Theorem sltval 13988

Description: The value of the surreal less than relationship.

Assertion

Ref	Expression
sltval	|- ((A e. No /\ B e. No ) -> (A <s B <-> E.x e. On (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))

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

Proof of Theorem sltval

Step	Hyp	Ref	Expression
1		eleq1 1957	. . . . 5 |- (f = A -> (f e. No <-> A e. No ))
2	1	anbi1d 679	. . . 4 |- (f = A -> ((f e. No /\ g e. No ) <-> (A e. No /\ g e. No )))
3		fveq1 4680	. . . . . . . 8 |- (f = A -> (f` y) = (A` y))
4	3	eqeq1d 1892	. . . . . . 7 |- (f = A -> ((f` y) = (g` y) <-> (A` y) = (g` y)))
5	4	ralbidv 2123	. . . . . 6 |- (f = A -> (A.y e. x (f` y) = (g` y) <-> A.y e. x (A` y) = (g` y)))
6		fveq1 4680	. . . . . . 7 |- (f = A -> (f` x) = (A` x))
7	6	breq1d 3348	. . . . . 6 |- (f = A -> ((f` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x) <-> (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)))
8	5, 7	anbi12d 690	. . . . 5 |- (f = A -> ((A.y e. x (f` y) = (g` y) /\ (f` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)) <-> (A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x))))
9	8	rexbidv 2124	. . . 4 |- (f = A -> (E.x e. On (A.y e. x (f` y) = (g` y) /\ (f` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)) <-> E.x e. On (A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x))))
10	2, 9	anbi12d 690	. . 3 |- (f = A -> (((f e. No /\ g e. No ) /\ E.x e. On (A.y e. x (f` y) = (g` y) /\ (f` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x))) <-> ((A e. No /\ g e. No ) /\ E.x e. On (A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)))))
11		eleq1 1957	. . . . 5 |- (g = B -> (g e. No <-> B e. No ))
12	11	anbi2d 678	. . . 4 |- (g = B -> ((A e. No /\ g e. No ) <-> (A e. No /\ B e. No )))
13		fveq1 4680	. . . . . . . 8 |- (g = B -> (g` y) = (B` y))
14	13	eqeq2d 1895	. . . . . . 7 |- (g = B -> ((A` y) = (g` y) <-> (A` y) = (B` y)))
15	14	ralbidv 2123	. . . . . 6 |- (g = B -> (A.y e. x (A` y) = (g` y) <-> A.y e. x (A` y) = (B` y)))
16		fveq1 4680	. . . . . . 7 |- (g = B -> (g` x) = (B` x))
17	16	breq2d 3350	. . . . . 6 |- (g = B -> ((A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x) <-> (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)))
18	15, 17	anbi12d 690	. . . . 5 |- (g = B -> ((A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)) <-> (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))
19	18	rexbidv 2124	. . . 4 |- (g = B -> (E.x e. On (A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)) <-> E.x e. On (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))
20	12, 19	anbi12d 690	. . 3 |- (g = B -> (((A e. No /\ g e. No ) /\ E.x e. On (A.y e. x (A` y) = (g` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x))) <-> ((A e. No /\ B e. No ) /\ E.x e. On (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)))))
21		df-slt 13985	. . 3 |- <s = {<.f, g>. | ((f e. No /\ g e. No ) /\ E.x e. On (A.y e. x (f` y) = (g` y) /\ (f` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (g` x)))}
22	10, 20, 21	brabg 3568	. 2 |- ((A e. No /\ B e. No ) -> (A <s B <-> ((A e. No /\ B e. No ) /\ E.x e. On (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x)))))
23	22	bianabs 715	1 |- ((A e. No /\ B e. No ) -> (A <s B <-> E.x e. On (A.y e. x (A` y) = (B` y) /\ (A` x){<.1o, (/)>., <.1o, 2o>., <.(/), 2o>.} (B` x))))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  (/)c0 2875  <.cop 3046  {ctp 3051   class class class wbr 3338  Oncon0 3657  ` cfv 3998  1oc1o 5172  2oc2o 5173   No csur 13981   <s cslt 13982

This theorem is referenced by:  sltval2 13997  axdense 14027  axfelem8 14038  axfelem9 14039  axfelem12 14042

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-slt 13985

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