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Mirrors > Home > MPE Home > Th. List > ismet2 | Structured version Visualization version Unicode version |
Description: An extended metric is a metric exactly when it takes real values for all values of the arguments. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
ismet2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvex 6221 | . 2 | |
2 | elfvex 6221 | . . 3 | |
3 | 2 | adantr 481 | . 2 |
4 | simpllr 799 | . . . . . . . . . . . 12 | |
5 | simpr 477 | . . . . . . . . . . . 12 | |
6 | simplrl 800 | . . . . . . . . . . . 12 | |
7 | 4, 5, 6 | fovrnd 6806 | . . . . . . . . . . 11 |
8 | simplrr 801 | . . . . . . . . . . . 12 | |
9 | 4, 5, 8 | fovrnd 6806 | . . . . . . . . . . 11 |
10 | rexadd 12063 | . . . . . . . . . . 11 | |
11 | 7, 9, 10 | syl2anc 693 | . . . . . . . . . 10 |
12 | 11 | breq2d 4665 | . . . . . . . . 9 |
13 | 12 | ralbidva 2985 | . . . . . . . 8 |
14 | 13 | anbi2d 740 | . . . . . . 7 |
15 | 14 | 2ralbidva 2988 | . . . . . 6 |
16 | simpr 477 | . . . . . . . 8 | |
17 | ressxr 10083 | . . . . . . . 8 | |
18 | fss 6056 | . . . . . . . 8 | |
19 | 16, 17, 18 | sylancl 694 | . . . . . . 7 |
20 | 19 | biantrurd 529 | . . . . . 6 |
21 | 15, 20 | bitr3d 270 | . . . . 5 |
22 | 21 | pm5.32da 673 | . . . 4 |
23 | ancom 466 | . . . 4 | |
24 | 22, 23 | syl6bb 276 | . . 3 |
25 | ismet 22128 | . . 3 | |
26 | isxmet 22129 | . . . 4 | |
27 | 26 | anbi1d 741 | . . 3 |
28 | 24, 25, 27 | 3bitr4d 300 | . 2 |
29 | 1, 3, 28 | pm5.21nii 368 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 cvv 3200 wss 3574 class class class wbr 4653 cxp 5112 wf 5884 cfv 5888 (class class class)co 6650 cr 9935 cc0 9936 caddc 9939 cxr 10073 cle 10075 cxad 11944 cxmt 19731 cme 19732 |
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-mulcl 9998 ax-i2m1 10004 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 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-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-xadd 11947 df-xmet 19739 df-met 19740 |
This theorem is referenced by: metxmet 22139 metres2 22168 prdsmet 22175 imasf1omet 22181 xmetresbl 22242 stdbdmet 22321 isbndx 33581 |
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