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Mirrors > Home > MPE Home > Th. List > xmeteq0 | Structured version Unicode version |
Description: The value of an extended metric is zero iff its arguments are equal. (Contributed by Mario Carneiro, 20-Aug-2015.) |
Ref | Expression |
---|---|
xmeteq0 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elfvdm 5828 |
. . . . . . 7
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
2 | isxmet 20034 |
. . . . . . 7
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3 | 1, 2 | syl 16 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
4 | 3 | ibi 241 |
. . . . 5
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5 | 4 | simprd 463 |
. . . 4
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6 | simpl 457 |
. . . . . 6
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7 | 6 | ralimi 2819 |
. . . . 5
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8 | 7 | ralimi 2819 |
. . . 4
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9 | 5, 8 | syl 16 |
. . 3
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10 | oveq1 6210 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
11 | 10 | eqeq1d 2456 |
. . . . 5
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | eqeq1 2458 |
. . . . 5
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13 | 11, 12 | bibi12d 321 |
. . . 4
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14 | oveq2 6211 |
. . . . . 6
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
15 | 14 | eqeq1d 2456 |
. . . . 5
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16 | eqeq2 2469 |
. . . . 5
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17 | 15, 16 | bibi12d 321 |
. . . 4
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18 | 13, 17 | rspc2v 3186 |
. . 3
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19 | 9, 18 | syl5com 30 |
. 2
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20 | 19 | 3impib 1186 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4524 ax-nul 4532 ax-pow 4581 ax-pr 4642 ax-un 6485 ax-cnex 9452 ax-resscn 9453 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-ral 2804 df-rex 2805 df-rab 2808 df-v 3080 df-sbc 3295 df-dif 3442 df-un 3444 df-in 3446 df-ss 3453 df-nul 3749 df-if 3903 df-pw 3973 df-sn 3989 df-pr 3991 df-op 3995 df-uni 4203 df-br 4404 df-opab 4462 df-mpt 4463 df-id 4747 df-xp 4957 df-rel 4958 df-cnv 4959 df-co 4960 df-dm 4961 df-rn 4962 df-iota 5492 df-fun 5531 df-fn 5532 df-f 5533 df-fv 5537 df-ov 6206 df-oprab 6207 df-mpt2 6208 df-map 7329 df-xr 9536 df-xmet 17938 |
This theorem is referenced by: meteq0 20049 xmet0 20052 xmetgt0 20068 xmetres2 20071 prdsxmetlem 20078 imasf1oxmet 20085 xblss2 20112 xmseq0 20174 comet 20223 |
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