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Mirrors > Home > ILE Home > Th. List > xmspropd | Unicode version |
Description: Property deduction for an extended metric space. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
xmspropd.1 |
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xmspropd.2 |
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xmspropd.3 |
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xmspropd.4 |
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Ref | Expression |
---|---|
xmspropd |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xmspropd.1 |
. . . . 5
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2 | xmspropd.2 |
. . . . 5
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3 | 1, 2 | eqtr3d 2212 |
. . . 4
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4 | xmspropd.4 |
. . . 4
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5 | 3, 4 | tpspropd 13167 |
. . 3
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6 | xmspropd.3 |
. . . . . . 7
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7 | 1 | sqxpeqd 4648 |
. . . . . . . 8
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8 | 7 | reseq2d 4902 |
. . . . . . 7
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9 | 6, 8 | eqtr3d 2212 |
. . . . . 6
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10 | 2 | sqxpeqd 4648 |
. . . . . . 7
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11 | 10 | reseq2d 4902 |
. . . . . 6
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12 | 9, 11 | eqtr3d 2212 |
. . . . 5
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13 | 12 | fveq2d 5514 |
. . . 4
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14 | 4, 13 | eqeq12d 2192 |
. . 3
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15 | 5, 14 | anbi12d 473 |
. 2
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16 | eqid 2177 |
. . 3
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17 | eqid 2177 |
. . 3
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18 | eqid 2177 |
. . 3
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19 | 16, 17, 18 | isxms 13584 |
. 2
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20 | eqid 2177 |
. . 3
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21 | eqid 2177 |
. . 3
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22 | eqid 2177 |
. . 3
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23 | 20, 21, 22 | isxms 13584 |
. 2
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24 | 15, 19, 23 | 3bitr4g 223 |
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-coll 4115 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 |
This theorem depends on definitions: df-bi 117 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-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 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-iun 3886 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-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-f1 5216 df-fo 5217 df-f1o 5218 df-fv 5219 df-ov 5871 df-oprab 5872 df-mpo 5873 df-1st 6134 df-2nd 6135 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-ndx 12435 df-slot 12436 df-base 12438 df-tset 12524 df-rest 12625 df-topn 12626 df-top 13129 df-topon 13142 df-topsp 13162 df-xms 13472 |
This theorem is referenced by: mspropd 13611 |
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