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Mirrors > Home > ILE Home > Th. List > blfvalps | Unicode version |
Description: The value of the ball function. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 11-Nov-2013.) (Revised by Thierry Arnoux, 11-Feb-2018.) |
Ref | Expression |
---|---|
blfvalps |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-bl 13876 |
. . 3
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2 | 1 | a1i 9 |
. 2
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3 | dmeq 4845 |
. . . . 5
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4 | 3 | dmeqd 4847 |
. . . 4
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5 | psmetdmdm 14301 |
. . . . 5
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6 | 5 | eqcomd 2195 |
. . . 4
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7 | 4, 6 | sylan9eqr 2244 |
. . 3
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8 | eqidd 2190 |
. . 3
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9 | simpr 110 |
. . . . . 6
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10 | 9 | oveqd 5914 |
. . . . 5
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11 | 10 | breq1d 4028 |
. . . 4
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12 | 7, 11 | rabeqbidv 2747 |
. . 3
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13 | 7, 8, 12 | mpoeq123dv 5959 |
. 2
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14 | elex 2763 |
. 2
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15 | ssrab2 3255 |
. . . . . 6
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16 | psmetrel 14299 |
. . . . . . . . 9
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17 | relelfvdm 5566 |
. . . . . . . . 9
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18 | 16, 17 | mpan 424 |
. . . . . . . 8
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19 | 18 | adantr 276 |
. . . . . . 7
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20 | elpw2g 4174 |
. . . . . . 7
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21 | 19, 20 | syl 14 |
. . . . . 6
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22 | 15, 21 | mpbiri 168 |
. . . . 5
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23 | 22 | ralrimivva 2572 |
. . . 4
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24 | eqid 2189 |
. . . . 5
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25 | 24 | fmpo 6227 |
. . . 4
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26 | 23, 25 | sylib 122 |
. . 3
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27 | xrex 9888 |
. . . 4
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28 | xpexg 4758 |
. . . 4
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29 | 18, 27, 28 | sylancl 413 |
. . 3
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30 | 18 | pwexd 4199 |
. . 3
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31 | fex2 5403 |
. . 3
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32 | 26, 29, 30, 31 | syl3anc 1249 |
. 2
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33 | 2, 13, 14, 32 | fvmptd 5618 |
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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 ax-un 4451 ax-setind 4554 ax-cnex 7933 ax-resscn 7934 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-ral 2473 df-rex 2474 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-id 4311 df-xp 4650 df-rel 4651 df-cnv 4652 df-co 4653 df-dm 4654 df-rn 4655 df-res 4656 df-ima 4657 df-iota 5196 df-fun 5237 df-fn 5238 df-f 5239 df-fv 5243 df-ov 5900 df-oprab 5901 df-mpo 5902 df-1st 6166 df-2nd 6167 df-map 6677 df-pnf 8025 df-mnf 8026 df-xr 8027 df-psmet 13873 df-bl 13876 |
This theorem is referenced by: blfval 14363 blvalps 14365 blfps 14386 |
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