Theorem metxpdval 7829

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 5512	. . . 4 |- < Or RR
2	1	supex 4577	. . 3 |- sup({(FBH), (GCJ)}, RR, < ) e. V
3		fveq2 3724	. . . . . . . 8 |- (x = R -> (1st` x) = (1st` R))
4		metxpval.8	. . . . . . . 8 |- F = (1st` R)
5	3, 4	syl6eqr 1525	. . . . . . 7 |- (x = R -> (1st` x) = F)
6	5	opreq1d 3975	. . . . . 6 |- (x = R -> ((1st` x)B(1st` y)) = (FB(1st` y)))
7		preq1 2448	. . . . . 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 3724	. . . . . . . 8 |- (x = R -> (2nd` x) = (2nd` R))
10		metxpval.9	. . . . . . . 8 |- G = (2nd` R)
11	9, 10	syl6eqr 1525	. . . . . . 7 |- (x = R -> (2nd` x) = G)
12	11	opreq1d 3975	. . . . . 6 |- (x = R -> ((2nd` x)C(2nd` y)) = (GC(2nd` y)))
13		preq2 2449	. . . . . 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 1507	. . . 4 |- (x = R -> {((1st` x)B(1st` y)), ((2nd` x)C(2nd` y))} = {(FB(1st` y)), (GC(2nd` y))})
16		supeq1 4575	. . . 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 3724	. . . . . . . 8 |- (y = S -> (1st` y) = (1st` S))
19		metxpval.10	. . . . . . . 8 |- H = (1st` S)
20	18, 19	syl6eqr 1525	. . . . . . 7 |- (y = S -> (1st` y) = H)
21	20	opreq2d 3976	. . . . . 6 |- (y = S -> (FB(1st` y)) = (FBH))
22		preq1 2448	. . . . . 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 3724	. . . . . . . 8 |- (y = S -> (2nd` y) = (2nd` S))
25		metxpval.11	. . . . . . . 8 |- J = (2nd` S)
26	24, 25	syl6eqr 1525	. . . . . . 7 |- (y = S -> (2nd` y) = J)
27	26	opreq2d 3976	. . . . . 6 |- (y = S -> (GC(2nd` y)) = (GCJ))
28		preq2 2449	. . . . . 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 1507	. . . 4 |- (y = S -> {(FB(1st` y)), (GC(2nd` y))} = {(FBH), (GCJ)})
31		supeq1 4575	. . . 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 4028	. 2 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = sup({(FBH), (GCJ)}, RR, < ))
35	1	suppr 4590	. . 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 7826	. . 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 7827	. . 3 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (GCJ) e. RR)
42	35, 38, 41	sylanc 471	. 2 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> sup({(FBH), (GCJ)}, RR, < ) = if((GCJ) < (FBH), (FBH), (GCJ)))
43	34, 42	eqtrd 1507	1 |- ((R e. (X X. Y) /\ S e. (X X. Y)) -> (RDS) = if((GCJ) < (FBH), (FBH), (GCJ)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  ifcif 2361  {cpr 2410   class class class wbr 2619   X. cxp 3168  dom cdm 3170  ` cfv 3182  (class class class)co 3963  {copab2 3964  1stc1st 4077  2ndc2nd 4078  supcsup 4573  RRcr 5233   < clt 5486  Metcme 7789

This theorem is referenced by:  metxptval 7830  metxpfval 7831  metxp 7834  xplm 7975  xpcn 7976

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-inf2 4625

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-nel 1588  df-ral 1649  df-rex 1650  df-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-csb 2002  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-pss 2055  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198  df-rdg 3932  df-opr 3965  df-oprab 3966  df-1st 4079  df-2nd 4080  df-1o 4133  df-oadd 4135  df-omul 4136  df-er 4261  df-ec 4263  df-qs 4266  df-en 4368  df-dom 4369  df-sdom 4370  df-sup 4574  df-ni 5000  df-pli 5001  df-mi 5002  df-lti 5003  df-plpq 5035  df-mpq 5036  df-enq 5037  df-nq 5038  df-plq 5039  df-mq 5040  df-rq 5041  df-ltq 5042  df-1q 5043  df-np 5086  df-1p 5087  df-plp 5088  df-ltp 5090  df-enr 5166  df-nr 5167  df-ltr 5170  df-0r 5171  df-c 5240  df-r 5244  df-lt 5247  df-pnf 5487  df-mnf 5488  df-xr 5489  df-ltxr 5490  df-met 7793

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