Theorem ntrk1k3eqk13 38348

Description: An interior function is both monotone and sub-linear if and only if it is finitely linear. (Contributed by RP, 18-Jun-2021.)

Assertion

Ref	Expression
ntrk1k3eqk13	|- ( ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) <-> A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) )

Distinct variable groups:   B, s, t   I, s, t

Proof of Theorem ntrk1k3eqk13

Step	Hyp	Ref	Expression
1		r19.26-2 3065	. 2 |- ( A. s e. ~P B A. t e. ~P B ( ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) /\ ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) <-> ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) )
2		eqss 3618	. . 3 |- ( ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> ( ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) /\ ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) )
3	2	2ralbii 2981	. 2 |- ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) /\ ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) )
4		isotone2 38347	. . 3 |- ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) <-> A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) )
5	4	anbi1i 731	. 2 |- ( ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) <-> ( A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) C_ ( ( I ` s ) i^i ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) )
6	1, 3, 5	3bitr4ri 293	1 |- ( ( A. s e. ~P B A. t e. ~P B ( s C_ t -> ( I ` s ) C_ ( I ` t ) ) /\ A. s e. ~P B A. t e. ~P B ( ( I ` s ) i^i ( I ` t ) ) C_ ( I ` ( s i^i t ) ) ) <-> A. s e. ~P B A. t e. ~P B ( I ` ( s i^i t ) ) = ( ( I ` s ) i^i ( I ` t ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   A.wral 2912   i^i cin 3573   C_ wss 3574   ~Pcpw 4158   ` cfv 5888

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by: (None)