Theorem prsss 29962

Description: Relation of a subpreset. (Contributed by Thierry Arnoux, 13-Sep-2018.)

Hypotheses

Ref	Expression
ordtNEW.b	|- B = ( Base ` K )
ordtNEW.l	|- .<_ = ( ( le ` K ) i^i ( B X. B ) )

Assertion

Ref	Expression
prsss	|- ( ( K e. Preset /\ A C_ B ) -> ( .<_ i^i ( A X. A ) ) = ( ( le ` K ) i^i ( A X. A ) ) )

Proof of Theorem prsss

Step	Hyp	Ref	Expression
1		ordtNEW.l	. . . . 5 |- .<_ = ( ( le ` K ) i^i ( B X. B ) )
2	1	ineq1i 3810	. . . 4 |- ( .<_ i^i ( A X. A ) ) = ( ( ( le ` K ) i^i ( B X. B ) ) i^i ( A X. A ) )
3		inass 3823	. . . 4 |- ( ( ( le ` K ) i^i ( B X. B ) ) i^i ( A X. A ) ) = ( ( le ` K ) i^i ( ( B X. B ) i^i ( A X. A ) ) )
4	2, 3	eqtri 2644	. . 3 |- ( .<_ i^i ( A X. A ) ) = ( ( le ` K ) i^i ( ( B X. B ) i^i ( A X. A ) ) )
5		xpss12 5225	. . . . . 6 |- ( ( A C_ B /\ A C_ B ) -> ( A X. A ) C_ ( B X. B ) )
6	5	anidms 677	. . . . 5 |- ( A C_ B -> ( A X. A ) C_ ( B X. B ) )
7		sseqin2 3817	. . . . 5 |- ( ( A X. A ) C_ ( B X. B ) <-> ( ( B X. B ) i^i ( A X. A ) ) = ( A X. A ) )
8	6, 7	sylib 208	. . . 4 |- ( A C_ B -> ( ( B X. B ) i^i ( A X. A ) ) = ( A X. A ) )
9	8	ineq2d 3814	. . 3 |- ( A C_ B -> ( ( le ` K ) i^i ( ( B X. B ) i^i ( A X. A ) ) ) = ( ( le ` K ) i^i ( A X. A ) ) )
10	4, 9	syl5eq 2668	. 2 |- ( A C_ B -> ( .<_ i^i ( A X. A ) ) = ( ( le ` K ) i^i ( A X. A ) ) )
11	10	adantl 482	1 |- ( ( K e. Preset /\ A C_ B ) -> ( .<_ i^i ( A X. A ) ) = ( ( le ` K ) i^i ( A X. A ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   i^i cin 3573   C_ wss 3574   X. cxp 5112   ` cfv 5888   Basecbs 15857   lecple 15948   Preset cpreset 16926

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

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-v 3202  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120

This theorem is referenced by:  prsssdm  29963  ordtrestNEW  29967  ordtrest2NEW  29969