Theorem oprpiece1res2 21973

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 9690	. . . . 5 |- A e. RR*
4		oprpiece1.2	. . . . . 6 |- B e. RR
5	4	rexri 9690	. . . . 5 |- B e. RR*
6		oprpiece1.3	. . . . 5 |- A <_ B
7		ubicc2 11746	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	3, 5, 6, 7	mp3an 1363	. . . 4 |- B e. ( A [,] B )
9		iccss2 11702	. . . 4 |- ( ( K e. ( A [,] B ) /\ B e. ( A [,] B ) ) -> ( K [,] B ) C_ ( A [,] B ) )
10	1, 8, 9	mp2an 677	. . 3 |- ( K [,] B ) C_ ( A [,] B )
11		ssid 3450	. . 3 |- C C_ C
12		resmpt2 6391	. . 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 677	. 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 5100	. 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 732	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> P = Q )
19		simpr 463	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x <_ K )
20	2, 4	elicc2i 11697	. . . . . . . . . . . . 13 |- ( K e. ( A [,] B ) <-> ( K e. RR /\ A <_ K /\ K <_ B ) )
21	20	simp1bi 1022	. . . . . . . . . . . 12 |- ( K e. ( A [,] B ) -> K e. RR )
22	1, 21	ax-mp 5	. . . . . . . . . . 11 |- K e. RR
23	22, 4	elicc2i 11697	. . . . . . . . . 10 |- ( x e. ( K [,] B ) <-> ( x e. RR /\ K <_ x /\ x <_ B ) )
24	23	simp2bi 1023	. . . . . . . . 9 |- ( x e. ( K [,] B ) -> K <_ x )
25	24	ad2antrr 731	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> K <_ x )
26	23	simp1bi 1022	. . . . . . . . . 10 |- ( x e. ( K [,] B ) -> x e. RR )
27	26	ad2antrr 731	. . . . . . . . 9 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x e. RR )
28		letri3 9716	. . . . . . . . 9 |- ( ( x e. RR /\ K e. RR ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
29	27, 22, 28	sylancl 667	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
30	19, 25, 29	mpbir2and 932	. . . . . . 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 2494	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = S )
36		eqidd 2451	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ -. x <_ K ) -> S = S )
37	35, 36	ifeqda 3913	. . . 4 |- ( ( x e. ( K [,] B ) /\ y e. C ) -> if ( x <_ K , R , S ) = S )
38	37	mpt2eq3ia 6353	. . 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 2475	. 2 |- G = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
40	13, 15, 39	3eqtr4i 2482	1 |- ( F |` ( ( K [,] B ) X. C ) ) = G

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1443   e. wcel 1886   _Vcvv 3044   C_ wss 3403   ifcif 3880   class class class wbr 4401   X. cxp 4831   |` cres 4835  (class class class)co 6288   |-> cmpt2 6290   RRcr 9535   RR*cxr 9671   <_ cle 9673   [,]cicc 11635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-pre-lttri 9610  ax-pre-lttrn 9611

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-id 4748  df-po 4754  df-so 4755  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-1st 6790  df-2nd 6791  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-icc 11639

This theorem is referenced by: (None)