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Mirrors > Home > ILE Home > Th. List > setsmsdsg | Unicode version |
Description: The distance function of a constructed metric space. (Contributed by Mario Carneiro, 28-Aug-2015.) |
Ref | Expression |
---|---|
setsms.x |
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setsms.d |
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setsms.k |
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setsmsbasg.m |
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setsmsbasg.d |
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Ref | Expression |
---|---|
setsmsdsg |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | setsmsbasg.m |
. . 3
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2 | setsmsbasg.d |
. . 3
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3 | dsslid 12614 |
. . . 4
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4 | 9re 8982 |
. . . . . 6
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5 | 1nn 8906 |
. . . . . . 7
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6 | 2nn0 9169 |
. . . . . . 7
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7 | 9nn0 9176 |
. . . . . . 7
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8 | 9lt10 9490 |
. . . . . . 7
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9 | 5, 6, 7, 8 | declti 9397 |
. . . . . 6
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10 | 4, 9 | gtneii 8030 |
. . . . 5
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11 | dsndx 12612 |
. . . . . 6
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12 | tsetndx 12595 |
. . . . . 6
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13 | 11, 12 | neeq12i 2364 |
. . . . 5
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14 | 10, 13 | mpbir 146 |
. . . 4
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15 | tsetslid 12597 |
. . . . 5
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16 | 15 | simpri 113 |
. . . 4
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17 | 3, 14, 16 | setsslnid 12483 |
. . 3
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18 | 1, 2, 17 | syl2anc 411 |
. 2
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19 | setsms.k |
. . 3
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20 | 19 | fveq2d 5514 |
. 2
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21 | 18, 20 | eqtr4d 2213 |
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-1cn 7882 ax-1re 7883 ax-icn 7884 ax-addcl 7885 ax-addrcl 7886 ax-mulcl 7887 ax-mulrcl 7888 ax-addcom 7889 ax-mulcom 7890 ax-addass 7891 ax-mulass 7892 ax-distr 7893 ax-i2m1 7894 ax-0lt1 7895 ax-1rid 7896 ax-0id 7897 ax-rnegex 7898 ax-precex 7899 ax-cnre 7900 ax-pre-ltirr 7901 ax-pre-ltwlin 7902 ax-pre-lttrn 7903 ax-pre-ltadd 7905 ax-pre-mulgt0 7906 |
This theorem depends on definitions: df-bi 117 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-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 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-iota 5173 df-fun 5213 df-fv 5219 df-riota 5824 df-ov 5871 df-oprab 5872 df-mpo 5873 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-sub 8107 df-neg 8108 df-inn 8896 df-2 8954 df-3 8955 df-4 8956 df-5 8957 df-6 8958 df-7 8959 df-8 8960 df-9 8961 df-n0 9153 df-z 9230 df-dec 9361 df-ndx 12435 df-slot 12436 df-sets 12439 df-tset 12524 df-ds 12527 |
This theorem is referenced by: (None) |
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