Theorem oprpiece1res2 21746

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 9678	. . . . 5 |- A e. RR*
4		oprpiece1.2	. . . . . 6 |- B e. RR
5	4	rexri 9678	. . . . 5 |- B e. RR*
6		oprpiece1.3	. . . . 5 |- A <_ B
7		ubicc2 11693	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	3, 5, 6, 7	mp3an 1328	. . . 4 |- B e. ( A [,] B )
9		iccss2 11651	. . . 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 3463	. . 3 |- C C_ C
12		resmpt2 6383	. . 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 5092	. 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		oprpiece1.11	. . . . . . 7 |- ( y e. C -> P = Q )
18	17	ad2antlr 727	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> P = Q )
19		simpr 461	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x <_ K )
20	2, 4	elicc2i 11646	. . . . . . . . . . . . 13 |- ( K e. ( A [,] B ) <-> ( K e. RR /\ A <_ K /\ K <_ B ) )
21	20	simp1bi 1014	. . . . . . . . . . . 12 |- ( K e. ( A [,] B ) -> K e. RR )
22	1, 21	ax-mp 5	. . . . . . . . . . 11 |- K e. RR
23	22, 4	elicc2i 11646	. . . . . . . . . 10 |- ( x e. ( K [,] B ) <-> ( x e. RR /\ K <_ x /\ x <_ B ) )
24	23	simp2bi 1015	. . . . . . . . 9 |- ( x e. ( K [,] B ) -> K <_ x )
25	24	ad2antrr 726	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> K <_ x )
26	23	simp1bi 1014	. . . . . . . . . 10 |- ( x e. ( K [,] B ) -> x e. RR )
27	26	ad2antrr 726	. . . . . . . . 9 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x e. RR )
28		letri3 9703	. . . . . . . . 9 |- ( ( x e. RR /\ K e. RR ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
29	27, 22, 28	sylancl 662	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
30	19, 25, 29	mpbir2and 925	. . . . . . 7 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x = K )
31		oprpiece1.9	. . . . . . 7 |- ( x = K -> R = P )
32	30, 31	syl 17	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = P )
33		oprpiece1.10	. . . . . . 7 |- ( x = K -> S = Q )
34	30, 33	syl 17	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> S = Q )
35	18, 32, 34	3eqtr4d 2455	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = S )
36		eqidd 2405	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ -. x <_ K ) -> S = S )
37	35, 36	ifeqda 3920	. . . 4 |- ( ( x e. ( K [,] B ) /\ y e. C ) -> if ( x <_ K , R , S ) = S )
38	37	mpt2eq3ia 6345	. . 3 |- ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) = ( x e. ( K [,] B ) , y e. C |-> S )
39	16, 38	eqtr4i 2436	. 2 |- G = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
40	13, 15, 39	3eqtr4i 2443	1 |- ( F |` ( ( K [,] B ) X. C ) ) = G

Syntax hints:   -. wn 3   -> wi 4   <-> wb 186   /\ wa 369   = wceq 1407   e. wcel 1844   _Vcvv 3061   C_ wss 3416   ifcif 3887   class class class wbr 4397   X. cxp 4823   |` cres 4827  (class class class)co 6280   |-> cmpt2 6282   RRcr 9523   RR*cxr 9659   <_ cle 9661   [,]cicc 11587

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632  ax-un 6576  ax-cnex 9580  ax-resscn 9581  ax-pre-lttri 9598  ax-pre-lttrn 9599

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3or 977  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-nel 2603  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-csb 3376  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-iun 4275  df-br 4398  df-opab 4456  df-mpt 4457  df-id 4740  df-po 4746  df-so 4747  df-xp 4831  df-rel 4832  df-cnv 4833  df-co 4834  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fun 5573  df-fn 5574  df-f 5575  df-f1 5576  df-fo 5577  df-f1o 5578  df-fv 5579  df-ov 6283  df-oprab 6284  df-mpt2 6285  df-1st 6786  df-2nd 6787  df-er 7350  df-en 7557  df-dom 7558  df-sdom 7559  df-pnf 9662  df-mnf 9663  df-xr 9664  df-ltxr 9665  df-le 9666  df-icc 11591

This theorem is referenced by: (None)