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| Mirrors > Home > MPE Home > Th. List > ssblps | Structured version Visualization version GIF version | ||
| Description: The size of a ball increases monotonically with its radius. (Contributed by NM, 20-Sep-2007.) (Revised by Mario Carneiro, 24-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
| Ref | Expression |
|---|---|
| ssblps | β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (π(ballβπ·)π ) β (π(ballβπ·)π)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simp1l 1196 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β π· β (PsMetβπ)) | |
| 2 | simp1r 1197 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β π β π) | |
| 3 | simp2l 1198 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β π β β*) | |
| 4 | simp2r 1199 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β π β β*) | |
| 5 | psmet0 24134 | . . . 4 β’ ((π· β (PsMetβπ) β§ π β π) β (ππ·π) = 0) | |
| 6 | 5 | 3ad2ant1 1132 | . . 3 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (ππ·π) = 0) |
| 7 | 0re 11223 | . . 3 β’ 0 β β | |
| 8 | 6, 7 | eqeltrdi 2840 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (ππ·π) β β) |
| 9 | simp3 1137 | . . . 4 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β π β€ π) | |
| 10 | xsubge0 13247 | . . . . 5 β’ ((π β β* β§ π β β*) β (0 β€ (π +π -ππ ) β π β€ π)) | |
| 11 | 4, 3, 10 | syl2anc 583 | . . . 4 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (0 β€ (π +π -ππ ) β π β€ π)) |
| 12 | 9, 11 | mpbird 257 | . . 3 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β 0 β€ (π +π -ππ )) |
| 13 | 6, 12 | eqbrtrd 5170 | . 2 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (ππ·π) β€ (π +π -ππ )) |
| 14 | 1, 2, 2, 3, 4, 8, 13 | xblss2ps 24227 | 1 β’ (((π· β (PsMetβπ) β§ π β π) β§ (π β β* β§ π β β*) β§ π β€ π) β (π(ballβπ·)π ) β (π(ballβπ·)π)) |
| Colors of variables: wff setvar class |
| Syntax hints: β wi 4 β wb 205 β§ wa 395 β§ w3a 1086 = wceq 1540 β wcel 2105 β wss 3948 class class class wbr 5148 βcfv 6543 (class class class)co 7412 βcr 11115 0cc0 11116 β*cxr 11254 β€ cle 11256 -πcxne 13096 +π cxad 13097 PsMetcpsmet 21217 ballcbl 21220 |
| 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 ax-pre-mulgt0 11193 |
| 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-rmo 3375 df-reu 3376 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-riota 7368 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-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-rp 12982 df-xneg 13099 df-xadd 13100 df-xmul 13101 df-psmet 21225 df-bl 21228 |
| This theorem is referenced by: blssps 24250 |
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