Theorem prdsdsval3 16145

Description: Value of the metric in a structure product. (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypotheses

Ref	Expression
prdsbasmpt2.y	|- Y = ( S X_s ( x e. I |-> R ) )
prdsbasmpt2.b	|- B = ( Base ` Y )
prdsbasmpt2.s	|- ( ph -> S e. V )
prdsbasmpt2.i	|- ( ph -> I e. W )
prdsbasmpt2.r	|- ( ph -> A. x e. I R e. X )
prdsdsval2.f	|- ( ph -> F e. B )
prdsdsval2.g	|- ( ph -> G e. B )
prdsdsval3.k	|- K = ( Base ` R )
prdsdsval3.e	|- E = ( ( dist ` R ) |` ( K X. K ) )
prdsdsval3.d	|- D = ( dist ` Y )

Assertion

Ref	Expression
prdsdsval3	|- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) , RR* , < ) )

Distinct variable groups:   x, F   x, G   x, I

Allowed substitution hints:   ph( x)   B( x)   D( x)   R( x)   S( x)   E( x)   K( x)   V( x)   W( x)   X( x)   Y( x)

Proof of Theorem prdsdsval3

Step	Hyp	Ref	Expression
1		prdsbasmpt2.y	. . 3 |- Y = ( S X_s ( x e. I |-> R ) )
2		prdsbasmpt2.b	. . 3 |- B = ( Base ` Y )
3		prdsbasmpt2.s	. . 3 |- ( ph -> S e. V )
4		prdsbasmpt2.i	. . 3 |- ( ph -> I e. W )
5		prdsbasmpt2.r	. . 3 |- ( ph -> A. x e. I R e. X )
6		prdsdsval2.f	. . 3 |- ( ph -> F e. B )
7		prdsdsval2.g	. . 3 |- ( ph -> G e. B )
8		eqid 2622	. . 3 |- ( dist ` R ) = ( dist ` R )
9		prdsdsval3.d	. . 3 |- D = ( dist ` Y )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	prdsdsval2 16144	. 2 |- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) )
11		eqidd 2623	. . . . . 6 |- ( ph -> I = I )
12		prdsdsval3.k	. . . . . . . 8 |- K = ( Base ` R )
13	1, 2, 3, 4, 5, 12, 6	prdsbascl 16143	. . . . . . 7 |- ( ph -> A. x e. I ( F ` x ) e. K )
14	1, 2, 3, 4, 5, 12, 7	prdsbascl 16143	. . . . . . 7 |- ( ph -> A. x e. I ( G ` x ) e. K )
15		prdsdsval3.e	. . . . . . . . . . 11 |- E = ( ( dist ` R ) |` ( K X. K ) )
16	15	oveqi 6663	. . . . . . . . . 10 |- ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( ( dist ` R ) |` ( K X. K ) ) ( G ` x ) )
17		ovres 6800	. . . . . . . . . 10 |- ( ( ( F ` x ) e. K /\ ( G ` x ) e. K ) -> ( ( F ` x ) ( ( dist ` R ) |` ( K X. K ) ) ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) )
18	16, 17	syl5eq 2668	. . . . . . . . 9 |- ( ( ( F ` x ) e. K /\ ( G ` x ) e. K ) -> ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) )
19	18	ex 450	. . . . . . . 8 |- ( ( F ` x ) e. K -> ( ( G ` x ) e. K -> ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) )
20	19	ral2imi 2947	. . . . . . 7 |- ( A. x e. I ( F ` x ) e. K -> ( A. x e. I ( G ` x ) e. K -> A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) )
21	13, 14, 20	sylc 65	. . . . . 6 |- ( ph -> A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) )
22		mpteq12 4736	. . . . . 6 |- ( ( I = I /\ A. x e. I ( ( F ` x ) E ( G ` x ) ) = ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) -> ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) )
23	11, 21, 22	syl2anc 693	. . . . 5 |- ( ph -> ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) )
24	23	rneqd 5353	. . . 4 |- ( ph -> ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) = ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) )
25	24	uneq1d 3766	. . 3 |- ( ph -> ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) = ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) )
26	25	supeq1d 8352	. 2 |- ( ph -> sup ( ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) , RR* , < ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) ( dist ` R ) ( G ` x ) ) ) u. { 0 } ) , RR* , < ) )
27	10, 26	eqtr4d 2659	1 |- ( ph -> ( F D G ) = sup ( ( ran ( x e. I |-> ( ( F ` x ) E ( G ` x ) ) ) u. { 0 } ) , RR* , < ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   u. cun 3572   {csn 4177   |-> cmpt 4729   X. cxp 5112   ran crn 5115   |` cres 5116   ` cfv 5888  (class class class)co 6650   supcsup 8346   0cc0 9936   RR*cxr 10073   < clt 10074   Basecbs 15857   distcds 15950   X__scprds 16106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-fz 12327  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-plusg 15954  df-mulr 15955  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-hom 15966  df-cco 15967  df-prds 16108

This theorem is referenced by:  prdsxmetlem  22173  prdsmet  22175  prdsbl  22296  prdsbnd  33592  rrnequiv  33634