Theorem metxpdval 7826

Description: Value of the distance function of the direct product of two metric spaces. Based on Definition 14-1.5 of [Gleason] p. 225.

Hypotheses

Ref	Expression
metxp.1	|- X = dom dom B
metxp.3	|- Y = dom dom C
metxp.5	|- B e. Met
metxp.6	|- C e. Met
metxp.7	|- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
metxpval.8	|- F = (1st` R)
metxpval.9	|- G = (2nd` R)
metxpval.10	|- H = (1st` S)
metxpval.11	|- J = (2nd` S)

Assertion

Ref	Expression
metxpdval	|- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = if((GCJ) < (FBH), (FBH), (GCJ)))

Distinct variable groups:   x,y,z,B   x,C,y,z   x,R,y,z   x,S,y,z   x,X,y,z   x,Y,y,z   x,F,y,z   x,G,y,z   y,H,z   y,J,z

Proof of Theorem metxpdval

Step	Hyp	Ref	Expression
1		ltso 5524	. . . 4 |- < Or RR
2	1	supex 4586	. . 3 |- sup({(FBH), (GCJ)}, RR, < ) e. V
3		fveq2 3730	. . . . . . . 8 |- (x = R -> (1st` x) = (1st` R))
4		metxpval.8	. . . . . . . 8 |- F = (1st` R)
5	3, 4	syl6eqr 1528	. . . . . . 7 |- (x = R -> (1st` x) = F)
6	5	opreq1d 3981	. . . . . 6 |- (x = R -> ((1st` x)B(1st` y)) = (FB(1st` y)))
7		preq1 2452	. . . . . 6 |- (((1st` x)B(1st` y)) = (FB(1st` y)) -> {((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), ((2nd` x)C(2nd` y))})
8	6, 7	syl 10	. . . . 5 |- (x = R -> {((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), ((2nd` x)C(2nd` y))})
9		fveq2 3730	. . . . . . . 8 |- (x = R -> (2nd` x) = (2nd` R))
10		metxpval.9	. . . . . . . 8 |- G = (2nd` R)
11	9, 10	syl6eqr 1528	. . . . . . 7 |- (x = R -> (2nd` x) = G)
12	11	opreq1d 3981	. . . . . 6 |- (x = R -> ((2nd` x)C(2nd` y)) = (GC(2nd` y)))
13		preq2 2453	. . . . . 6 |- (((2nd` x)C(2nd` y)) = (GC(2nd` y)) -> {(FB(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), (GC(2nd` y))})
14	12, 13	syl 10	. . . . 5 |- (x = R -> {(FB(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), (GC(2nd` y))})
15	8, 14	eqtrd 1510	. . . 4 |- (x = R -> {((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), (GC(2nd` y))})
16		supeq1 4584	. . . 4 |- ({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), (GC(2nd` y))} -> sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) = sup({(FB(1st` y)), (GC(2nd` y))}, RR, < ))
17	15, 16	syl 10	. . 3 |- (x = R -> sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ) = sup({(FB(1st` y)), (GC(2nd` y))}, RR, < ))
18		fveq2 3730	. . . . . . . 8 |- (y = S -> (1st` y) = (1st` S))
19		metxpval.10	. . . . . . . 8 |- H = (1st` S)
20	18, 19	syl6eqr 1528	. . . . . . 7 |- (y = S -> (1st` y) = H)
21	20	opreq2d 3982	. . . . . 6 |- (y = S -> (FB(1st` y)) = (FBH))
22		preq1 2452	. . . . . 6 |- ((FB(1st` y)) = (FBH) -> {(FB(1st` y)), (GC(2nd` y))} = {(FBH), (GC(2nd` y))})
23	21, 22	syl 10	. . . . 5 |- (y = S -> {(FB(1st` y)), (GC(2nd` y))} = {(FBH), (GC(2nd` y))})
24		fveq2 3730	. . . . . . . 8 |- (y = S -> (2nd` y) = (2nd` S))
