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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 |
|
| setsms.d |
|
| setsms.k |
|
| setsmsbasg.m |
|
| setsmsbasg.d |
|
| Ref | Expression |
|---|---|
| setsmsdsg |
|
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | setsmsbasg.m |
. . 3
| |
| 2 | setsmsbasg.d |
. . 3
| |
| 3 | dsslid 13049 |
. . . 4
| |
| 4 | 9re 9123 |
. . . . . 6
| |
| 5 | 1nn 9047 |
. . . . . . 7
| |
| 6 | 2nn0 9312 |
. . . . . . 7
| |
| 7 | 9nn0 9319 |
. . . . . . 7
| |
| 8 | 9lt10 9634 |
. . . . . . 7
| |
| 9 | 5, 6, 7, 8 | declti 9541 |
. . . . . 6
|
| 10 | 4, 9 | gtneii 8168 |
. . . . 5
|
| 11 | dsndx 13047 |
. . . . . 6
| |
| 12 | tsetndx 13018 |
. . . . . 6
| |
| 13 | 11, 12 | neeq12i 2393 |
. . . . 5
|
| 14 | 10, 13 | mpbir 146 |
. . . 4
|
| 15 | tsetslid 13020 |
. . . . 5
| |
| 16 | 15 | simpri 113 |
. . . 4
|
| 17 | 3, 14, 16 | setsslnid 12884 |
. . 3
|
| 18 | 1, 2, 17 | syl2anc 411 |
. 2
|
| 19 | setsms.k |
. . 3
| |
| 20 | 19 | fveq2d 5580 |
. 2
|
| 21 | 18, 20 | eqtr4d 2241 |
1
|
| 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 615 ax-in2 616 ax-io 711 ax-5 1470 ax-7 1471 ax-gen 1472 ax-ie1 1516 ax-ie2 1517 ax-8 1527 ax-10 1528 ax-11 1529 ax-i12 1530 ax-bndl 1532 ax-4 1533 ax-17 1549 ax-i9 1553 ax-ial 1557 ax-i5r 1558 ax-13 2178 ax-14 2179 ax-ext 2187 ax-sep 4162 ax-pow 4218 ax-pr 4253 ax-un 4480 ax-setind 4585 ax-cnex 8016 ax-resscn 8017 ax-1cn 8018 ax-1re 8019 ax-icn 8020 ax-addcl 8021 ax-addrcl 8022 ax-mulcl 8023 ax-mulrcl 8024 ax-addcom 8025 ax-mulcom 8026 ax-addass 8027 ax-mulass 8028 ax-distr 8029 ax-i2m1 8030 ax-0lt1 8031 ax-1rid 8032 ax-0id 8033 ax-rnegex 8034 ax-precex 8035 ax-cnre 8036 ax-pre-ltirr 8037 ax-pre-ltwlin 8038 ax-pre-lttrn 8039 ax-pre-ltadd 8041 ax-pre-mulgt0 8042 |
| This theorem depends on definitions: df-bi 117 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1484 df-sb 1786 df-eu 2057 df-mo 2058 df-clab 2192 df-cleq 2198 df-clel 2201 df-nfc 2337 df-ne 2377 df-nel 2472 df-ral 2489 df-rex 2490 df-reu 2491 df-rab 2493 df-v 2774 df-sbc 2999 df-dif 3168 df-un 3170 df-in 3172 df-ss 3179 df-nul 3461 df-pw 3618 df-sn 3639 df-pr 3640 df-op 3642 df-uni 3851 df-int 3886 df-br 4045 df-opab 4106 df-mpt 4107 df-id 4340 df-xp 4681 df-rel 4682 df-cnv 4683 df-co 4684 df-dm 4685 df-rn 4686 df-res 4687 df-iota 5232 df-fun 5273 df-fv 5279 df-riota 5899 df-ov 5947 df-oprab 5948 df-mpo 5949 df-pnf 8109 df-mnf 8110 df-xr 8111 df-ltxr 8112 df-le 8113 df-sub 8245 df-neg 8246 df-inn 9037 df-2 9095 df-3 9096 df-4 9097 df-5 9098 df-6 9099 df-7 9100 df-8 9101 df-9 9102 df-n0 9296 df-z 9373 df-dec 9505 df-ndx 12835 df-slot 12836 df-sets 12839 df-tset 12928 df-ds 12931 |
| This theorem is referenced by: (None) |
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