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Mirrors > Home > MPE Home > Th. List > blss2ps | Structured version Unicode version |
Description: One ball is contained in another if the center-to-center distance is less than the difference of the radii. (Contributed by Mario Carneiro, 15-Jan-2014.) (Revised by Mario Carneiro, 23-Aug-2015.) (Revised by Thierry Arnoux, 11-Mar-2018.) |
Ref | Expression |
---|---|
blss2ps |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl1 991 |
. 2
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2 | simpl2 992 |
. 2
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3 | simpl3 993 |
. 2
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4 | simpr1 994 |
. . 3
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5 | 4 | rexrd 9545 |
. 2
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6 | simpr2 995 |
. . 3
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7 | 6 | rexrd 9545 |
. 2
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8 | 6, 4 | resubcld 9888 |
. . 3
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9 | simpr3 996 |
. . 3
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10 | psmetlecl 20024 |
. . 3
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11 | 1, 2, 3, 8, 9, 10 | syl122anc 1228 |
. 2
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12 | rexsub 11315 |
. . . 4
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13 | 6, 4, 12 | syl2anc 661 |
. . 3
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14 | 9, 13 | breqtrrd 4427 |
. 2
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15 | 1, 2, 3, 5, 7, 11, 14 | xblss2ps 20109 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1592 ax-4 1603 ax-5 1671 ax-6 1710 ax-7 1730 ax-8 1760 ax-9 1762 ax-10 1777 ax-11 1782 ax-12 1794 ax-13 1955 ax-ext 2432 ax-sep 4522 ax-nul 4530 ax-pow 4579 ax-pr 4640 ax-un 6483 ax-cnex 9450 ax-resscn 9451 ax-1cn 9452 ax-icn 9453 ax-addcl 9454 ax-addrcl 9455 ax-mulcl 9456 ax-mulrcl 9457 ax-mulcom 9458 ax-addass 9459 ax-mulass 9460 ax-distr 9461 ax-i2m1 9462 ax-1ne0 9463 ax-1rid 9464 ax-rnegex 9465 ax-rrecex 9466 ax-cnre 9467 ax-pre-lttri 9468 ax-pre-lttrn 9469 ax-pre-ltadd 9470 ax-pre-mulgt0 9471 |
This theorem depends on definitions: df-bi 185 df-or 370 df-an 371 df-3or 966 df-3an 967 df-tru 1373 df-ex 1588 df-nf 1591 df-sb 1703 df-eu 2266 df-mo 2267 df-clab 2440 df-cleq 2446 df-clel 2449 df-nfc 2604 df-ne 2650 df-nel 2651 df-ral 2804 df-rex 2805 df-reu 2806 df-rmo 2807 df-rab 2808 df-v 3080 df-sbc 3295 df-csb 3397 df-dif 3440 df-un 3442 df-in 3444 df-ss 3451 df-nul 3747 df-if 3901 df-pw 3971 df-sn 3987 df-pr 3989 df-op 3993 df-uni 4201 df-iun 4282 df-br 4402 df-opab 4460 df-mpt 4461 df-id 4745 df-po 4750 df-so 4751 df-xp 4955 df-rel 4956 df-cnv 4957 df-co 4958 df-dm 4959 df-rn 4960 df-res 4961 df-ima 4962 df-iota 5490 df-fun 5529 df-fn 5530 df-f 5531 df-f1 5532 df-fo 5533 df-f1o 5534 df-fv 5535 df-riota 6162 df-ov 6204 df-oprab 6205 df-mpt2 6206 df-1st 6688 df-2nd 6689 df-er 7212 df-map 7327 df-en 7422 df-dom 7423 df-sdom 7424 df-pnf 9532 df-mnf 9533 df-xr 9534 df-ltxr 9535 df-le 9536 df-sub 9709 df-neg 9710 df-div 10106 df-2 10492 df-rp 11104 df-xneg 11201 df-xadd 11202 df-xmul 11203 df-psmet 17935 df-bl 17938 |
This theorem is referenced by: blssps 20132 |
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