Theorem fourierdlem17 40341

Description: The defined L is actually a function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Hypotheses

Ref	Expression
fourierdlem17.a	|- ( ph -> A e. RR )
fourierdlem17.b	|- ( ph -> B e. RR )
fourierdlem17.altb	|- ( ph -> A < B )
fourierdlem17.l	|- L = ( x e. ( A (,] B ) |-> if ( x = B , A , x ) )

Assertion

Ref	Expression
fourierdlem17	|- ( ph -> L : ( A (,] B ) --> ( A [,] B ) )

Distinct variable groups:   x, A   x, B   ph, x

Allowed substitution hint:   L( x)

Proof of Theorem fourierdlem17

Step	Hyp	Ref	Expression
1		fourierdlem17.a	. . . . 5 |- ( ph -> A e. RR )
2		fourierdlem17.b	. . . . 5 |- ( ph -> B e. RR )
3	1	leidd 10594	. . . . 5 |- ( ph -> A <_ A )
4		fourierdlem17.altb	. . . . . 6 |- ( ph -> A < B )
5	1, 2, 4	ltled 10185	. . . . 5 |- ( ph -> A <_ B )
6	1, 2, 1, 3, 5	eliccd 39726	. . . 4 |- ( ph -> A e. ( A [,] B ) )
7	6	ad2antrr 762	. . 3 |- ( ( ( ph /\ x e. ( A (,] B ) ) /\ x = B ) -> A e. ( A [,] B ) )
8		iocssicc 12261	. . . . 5 |- ( A (,] B ) C_ ( A [,] B )
9	8	sseli 3599	. . . 4 |- ( x e. ( A (,] B ) -> x e. ( A [,] B ) )
10	9	ad2antlr 763	. . 3 |- ( ( ( ph /\ x e. ( A (,] B ) ) /\ -. x = B ) -> x e. ( A [,] B ) )
11	7, 10	ifclda 4120	. 2 |- ( ( ph /\ x e. ( A (,] B ) ) -> if ( x = B , A , x ) e. ( A [,] B ) )
12		fourierdlem17.l	. 2 |- L = ( x e. ( A (,] B ) |-> if ( x = B , A , x ) )
13	11, 12	fmptd 6385	1 |- ( ph -> L : ( A (,] B ) --> ( A [,] B ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   ifcif 4086   class class class wbr 4653   |-> cmpt 4729   -->wf 5884  (class class class)co 6650   RRcr 9935   < clt 10074   (,]cioc 12176   [,]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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-ioc 12180  df-icc 12182

This theorem is referenced by:  fourierdlem79  40402  fourierdlem89  40412  fourierdlem90  40413  fourierdlem91  40414