Theorem metideq 29936

Description: Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)

Assertion

Ref	Expression
metideq	|- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) = ( B D F ) )

Proof of Theorem metideq

Step	Hyp	Ref	Expression
1		simpl 473	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> D e. (PsMet ` X ) )
2		metidss 29934	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( X X. X ) )
3		dmss 5323	. . . . . . . . 9 |- ( (~Met ` D ) C_ ( X X. X ) -> dom (~Met ` D ) C_ dom ( X X. X ) )
4	2, 3	syl 17	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> dom (~Met ` D ) C_ dom ( X X. X ) )
5		dmxpid 5345	. . . . . . . 8 |- dom ( X X. X ) = X
6	4, 5	syl6sseq 3651	. . . . . . 7 |- ( D e. (PsMet ` X ) -> dom (~Met ` D ) C_ X )
7	1, 6	syl 17	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> dom (~Met ` D ) C_ X )
8		xpss 5226	. . . . . . . . . 10 |- ( X X. X ) C_ ( _V X. _V )
9	2, 8	syl6ss 3615	. . . . . . . . 9 |- ( D e. (PsMet ` X ) -> (~Met ` D ) C_ ( _V X. _V ) )
10		df-rel 5121	. . . . . . . . 9 |- ( Rel (~Met ` D ) <-> (~Met ` D ) C_ ( _V X. _V ) )
11	9, 10	sylibr 224	. . . . . . . 8 |- ( D e. (PsMet ` X ) -> Rel (~Met ` D ) )
12	1, 11	syl 17	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> Rel (~Met ` D ) )
13		simprl 794	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> A (~Met ` D ) B )
14		releldm 5358	. . . . . . 7 |- ( ( Rel (~Met ` D ) /\ A (~Met ` D ) B ) -> A e. dom (~Met ` D ) )
15	12, 13, 14	syl2anc 693	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> A e. dom (~Met ` D ) )
16	7, 15	sseldd 3604	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> A e. X )
17		simprr 796	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> E (~Met ` D ) F )
18		releldm 5358	. . . . . . 7 |- ( ( Rel (~Met ` D ) /\ E (~Met ` D ) F ) -> E e. dom (~Met ` D ) )
19	12, 17, 18	syl2anc 693	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> E e. dom (~Met ` D ) )
20	7, 19	sseldd 3604	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> E e. X )
21		psmetsym 22115	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ E e. X ) -> ( A D E ) = ( E D A ) )
22	1, 16, 20, 21	syl3anc 1326	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) = ( E D A ) )
23		psmetf 22111	. . . . . 6 |- ( D e. (PsMet ` X ) -> D : ( X X. X ) --> RR* )
24	23	fovrnda 6805	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( E e. X /\ A e. X ) ) -> ( E D A ) e. RR* )
25	1, 20, 16, 24	syl12anc 1324	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( E D A ) e. RR* )
26	22, 25	eqeltrd 2701	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) e. RR* )
27		rnss 5354	. . . . . . . 8 |- ( (~Met ` D ) C_ ( X X. X ) -> ran (~Met ` D ) C_ ran ( X X. X ) )
28	2, 27	syl 17	. . . . . . 7 |- ( D e. (PsMet ` X ) -> ran (~Met ` D ) C_ ran ( X X. X ) )
29		rnxpid 5567	. . . . . . 7 |- ran ( X X. X ) = X
30	28, 29	syl6sseq 3651	. . . . . 6 |- ( D e. (PsMet ` X ) -> ran (~Met ` D ) C_ X )
31	1, 30	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ran (~Met ` D ) C_ X )
32		relelrn 5359	. . . . . 6 |- ( ( Rel (~Met ` D ) /\ A (~Met ` D ) B ) -> B e. ran (~Met ` D ) )
33	12, 13, 32	syl2anc 693	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> B e. ran (~Met ` D ) )
34	31, 33	sseldd 3604	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> B e. X )
35	23	fovrnda 6805	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( B e. X /\ E e. X ) ) -> ( B D E ) e. RR* )
36	1, 34, 20, 35	syl12anc 1324	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D E ) e. RR* )
