Nearby theorems
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| Mirrors > Home > ILE Home > Th. List > blcomps | GIF version | ||
| Description: Commute the arguments to the ball function. (Contributed by Mario Carneiro, 22-Jan-2014.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| blcomps | β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β π β (π΄(ballβπ·)π ))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | elbl2ps 13754 | . 2 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β (ππ·π΄) < π )) | |
| 2 | elbl3ps 13756 | . . 3 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π΄ β π β§ π β π)) β (π β (π΄(ballβπ·)π ) β (ππ·π΄) < π )) | |
| 3 | 2 | ancom2s 566 | . 2 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π β (π΄(ballβπ·)π ) β (ππ·π΄) < π )) |
| 4 | 1, 3 | bitr4d 191 | 1 β’ (((π· β (PsMetβπ) β§ π β β*) β§ (π β π β§ π΄ β π)) β (π΄ β (π(ballβπ·)π ) β π β (π΄(ballβπ·)π ))) |
| Colors of variables: wff set class |
| Syntax hints: β wi 4 β§ wa 104 β wb 105 β wcel 2148 class class class wbr 4002 βcfv 5214 (class class class)co 5871 β*cxr 7986 < clt 7987 PsMetcpsmet 13299 ballcbl 13302 |
| 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-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-cnex 7898 ax-resscn 7899 ax-1re 7901 ax-addrcl 7904 ax-0id 7915 ax-rnegex 7916 ax-pre-ltirr 7919 ax-pre-apti 7922 |
| This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 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-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 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-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-iun 3888 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5176 df-fun 5216 df-fn 5217 df-f 5218 df-fv 5222 df-ov 5874 df-oprab 5875 df-mpo 5876 df-1st 6137 df-2nd 6138 df-map 6646 df-pnf 7989 df-mnf 7990 df-xr 7991 df-ltxr 7992 df-le 7993 df-xadd 9768 df-psmet 13307 df-bl 13310 |
| This theorem is referenced by: (None) |
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