Theorem xmetsym 15392

Description: The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)

Assertion

Ref	Expression
xmetsym	|- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )

Proof of Theorem xmetsym

Step	Hyp	Ref	Expression
1		simp1 1028	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> D e. ( *Met ` X ) )
2		simp3 1030	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3		simp2 1029	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4		xmettri2 15385	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( B e. X /\ A e. X /\ B e. X ) ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
5	1, 2, 3, 2, 4	syl13anc 1280	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6		xmet0 15387	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X ) -> ( B D B ) = 0 )
7	6	3adant2 1047	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
8	7	oveq2d 6091	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( ( B D A ) +e 0 ) )
9		xmetcl 15376	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10		xaddid1 10243	. . . . . 6 |- ( ( B D A ) e. RR* -> ( ( B D A ) +e 0 ) = ( B D A ) )
11	9, 10	syl 14	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
12	11	3com23 1240	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
13	8, 12	eqtrd 2271	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
14	5, 13	breqtrd 4151	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15		xmettri2 15385	. . . 4 |- ( ( D e. ( *Met ` X ) /\ ( A e. X /\ B e. X /\ A e. X ) ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
16	1, 3, 2, 3, 15	syl13anc 1280	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17		xmet0 15387	. . . . . 6 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18	17	3adant3 1048	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
19	18	oveq2d 6091	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( ( A D B ) +e 0 ) )
20		xmetcl 15376	. . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21		xaddid1 10243	. . . . 5 |- ( ( A D B ) e. RR* -> ( ( A D B ) +e 0 ) = ( A D B ) )
22	20, 21	syl 14	. . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e 0 ) = ( A D B ) )
23	19, 22	eqtrd 2271	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
24	16, 23	breqtrd 4151	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
25	9	3com23 1240	. . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26		xrletri3 10185	. . 3 |- ( ( ( A D B ) e. RR* /\ ( B D A ) e. RR* ) -> ( ( A D B ) = ( B D A ) <-> ( ( A D B ) <_ ( B D A ) /\ ( B D A ) <_ ( A D B ) ) ) )
27	20, 25, 26	syl2anc 415	. 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) = ( B D A ) <-> ( ( A D B ) <_ ( B D A ) /\ ( B D A ) <_ ( A D B ) ) ) )
28	14, 24, 27	mpbir2and 957	1 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   class class class wbr 4125   ` cfv 5372  (class class class)co 6075   0cc0 8169   RR*cxr 8349   <_ cle 8351   +ecxad 10151   *Metcxmet 14845

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261  ax-1re 8263  ax-addrcl 8266  ax-0id 8277  ax-rnegex 8278  ax-pre-ltirr 8281  ax-pre-apti 8284

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-if 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-map 6914  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-xadd 10154  df-xmet 14853

This theorem is referenced by:  xmettpos  15394  metsym  15395  xmettri  15396  xmettri3  15398  elbl3  15419  blss  15452  xmeter  15460  xmssym  15493  metcnp2  15537