Theorem hoidifhspval2 40829

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
hoidifhspval2.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 ) ) ) ) )
hoidifhspval2.y	|- ( ph -> Y e. RR )
hoidifhspval2.x	|- ( ph -> X e. V )
hoidifhspval2.a	|- ( ph -> A : X --> RR )

Assertion

Ref	Expression
hoidifhspval2	|- ( ph -> ( ( D ` Y ) ` A ) = ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) )

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

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

Proof of Theorem hoidifhspval2

Step	Hyp	Ref	Expression
1		hoidifhspval2.d	. . 3 |- D = ( x e. RR |-> ( a e. ( RR ^m X ) |-> ( k e. X |-> if ( k = K , if ( x <_ ( a ` k ) , ( a ` k ) , x ) , ( a ` k ) ) ) ) )
2		hoidifhspval2.y	. . 3 |- ( ph -> Y e. RR )
3	1, 2	hoidifhspval 40822	. 2 |- ( ph -> ( D ` Y ) = ( a e. ( RR ^m X ) |-> ( k e. X |-> if ( k = K , if ( Y <_ ( a ` k ) , ( a ` k ) , Y ) , ( a ` k ) ) ) ) )
4		fveq1 6190	. . . . . . 7 |- ( a = A -> ( a ` k ) = ( A ` k ) )
5	4	breq2d 4665	. . . . . 6 |- ( a = A -> ( Y <_ ( a ` k ) <-> Y <_ ( A ` k ) ) )
6	5, 4	ifbieq1d 4109	. . . . 5 |- ( a = A -> if ( Y <_ ( a ` k ) , ( a ` k ) , Y ) = if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) )
7	6, 4	ifeq12d 4106	. . . 4 |- ( a = A -> if ( k = K , if ( Y <_ ( a ` k ) , ( a ` k ) , Y ) , ( a ` k ) ) = if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) )
8	7	mpteq2dv 4745	. . 3 |- ( a = A -> ( 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 ) ) ) )
9	8	adantl 482	. 2 |- ( ( ph /\ a = A ) -> ( 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 ) ) ) )
10		hoidifhspval2.a	. . 3 |- ( ph -> A : X --> RR )
11		reex 10027	. . . . . 6 |- RR e. _V
12	11	a1i 11	. . . . 5 |- ( ph -> RR e. _V )
13		hoidifhspval2.x	. . . . 5 |- ( ph -> X e. V )
14	12, 13	jca 554	. . . 4 |- ( ph -> ( RR e. _V /\ X e. V ) )
15		elmapg 7870	. . . 4 |- ( ( RR e. _V /\ X e. V ) -> ( A e. ( RR ^m X ) <-> A : X --> RR ) )
16	14, 15	syl 17	. . 3 |- ( ph -> ( A e. ( RR ^m X ) <-> A : X --> RR ) )
17	10, 16	mpbird 247	. 2 |- ( ph -> A e. ( RR ^m X ) )
18		mptexg 6484	. . 3 |- ( X e. V -> ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) e. _V )
19	13, 18	syl 17	. 2 |- ( ph -> ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) e. _V )
20	3, 9, 17, 19	fvmptd 6288	1 |- ( ph -> ( ( D ` Y ) ` A ) = ( k e. X |-> if ( k = K , if ( Y <_ ( A ` k ) , ( A ` k ) , Y ) , ( A ` k ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   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:  hoidifhspf  40832  hoidifhspval3  40833