37		relelrn 5359	. . . . . . 7 |- ( ( Rel (~Met ` D ) /\ E (~Met ` D ) F ) -> F e. ran (~Met ` D ) )
38	12, 17, 37	syl2anc 693	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> F e. ran (~Met ` D ) )
39	31, 38	sseldd 3604	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> F e. X )
40		psmetsym 22115	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ F e. X /\ B e. X ) -> ( F D B ) = ( B D F ) )
41	1, 39, 34, 40	syl3anc 1326	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( F D B ) = ( B D F ) )
42	23	fovrnda 6805	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( F e. X /\ B e. X ) ) -> ( F D B ) e. RR* )
43	1, 39, 34, 42	syl12anc 1324	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( F D B ) e. RR* )
44	41, 43	eqeltrrd 2702	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D F ) e. RR* )
45		psmettri2 22114	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( B e. X /\ A e. X /\ E e. X ) ) -> ( A D E ) <_ ( ( B D A ) +e ( B D E ) ) )
46	1, 34, 16, 20, 45	syl13anc 1328	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) <_ ( ( B D A ) +e ( B D E ) ) )
47		psmetsym 22115	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
48	1, 16, 34, 47	syl3anc 1326	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D B ) = ( B D A ) )
49	16, 34	jca 554	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A e. X /\ B e. X ) )
50		metidv 29935	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) -> ( A (~Met ` D ) B <-> ( A D B ) = 0 ) )
51	50	biimpa 501	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X ) ) /\ A (~Met ` D ) B ) -> ( A D B ) = 0 )
52	1, 49, 13, 51	syl21anc 1325	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D B ) = 0 )
53	48, 52	eqtr3d 2658	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D A ) = 0 )
54	53	oveq1d 6665	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( B D A ) +e ( B D E ) ) = ( 0 +e ( B D E ) ) )
55		xaddid2 12073	. . . . . 6 |- ( ( B D E ) e. RR* -> ( 0 +e ( B D E ) ) = ( B D E ) )
56	36, 55	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( 0 +e ( B D E ) ) = ( B D E ) )
57	54, 56	eqtrd 2656	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( B D A ) +e ( B D E ) ) = ( B D E ) )
58	46, 57	breqtrd 4679	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) <_ ( B D E ) )
59		psmettri2 22114	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( F e. X /\ B e. X /\ E e. X ) ) -> ( B D E ) <_ ( ( F D B ) +e ( F D E ) ) )
60	1, 39, 34, 20, 59	syl13anc 1328	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D E ) <_ ( ( F D B ) +e ( F D E ) ) )
61		psmetsym 22115	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ F e. X /\ E e. X ) -> ( F D E ) = ( E D F ) )
62	1, 39, 20, 61	syl3anc 1326	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( F D E ) = ( E D F ) )
63	20, 39	jca 554	. . . . . . . 8 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( E e. X /\ F e. X ) )
64		metidv 29935	. . . . . . . . 9 |- ( ( D e. (PsMet ` X ) /\ ( E e. X /\ F e. X ) ) -> ( E (~Met ` D ) F <-> ( E D F ) = 0 ) )
65	64	biimpa 501	. . . . . . . 8 |- ( ( ( D e. (PsMet ` X ) /\ ( E e. X /\ F e. X ) ) /\ E (~Met ` D ) F ) -> ( E D F ) = 0 )
66	1, 63, 17, 65	syl21anc 1325	. . . . . . 7 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( E D F ) = 0 )
67	62, 66	eqtrd 2656	. . . . . 6 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( F D E ) = 0 )
68	67	oveq2d 6666	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( F D B ) +e ( F D E ) ) = ( ( F D B ) +e 0 ) )