25		metxpval.11	. . . . . . . 8 |- J = (2nd` S)
26	24, 25	syl6eqr 1528	. . . . . . 7 |- (y = S -> (2nd` y) = J)
27	26	opreq2d 3982	. . . . . 6 |- (y = S -> (GC(2nd` y)) = (GCJ))
28		preq2 2453	. . . . . 6 |- ((GC(2nd` y)) = (GCJ) -> {(FBH), (GC(2nd` y))} = {(FBH), (GCJ)})
29	27, 28	syl 10	. . . . 5 |- (y = S -> {(FBH), (GC(2nd` y))} = {(FBH), (GCJ)})
30	23, 29	eqtrd 1510	. . . 4 |- (y = S -> {(FB(1st` y)), (GC(2nd` y))} = {(FBH), (GCJ)})
31		supeq1 4584	. . . 4 |- ({(FB(1st` y)), (GC(2nd` y))} = {(FBH), (GCJ)} -> sup({(FB(1st` y)), (GC(2nd` y))}, RR, < ) = sup({(FBH), (GCJ)}, RR, < ))
32	30, 31	syl 10	. . 3 |- (y = S -> sup({(FB(1st` y)), (GC(2nd` y))}, RR, < ) = sup({(FBH), (GCJ)}, RR, < ))
33		metxp.7	. . 3 |- D = {<.<.x, y>., z>. | ((x e. (X X. Y) /\ y e. (X X. Y)) /\ z = sup({((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))}, RR, < ))}
34	2, 17, 32, 33	oprabval2 4034	. 2 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = sup({(FBH), (GCJ)}, RR, < ))
35	1	suppr 4599	. . 3 |- (((FBH) e. RR /\ (GCJ) e. RR) -> sup({(FBH), (GCJ)}, RR, < ) = if((GCJ) < (FBH), (FBH), (GCJ)))
36		metxp.5	. . . 4 |- B e. Met
37		metxp.1	. . . 4 |- X = dom dom B
38	36, 37, 4, 19	metxplem1 7823	. . 3 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (FBH) e. RR)
39		metxp.6	. . . 4 |- C e. Met
40		metxp.3	. . . 4 |- Y = dom dom C
41	39, 40, 10, 25	metxplem2 7824	. . 3 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (GCJ) e. RR)
42	35, 38, 41	sylanc 473	. 2 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> sup({(FBH), (GCJ)}, RR, < ) = if((GCJ) < (FBH), (FBH), (GCJ)))
43	34, 42	eqtrd 1510	1 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = if((GCJ) < (FBH), (FBH), (GCJ)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  ifcif 2365  {cpr 2414   class class class wbr 2624   X. cxp 3174  dom cdm 3176  ` cfv 3188  (class class class)co 3969  {copab2 3970  1stc1st 4083  2ndc2nd 4084  supcsup 4582  RRcr 5245   < clt 5498  Metcme 7786

This theorem is referenced by:  metxptval 7827  metxpfval 7828  metxp 7831  xplm 7972  xpcn 7973

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-inf2 4634

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-nel 1591  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-1st 4085  df-2nd 4086  df-1o 4139  df-oadd 4141  df-omul 4142  df-er 4267  df-ec 4269  df-qs 4272  df-en 4374  df-dom 4375  df-sdom 4376  df-sup 4583  df-ni 5012  df-pli 5013  df-mi 5014  df-lti 5015  df-plpq 5047  df-mpq 5048  df-enq 5049  df-nq 5050  df-plq 5051  df-mq 5052  df-rq 5053  df-ltq 5054  df-1q 5055  df-np 5098  df-1p 5099  df-plp 5100  df-ltp 5102  df-enr 5178  df-nr 5179  df-ltr 5182  df-0r 5183  df-c 5252  df-r 5256  df-lt 5259  df-pnf 5499  df-mnf 5500  df-xr 5501  df-ltxr 5502  df-met 7790

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