Theorem oprpiece1res2 20649

Description: Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
oprpiece1.1	|- A e. RR
oprpiece1.2	|- B e. RR
oprpiece1.3	|- A <_ B
oprpiece1.4	|- R e. _V
oprpiece1.5	|- S e. _V
oprpiece1.6	|- K e. ( A [,] B )
oprpiece1.7	|- F = ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
oprpiece1.9	|- ( x = K -> R = P )
oprpiece1.10	|- ( x = K -> S = Q )
oprpiece1.11	|- ( y e. C -> P = Q )
oprpiece1.12	|- G = ( x e. ( K [,] B ) , y e. C |-> S )

Assertion

Ref	Expression
oprpiece1res2	|- ( F |` ( ( K [,] B ) X. C ) ) = G

Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, K, y   x, P   x, Q

Allowed substitution hints:   P( y)   Q( y)   R( x, y)   S( x, y)   F( x, y)   G( x, y)

Proof of Theorem oprpiece1res2

Step	Hyp	Ref	Expression
1		oprpiece1.6	. . . 4 |- K e. ( A [,] B )
2		oprpiece1.1	. . . . . 6 |- A e. RR
3	2	rexri 9540	. . . . 5 |- A e. RR*
4		oprpiece1.2	. . . . . 6 |- B e. RR
5	4	rexri 9540	. . . . 5 |- B e. RR*
6		oprpiece1.3	. . . . 5 |- A <_ B
7		ubicc2 11512	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	3, 5, 6, 7	mp3an 1315	. . . 4 |- B e. ( A [,] B )
9		iccss2 11470	. . . 4 |- ( ( K e. ( A [,] B ) /\ B e. ( A [,] B ) ) -> ( K [,] B ) C_ ( A [,] B ) )
10	1, 8, 9	mp2an 672	. . 3 |- ( K [,] B ) C_ ( A [,] B )
11		ssid 3476	. . 3 |- C C_ C
12		resmpt2 6291	. . 3 |- ( ( ( K [,] B ) C_ ( A [,] B ) /\ C C_ C ) -> ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) ) = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) )
13	10, 11, 12	mp2an 672	. 2 |- ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) ) = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
14		oprpiece1.7	. . 3 |- F = ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
15	14	reseq1i 5207	. 2 |- ( F |` ( ( K [,] B ) X. C ) ) = ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) )
16		oprpiece1.12	. . 3 |- G = ( x e. ( K [,] B ) , y e. C |-> S )
17		eqeq1 2455	. . . . 5 |- ( R = if ( x <_ K , R , S ) -> ( R = S <-> if ( x <_ K , R , S ) = S ) )
18		eqeq1 2455	. . . . 5 |- ( S = if ( x <_ K , R , S ) -> ( S = S <-> if ( x <_ K , R , S ) = S ) )
19		oprpiece1.11	. . . . . . 7 |- ( y e. C -> P = Q )
20	19	ad2antlr 726	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> P = Q )
21		simpr 461	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x <_ K )
22	2, 4	elicc2i 11465	. . . . . . . . . . . . 13 |- ( K e. ( A [,] B ) <-> ( K e. RR /\ A <_ K /\ K <_ B ) )
23	22	simp1bi 1003	. . . . . . . . . . . 12 |- ( K e. ( A [,] B ) -> K e. RR )
24	1, 23	ax-mp 5	. . . . . . . . . . 11 |- K e. RR
25	24, 4	elicc2i 11465	. . . . . . . . . 10 |- ( x e. ( K [,] B ) <-> ( x e. RR /\ K <_ x /\ x <_ B ) )
26	25	simp2bi 1004	. . . . . . . . 9 |- ( x e. ( K [,] B ) -> K <_ x )
27	26	ad2antrr 725	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> K <_ x )
28	25	simp1bi 1003	. . . . . . . . . 10 |- ( x e. ( K [,] B ) -> x e. RR )
29	28	ad2antrr 725	. . . . . . . . 9 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x e. RR )
30		letri3 9564	. . . . . . . . 9 |- ( ( x e. RR /\ K e. RR ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
31	29, 24, 30	sylancl 662	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
32	21, 27, 31	mpbir2and 913	. . . . . . 7 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x = K )
33		oprpiece1.9	. . . . . . 7 |- ( x = K -> R = P )
34	32, 33	syl 16	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = P )
35		oprpiece1.10	. . . . . . 7 |- ( x = K -> S = Q )
36	32, 35	syl 16	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> S = Q )
37	20, 34, 36	3eqtr4d 2502	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = S )
38		eqidd 2452	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ -. x <_ K ) -> S = S )
39	17, 18, 37, 38	ifbothda 3925	. . . 4 |- ( ( x e. ( K [,] B ) /\ y e. C ) -> if ( x <_ K , R , S ) = S )
40	39	mpt2eq3ia 6253	. . 3 |- ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) = ( x e. ( K [,] B ) , y e. C |-> S )
41	16, 40	eqtr4i 2483	. 2 |- G = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
42	13, 15, 41	3eqtr4i 2490	1 |- ( F |` ( ( K [,] B ) X. C ) ) = G

Syntax hints:   -. wn 3   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1370   e. wcel 1758   _Vcvv 3071   C_ wss 3429   ifcif 3892   class class class wbr 4393   X. cxp 4939   |` cres 4943  (class class class)co 6193   |-> cmpt2 6195   RRcr 9385   RR*cxr 9521   <_ cle 9523   [,]cicc 11407

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-pre-lttri 9460  ax-pre-lttrn 9461

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-po 4742  df-so 4743  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-1st 6680  df-2nd 6681  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-icc 11411

This theorem is referenced by: (None)