69		xaddid1 12072	. . . . . 6 |- ( ( F D B ) e. RR* -> ( ( F D B ) +e 0 ) = ( F D B ) )
70	43, 69	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( F D B ) +e 0 ) = ( F D B ) )
71	68, 70, 41	3eqtrd 2660	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( F D B ) +e ( F D E ) ) = ( B D F ) )
72	60, 71	breqtrd 4679	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D E ) <_ ( B D F ) )
73	26, 36, 44, 58, 72	xrletrd 11993	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) <_ ( B D F ) )
74	23	fovrnda 6805	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ F e. X ) ) -> ( A D F ) e. RR* )
75	1, 16, 39, 74	syl12anc 1324	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D F ) e. RR* )
76		psmettri2 22114	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A e. X /\ B e. X /\ F e. X ) ) -> ( B D F ) <_ ( ( A D B ) +e ( A D F ) ) )
77	1, 16, 34, 39, 76	syl13anc 1328	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D F ) <_ ( ( A D B ) +e ( A D F ) ) )
78	52	oveq1d 6665	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( A D B ) +e ( A D F ) ) = ( 0 +e ( A D F ) ) )
79		xaddid2 12073	. . . . . 6 |- ( ( A D F ) e. RR* -> ( 0 +e ( A D F ) ) = ( A D F ) )
80	75, 79	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( 0 +e ( A D F ) ) = ( A D F ) )
81	78, 80	eqtrd 2656	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( A D B ) +e ( A D F ) ) = ( A D F ) )
82	77, 81	breqtrd 4679	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D F ) <_ ( A D F ) )
83		psmettri2 22114	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( E e. X /\ A e. X /\ F e. X ) ) -> ( A D F ) <_ ( ( E D A ) +e ( E D F ) ) )
84	1, 20, 16, 39, 83	syl13anc 1328	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D F ) <_ ( ( E D A ) +e ( E D F ) ) )
85		xaddid1 12072	. . . . . 6 |- ( ( E D A ) e. RR* -> ( ( E D A ) +e 0 ) = ( E D A ) )
86	25, 85	syl 17	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( E D A ) +e 0 ) = ( E D A ) )
87	66	oveq2d 6666	. . . . 5 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( E D A ) +e ( E D F ) ) = ( ( E D A ) +e 0 ) )
88	86, 87, 22	3eqtr4d 2666	. . . 4 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( E D A ) +e ( E D F ) ) = ( A D E ) )
89	84, 88	breqtrd 4679	. . 3 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D F ) <_ ( A D E ) )
90	44, 75, 26, 82, 89	xrletrd 11993	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( B D F ) <_ ( A D E ) )
91		xrletri3 11985	. . 3 |- ( ( ( A D E ) e. RR* /\ ( B D F ) e. RR* ) -> ( ( A D E ) = ( B D F ) <-> ( ( A D E ) <_ ( B D F ) /\ ( B D F ) <_ ( A D E ) ) ) )
92	26, 44, 91	syl2anc 693	. 2 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( ( A D E ) = ( B D F ) <-> ( ( A D E ) <_ ( B D F ) /\ ( B D F ) <_ ( A D E ) ) ) )
93	73, 90, 92	mpbir2and 957	1 |- ( ( D e. (PsMet ` X ) /\ ( A (~Met ` D ) B /\ E (~Met ` D ) F ) ) -> ( A D E ) = ( B D F ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   class class class wbr 4653   X. cxp 5112   dom cdm 5114   ran crn 5115   Rel wrel 5119   ` cfv 5888  (class class class)co 6650   0cc0 9936   RR*cxr 10073   <_ cle 10075   +ecxad 11944  PsMetcpsmet 19730  ~Metcmetid 29929

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-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

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-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-xadd 11947  df-psmet 19738  df-metid 29931

This theorem is referenced by:  pstmfval  29939