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Mirrors > Home > ILE Home > Th. List > xmetsym | Unicode version |
Description: The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmetsym |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 997 |
. . . 4
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2 | simp3 999 |
. . . 4
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3 | simp2 998 |
. . . 4
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4 | xmettri2 13494 |
. . . 4
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5 | 1, 2, 3, 2, 4 | syl13anc 1240 |
. . 3
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6 | xmet0 13496 |
. . . . . 6
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7 | 6 | 3adant2 1016 |
. . . . 5
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8 | 7 | oveq2d 5884 |
. . . 4
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9 | xmetcl 13485 |
. . . . . 6
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10 | xaddid1 9836 |
. . . . . 6
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11 | 9, 10 | syl 14 |
. . . . 5
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12 | 11 | 3com23 1209 |
. . . 4
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13 | 8, 12 | eqtrd 2210 |
. . 3
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14 | 5, 13 | breqtrd 4026 |
. 2
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15 | xmettri2 13494 |
. . . 4
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16 | 1, 3, 2, 3, 15 | syl13anc 1240 |
. . 3
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17 | xmet0 13496 |
. . . . . 6
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18 | 17 | 3adant3 1017 |
. . . . 5
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19 | 18 | oveq2d 5884 |
. . . 4
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20 | xmetcl 13485 |
. . . . 5
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21 | xaddid1 9836 |
. . . . 5
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22 | 20, 21 | syl 14 |
. . . 4
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23 | 19, 22 | eqtrd 2210 |
. . 3
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24 | 16, 23 | breqtrd 4026 |
. 2
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25 | 9 | 3com23 1209 |
. . 3
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26 | xrletri3 9778 |
. . 3
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27 | 20, 25, 26 | syl2anc 411 |
. 2
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28 | 14, 24, 27 | mpbir2and 944 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
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 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-apti 7904 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-if 3535 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-iun 3886 df-br 4001 df-opab 4062 df-mpt 4063 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 df-map 6643 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-xadd 9747 df-xmet 13121 |
This theorem is referenced by: xmettpos 13503 metsym 13504 xmettri 13505 xmettri3 13507 elbl3 13528 blss 13561 xmeter 13569 xmssym 13602 metcnp2 13646 |
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