y <s b /\ A. c e. C c <s X ) <-> ( A. c e. C c <s X /\ A. b e. B Y <s b ) )
8	6, 7	syl6bb 276	. . 3 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( Y ≤s X <-> ( A. c e. C c <s X /\ A. b e. B Y <s b ) ) )
9		scutcut 31912	. . . . . . 7 |- ( C < <s D -> ( ( C |s D ) e. No /\ C < <s { ( C |s D ) } /\ { ( C |s D ) } < <s D ) )
10	9	simp1d 1073	. . . . . 6 |- ( C < <s D -> ( C |s D ) e. No )
11	10	ad2antlr 763	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( C |s D ) e. No )
12	3, 11	eqeltrd 2701	. . . 4 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> Y e. No )
13		scutcut 31912	. . . . . . 7 |- ( A < <s B -> ( ( A |s B ) e. No /\ A < <s { ( A |s B ) } /\ { ( A |s B ) } < <s B ) )
14	13	simp1d 1073	. . . . . 6 |- ( A < <s B -> ( A |s B ) e. No )
15	14	ad2antrr 762	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( A |s B ) e. No )
16	4, 15	eqeltrd 2701	. . . 4 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> X e. No )
17		slenlt 31877	. . . 4 |- ( ( Y e. No /\ X e. No ) -> ( Y ≤s X <-> -. X <s Y ) )
18	12, 16, 17	syl2anc 693	. . 3 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( Y ≤s X <-> -. X <s Y ) )
19		ssltss1 31903	. . . . . . . . 9 |- ( C < <s D -> C C_ No )
20	19	ad2antlr 763	. . . . . . . 8 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> C C_ No )
21	20	sselda 3603	. . . . . . 7 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ c e. C ) -> c e. No )
22	16	adantr 481	. . . . . . 7 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ c e. C ) -> X e. No )
23		sltnle 31878	. . . . . . 7 |- ( ( c e. No /\ X e. No ) -> ( c <s X <-> -. X ≤s c ) )
24	21, 22, 23	syl2anc 693	. . . . . 6 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ c e. C ) -> ( c <s X <-> -. X ≤s c ) )
25	24	ralbidva 2985	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( A. c e. C c <s X <-> A. c e. C -. X ≤s c ) )
26	12	adantr 481	. . . . . . 7 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ b e. B ) -> Y e. No )
27		ssltss2 31904	. . . . . . . . 9 |- ( A < <s B -> B C_ No )
28	27	ad2antrr 762	. . . . . . . 8 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> B C_ No )
29	28	sselda 3603	. . . . . . 7 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ b e. B ) -> b e. No )
30		sltnle 31878	. . . . . . 7 |- ( ( Y e. No /\ b e. No ) -> ( Y <s b <-> -. b ≤s Y ) )
31	26, 29, 30	syl2anc 693	. . . . . 6 |- ( ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) /\ b e. B ) -> ( Y <s b <-> -. b ≤s Y ) )
32	31	ralbidva 2985	. . . . 5 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( A. b e. B Y <s b <-> A. b e. B -. b ≤s Y ) )
33	25, 32	anbi12d 747	. . . 4 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( ( A. c e. C c <s X /\ A. b e. B Y <s b ) <-> ( A. c e. C -. X ≤s c /\ A. b e. B -. b ≤s Y ) ) )
34		ralnex 2992	. . . . . 6 |- ( A. c e. C -. X ≤s c <-> -. E. c e. C X ≤s c )
35		ralnex 2992	. . . . . 6 |- ( A. b e. B -. b ≤s Y <-> -. E. b e. B b ≤s Y )
36	34, 35	anbi12i 733	. . . . 5 |- ( ( A. c e. C -. X ≤s c /\ A. b e. B -. b ≤s Y ) <-> ( -. E. c e. C X ≤s c /\ -. E. b e. B b ≤s Y ) )
37		ioran 511	. . . . 5 |- ( -. ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) <-> ( -. E. c e. C X ≤s c /\ -. E. b e. B b ≤s Y ) )
38	36, 37	bitr4i 267	. . . 4 |- ( ( A. c e. C -. X ≤s c /\ A. b e. B -. b ≤s Y ) <-> -. ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) )
39	33, 38	syl6bb 276	. . 3 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( ( A. c e. C c <s X /\ A. b e. B Y <s b ) <-> -. ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) ) )
40	8, 18, 39	3bitr3d 298	. 2 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( -. X <s Y <-> -. ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) ) )
41	40	con4bid 307	1 |- ( ( ( A < <s B /\ C < <s D ) /\ ( X = ( A |s B ) /\ Y = ( C |s D ) ) ) -> ( X <s Y <-> ( E. c e. C X ≤s c \/ E. b e. B b ≤s Y ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   C_ wss 3574   {csn 4177   class class class wbr 4653  (class class class)co 6650   Nocsur 31793   <scslt 31794   ≤scsle 31869   < <scsslt 31896   |scscut 31898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1o 7560  df-2o 7561  df-no 31796  df-slt 31797  df-bday 31798  df-sle 31870  df-sslt 31897  df-scut 31899

This theorem is referenced by: (None)