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Mirrors > Home > MPE Home > Th. List > cmspropd | Structured version Visualization version Unicode version |
Description: Property deduction for a complete metric space. (Contributed by Mario Carneiro, 15-Oct-2015.) |
Ref | Expression |
---|---|
cmspropd.1 | |
cmspropd.2 | |
cmspropd.3 | |
cmspropd.4 |
Ref | Expression |
---|---|
cmspropd | CMetSp CMetSp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cmspropd.1 | . . . 4 | |
2 | cmspropd.2 | . . . 4 | |
3 | cmspropd.3 | . . . 4 | |
4 | cmspropd.4 | . . . 4 | |
5 | 1, 2, 3, 4 | mspropd 22279 | . . 3 |
6 | 1 | sqxpeqd 5141 | . . . . . . 7 |
7 | 6 | reseq2d 5396 | . . . . . 6 |
8 | 3, 7 | eqtr3d 2658 | . . . . 5 |
9 | 2 | sqxpeqd 5141 | . . . . . 6 |
10 | 9 | reseq2d 5396 | . . . . 5 |
11 | 8, 10 | eqtr3d 2658 | . . . 4 |
12 | 1, 2 | eqtr3d 2658 | . . . . 5 |
13 | 12 | fveq2d 6195 | . . . 4 |
14 | 11, 13 | eleq12d 2695 | . . 3 |
15 | 5, 14 | anbi12d 747 | . 2 |
16 | eqid 2622 | . . 3 | |
17 | eqid 2622 | . . 3 | |
18 | 16, 17 | iscms 23142 | . 2 CMetSp |
19 | eqid 2622 | . . 3 | |
20 | eqid 2622 | . . 3 | |
21 | 19, 20 | iscms 23142 | . 2 CMetSp |
22 | 15, 18, 21 | 3bitr4g 303 | 1 CMetSp CMetSp |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cxp 5112 cres 5116 cfv 5888 cbs 15857 cds 15950 ctopn 16082 cmt 22123 cms 23052 CMetSpccms 23129 |
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 |
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-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 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-res 5126 df-iota 5851 df-fun 5890 df-fv 5896 df-top 20699 df-topon 20716 df-topsp 20737 df-xms 22125 df-ms 22126 df-cms 23132 |
This theorem is referenced by: srabn 23156 |
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