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| Mirrors > Home > MPE Home > Th. List > blcomps | Structured version Visualization version 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 24216 | . 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 24218 | . . 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 647 | . 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 282 | 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 setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β wcel 2105 class class class wbr 5148 βcfv 6543 (class class class)co 7412 β*cxr 11254 < clt 11255 PsMetcpsmet 21218 ballcbl 21221 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7415 df-oprab 7416 df-mpo 7417 df-1st 7979 df-2nd 7980 df-er 8709 df-map 8828 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-xadd 13100 df-psmet 21226 df-bl 21229 |
| This theorem is referenced by: elbl4 24393 |
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