Theorem ssbl 7838

Description: The size of a ball increases monotonically with its radius.

Hypothesis

Ref	Expression
ssbl.1	|- X = dom dom D

Assertion

Ref	Expression
ssbl	|- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (P( ball ` D)R) (_ (P( ball ` D)S))

Proof of Theorem ssbl

Step	Hyp	Ref	Expression
1		ltletrt 5511	. . . . . . . . . . . . . . . 16 |- (((PDx) e. RR /\ R e. RR /\ S e. RR) -> (((PDx) < R /\ R <_ S) -> (PDx) < S))
2		ssbl.1	. . . . . . . . . . . . . . . . . . 19 |- X = dom dom D
3	2	metcl 7790	. . . . . . . . . . . . . . . . . 18 |- ((D e. Met /\ P e. X /\ x e. X) -> (PDx) e. RR)
4	3	3expa 832	. . . . . . . . . . . . . . . . 17 |- (((D e. Met /\ P e. X) /\ x e. X) -> (PDx) e. RR)
5	4	ancoms 436	. . . . . . . . . . . . . . . 16 |- ((x e. X /\ (D e. Met /\ P e. X)) -> (PDx) e. RR)
6	1, 5	syl3an1 858	. . . . . . . . . . . . . . 15 |- (((x e. X /\ (D e. Met /\ P e. X)) /\ R e. RR /\ S e. RR) -> (((PDx) < R /\ R <_ S) -> (PDx) < S))
7	6	exp3a 375	. . . . . . . . . . . . . 14 |- (((x e. X /\ (D e. Met /\ P e. X)) /\ R e. RR /\ S e. RR) -> ((PDx) < R -> (R <_ S -> (PDx) < S)))
8	7	com23 32	. . . . . . . . . . . . 13 |- (((x e. X /\ (D e. Met /\ P e. X)) /\ R e. RR /\ S e. RR) -> (R <_ S -> ((PDx) < R -> (PDx) < S)))
9	8	3expb 833	. . . . . . . . . . . 12 |- (((x e. X /\ (D e. Met /\ P e. X)) /\ (R e. RR /\ S e. RR)) -> (R <_ S -> ((PDx) < R -> (PDx) < S)))
10	9	exp31 376	. . . . . . . . . . 11 |- (x e. X -> ((D e. Met /\ P e. X) -> ((R e. RR /\ S e. RR) -> (R <_ S -> ((PDx) < R -> (PDx) < S)))))
11	10	com4l 39	. . . . . . . . . 10 |- ((D e. Met /\ P e. X) -> ((R e. RR /\ S e. RR) -> (R <_ S -> (x e. X -> ((PDx) < R -> (PDx) < S)))))
12	11	imp31 362	. . . . . . . . 9 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ S e. RR)) /\ R <_ S) -> (x e. X -> ((PDx) < R -> (PDx) < S)))
13	12	imdistand 445	. . . . . . . 8 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ S e. RR)) /\ R <_ S) -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S)))
14	13	ex 373	. . . . . . 7 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ S e. RR)) -> (R <_ S -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S))))
15	14	3impb 828	. . . . . 6 |- (((D e. Met /\ P e. X) /\ R e. RR /\ S e. RR) -> (R <_ S -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S))))
16	15	3adant2r 854	. . . . 5 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ S e. RR) -> (R <_ S -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S))))
17	16	3adant3r 856	. . . 4 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> (R <_ S -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S))))
18	17	imp 350	. . 3 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> ((x e. X /\ (PDx) < R) -> (x e. X /\ (PDx) < S)))
19	2	elbl 7819	. . . . 5 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R)) -> (x e. (P( ball ` D)R) <-> (x e. X /\ (PDx) < R)))
20	19	3adant3 798	. . . 4 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> (x e. (P( ball ` D)R) <-> (x e. X /\ (PDx) < R)))
21	20	adantr 389	. . 3 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (x e. (P( ball ` D)R) <-> (x e. X /\ (PDx) < R)))
22	2	elbl 7819	. . . . 5 |- (((D e. Met /\ P e. X) /\ (S e. RR /\ 0 < S)) -> (x e. (P( ball ` D)S) <-> (x e. X /\ (PDx) < S)))
23	22	3adant2 797	. . . 4 |- (((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) -> (x e. (P( ball ` D)S) <-> (x e. X /\ (PDx) < S)))
24	23	adantr 389	. . 3 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (x e. (P( ball ` D)S) <-> (x e. X /\ (PDx) < S)))
25	18, 21, 24	3imtr4d 542	. 2 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (x e. (P( ball ` D)R) -> x e. (P( ball ` D)S)))
26	25	ssrdv 2068	1 |- ((((D e. Met /\ P e. X) /\ (R e. RR /\ 0 < R) /\ (S e. RR /\ 0 < S)) /\ R <_ S) -> (P( ball ` D)R) (_ (P( ball ` D)S))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 774   = wceq 955   e. wcel 957   (_ wss 2045   class class class wbr 2616  dom cdm 3167  ` cfv 3179  (class class class)co 3960  RRcr 5220  0cc0 5221   <_ cle 5282   < clt 5473  Metcme 7768   ball cbl 7770

This theorem is referenced by:  ssblex 7839  bcthlem18 7999

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776  ax-un 2863  ax-inf2 4612

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-nel 1587  df-ral 1648  df-rex 1649  df-reu 1650  df-rab 1651  df-v 1810  df-sbc 1940  df-csb 2000  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-pss 2053  df-nul 2279  df-if 2360  df-pw 2400  df-sn 2410  df-pr 2411  df-tp 2413  df-op 2414  df-uni 2501  df-int 2531  df-iun 2565  df-br 2617  df-opab 2664  df-tr 2678  df-eprel 2829  df-id 2832  df-po 2837  df-so 2847  df-fr 2914  df-we 2931  df-ord 2948  df-on 2949  df-lim 2950  df-suc 2951  df-om 3129  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-fv 3195  df-rdg 3929  df-opr 3962  df-oprab 3963  df-1st 4076  df-2nd 4077  df-1o 4130  df-oadd 4132  df-omul 4133  df-er 4258  df-ec 4260  df-qs 4263  df-en 4364  df-dom 4365  df-sdom 4366  df-ni 4987  df-pli 4988  df-mi 4989  df-lti 4990  df-plpq 5022  df-mpq 5023  df-enq 5024  df-nq 5025  df-plq 5026  df-mq 5027  df-rq 5028  df-ltq 5029  df-1q 5030  df-np 5073  df-1p 5074  df-plp 5075  df-ltp 5077  df-enr 5153  df-nr 5154  df-ltr 5157  df-0r 5158  df-c 5227  df-r 5231  df-lt 5234  df-pnf 5474  df-mnf 5475  df-xr 5476  df-ltxr 5477  df-le 5478  df-met 7772  df-bl 7774

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