Theorem carsgmon 30376

Description: Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	|- ( ph -> O e. V )
carsgval.2	|- ( ph -> M : ~P O --> ( 0 [,] +oo ) )
carsgmon.1	|- ( ph -> A C_ B )
carsgmon.2	|- ( ph -> B e. ~P O )
carsgmon.3	|- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )

Assertion

Ref	Expression
carsgmon	|- ( ph -> ( M ` A ) <_ ( M ` B ) )

Distinct variable groups:   x, A, y   y, B   x, M, y   x, O, y   ph, x, y

Allowed substitution hints:   B( x)   V( x, y)

Proof of Theorem carsgmon

Step	Hyp	Ref	Expression
1		carsgmon.2	. . 3 |- ( ph -> B e. ~P O )
2		carsgmon.1	. . 3 |- ( ph -> A C_ B )
3	1, 2	ssexd 4805	. 2 |- ( ph -> A e. _V )
4		id 22	. 2 |- ( ph -> ph )
5		sseq1 3626	. . . . . 6 |- ( x = A -> ( x C_ y <-> A C_ y ) )
6	5	3anbi2d 1404	. . . . 5 |- ( x = A -> ( ( ph /\ x C_ y /\ y e. ~P O ) <-> ( ph /\ A C_ y /\ y e. ~P O ) ) )
7		fveq2 6191	. . . . . 6 |- ( x = A -> ( M ` x ) = ( M ` A ) )
8	7	breq1d 4663	. . . . 5 |- ( x = A -> ( ( M ` x ) <_ ( M ` y ) <-> ( M ` A ) <_ ( M ` y ) ) )
9	6, 8	imbi12d 334	. . . 4 |- ( x = A -> ( ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) ) <-> ( ( ph /\ A C_ y /\ y e. ~P O ) -> ( M ` A ) <_ ( M ` y ) ) ) )
10		sseq2 3627	. . . . . 6 |- ( y = B -> ( A C_ y <-> A C_ B ) )
11		eleq1 2689	. . . . . 6 |- ( y = B -> ( y e. ~P O <-> B e. ~P O ) )
12	10, 11	3anbi23d 1402	. . . . 5 |- ( y = B -> ( ( ph /\ A C_ y /\ y e. ~P O ) <-> ( ph /\ A C_ B /\ B e. ~P O ) ) )
13		fveq2 6191	. . . . . 6 |- ( y = B -> ( M ` y ) = ( M ` B ) )
14	13	breq2d 4665	. . . . 5 |- ( y = B -> ( ( M ` A ) <_ ( M ` y ) <-> ( M ` A ) <_ ( M ` B ) ) )
15	12, 14	imbi12d 334	. . . 4 |- ( y = B -> ( ( ( ph /\ A C_ y /\ y e. ~P O ) -> ( M ` A ) <_ ( M ` y ) ) <-> ( ( ph /\ A C_ B /\ B e. ~P O ) -> ( M ` A ) <_ ( M ` B ) ) ) )
16		carsgmon.3	. . . 4 |- ( ( ph /\ x C_ y /\ y e. ~P O ) -> ( M ` x ) <_ ( M ` y ) )
17	9, 15, 16	vtocl2g 3270	. . 3 |- ( ( A e. _V /\ B e. ~P O ) -> ( ( ph /\ A C_ B /\ B e. ~P O ) -> ( M ` A ) <_ ( M ` B ) ) )
18	17	imp 445	. 2 |- ( ( ( A e. _V /\ B e. ~P O ) /\ ( ph /\ A C_ B /\ B e. ~P O ) ) -> ( M ` A ) <_ ( M ` B ) )
19	3, 1, 4, 2, 1, 18	syl23anc 1333	1 |- ( ph -> ( M ` A ) <_ ( M ` B ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ~Pcpw 4158   class class class wbr 4653   -->wf 5884   ` cfv 5888  (class class class)co 6650   0cc0 9936   +oocpnf 10071   <_ cle 10075   [,]cicc 12178

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-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-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:  carsggect  30380  carsgclctunlem2  30381