Theorem hoidifhspf 40832

Description: D is a function that returns the representation of the left side of the difference of a half-open interval and a half-space. Used in Lemma 115F of [Fremlin1] p. 31 . (Contributed by Glauco Siliprandi, 24-Dec-2020.)

Hypotheses

Ref	Expression
hoidifhspf.d	|- D = ( x e. RR |-> ( a e. ( RR ^m X ) |-> ( k e. X |-> if ( k = K , if ( x <_ ( a ` k ) , ( a ` k ) , x ) , ( a ` k ) ) ) ) )
hoidifhspf.y	|- ( ph -> Y e. RR )
hoidifhspf.x	|- ( ph -> X e. V )
hoidifhspf.a	|- ( ph -> A : X --> RR )

Assertion

Ref	Expression
hoidifhspf	|- ( ph -> ( ( D ` Y ) ` A ) : X --> RR )

Distinct variable groups:   x, k   A, a, k   K, a, x   X, a, k, x   Y, a, k, x   ph, a, k, x

Allowed substitution hints:   A( x)   D( x, k, a)   K( k)   V( x, k, a)

Proof of Theorem hoidifhspf

Step	Hyp	Ref	Expression
1		hoidifhspf.a	. . . . . 6 |- ( ph -> A : X --> RR )
2	1	ffvelrnda 6359	. . . . 5 |- ( ( ph /\ k e. X ) -> ( A ` k ) e. RR )
3		hoidifhspf.y	. . . . . 6 |- ( ph -> Y e. RR )
4	3	adantr 481	. . . . 5 |- ( ( ph /\ k e. X ) -> Y e. RR )
5	2, 4	ifcld 4131	. . . 4 |- ( ( ph /\ k e. X ) -> if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) e. RR )
6	5, 2	ifcld 4131	. . 3 |- ( ( ph /\ k e. X ) -> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) e. RR )
7		eqid 2622	. . 3 |- ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) = ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) )
8	6, 7	fmptd 6385	. 2 |- ( ph -> ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) : X --> RR )
9		hoidifhspf.d	. . . 4 |- D = ( x e. RR |-> ( a e. ( RR ^m X ) |-> ( k e. X |-> if ( k = K , if ( x <_ ( a ` k ) , ( a ` k ) , x ) , ( a ` k ) ) ) ) )
10		hoidifhspf.x	. . . 4 |- ( ph -> X e. V )
11	9, 3, 10, 1	hoidifhspval2 40829	. . 3 |- ( ph -> ( ( D ` Y ) ` A ) = ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) )
12	11	feq1d 6030	. 2 |- ( ph -> ( ( ( D ` Y ) ` A ) : X --> RR <-> ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) : X --> RR ) )
13	8, 12	mpbird 247	1 |- ( ph -> ( ( D ` Y ) ` A ) : X --> RR )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   RRcr 9935   <_ cle 10075

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  ax-cnex 9992  ax-resscn 9993

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-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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859

This theorem is referenced by:  hoidifhspdmvle  40834  hspmbllem1  40840  hspmbllem2  40841