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Theorem abs3lem 8159
Description: Lemma involving absolute value of differences.
Assertion
Ref Expression
abs3lem |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. RR)) -> (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D))
Proof of Theorem abs3lem
StepHypRef Expression
1 opreq1 4889 . . . . . 6 |- (A = if(A e. CC, A, 0) -> (A - C) = (if(A e. CC, A, 0) - C))
21fveq2d 4685 . . . . 5 |- (A = if(A e. CC, A, 0) -> (abs` (A - C)) = (abs` (if(A e. CC, A, 0) - C)))
32breq1d 3348 . . . 4 |- (A = if(A e. CC, A, 0) -> ((abs` (A - C)) < (D / 2) <-> (abs` (if(A e. CC, A, 0) - C)) < (D / 2)))
43anbi1d 679 . . 3 |- (A = if(A e. CC, A, 0) -> (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) <-> ((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2))))
5 opreq1 4889 . . . . 5 |- (A = if(A e. CC, A, 0) -> (A - B) = (if(A e. CC, A, 0) - B))
65fveq2d 4685 . . . 4 |- (A = if(A e. CC, A, 0) -> (abs` (A - B)) = (abs` (if(A e. CC, A, 0) - B)))
76breq1d 3348 . . 3 |- (A = if(A e. CC, A, 0) -> ((abs` (A - B)) < D <-> (abs` (if(A e. CC, A, 0) - B)) < D))
84, 7imbi12d 688 . 2 |- (A = if(A e. CC, A, 0) -> ((((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D) <-> (((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - B)) < D)))
9 opreq2 4890 . . . . . 6 |- (B = if(B e. CC, B, 0) -> (C - B) = (C - if(B e. CC, B, 0)))
109fveq2d 4685 . . . . 5 |- (B = if(B e. CC, B, 0) -> (abs` (C - B)) = (abs` (C - if(B e. CC, B, 0))))
1110breq1d 3348 . . . 4 |- (B = if(B e. CC, B, 0) -> ((abs` (C - B)) < (D / 2) <-> (abs` (C - if(B e. CC, B, 0))) < (D / 2)))
1211anbi2d 678 . . 3 |- (B = if(B e. CC, B, 0) -> (((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) <-> ((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - if(B e. CC, B, 0))) < (D / 2))))
13 opreq2 4890 . . . . 5 |- (B = if(B e. CC, B, 0) -> (if(A e. CC, A, 0) - B) = (if(A e. CC, A, 0) - if(B e. CC, B, 0)))
1413fveq2d 4685 . . . 4 |- (B = if(B e. CC, B, 0) -> (abs` (if(A e. CC, A, 0) - B)) = (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))))
1514breq1d 3348 . . 3 |- (B = if(B e. CC, B, 0) -> ((abs` (if(A e. CC, A, 0) - B)) < D <-> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D))
1612, 15imbi12d 688 . 2 |- (B = if(B e. CC, B, 0) -> ((((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - B)) < D) <-> (((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - if(B e. CC, B, 0))) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D)))
17 opreq2 4890 . . . . . 6 |- (C = if(C e. CC, C, 0) -> (if(A e. CC, A, 0) - C) = (if(A e. CC, A, 0) - if(C e. CC, C, 0)))
1817fveq2d 4685 . . . . 5 |- (C = if(C e. CC, C, 0) -> (abs` (if(A e. CC, A, 0) - C)) = (abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))))
1918breq1d 3348 . . . 4 |- (C = if(C e. CC, C, 0) -> ((abs` (if(A e. CC, A, 0) - C)) < (D / 2) <-> (abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2)))
20 opreq1 4889 . . . . . 6 |- (C = if(C e. CC, C, 0) -> (C - if(B e. CC, B, 0)) = (if(C e. CC, C, 0) - if(B e. CC, B, 0)))
2120fveq2d 4685 . . . . 5 |- (C = if(C e. CC, C, 0) -> (abs` (C - if(B e. CC, B, 0))) = (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))))
2221breq1d 3348 . . . 4 |- (C = if(C e. CC, C, 0) -> ((abs` (C - if(B e. CC, B, 0))) < (D / 2) <-> (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2)))
2319, 22anbi12d 690 . . 3 |- (C = if(C e. CC, C, 0) -> (((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - if(B e. CC, B, 0))) < (D / 2)) <-> ((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2))))
2423imbi1d 675 . 2 |- (C = if(C e. CC, C, 0) -> ((((abs` (if(A e. CC, A, 0) - C)) < (D / 2) /\ (abs` (C - if(B e. CC, B, 0))) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D) <-> (((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D)))
25 opreq1 4889 . . . . 5 |- (D = if(D e. RR, D, 0) -> (D / 2) = (if(D e. RR, D, 0) / 2))
2625breq2d 3350 . . . 4 |- (D = if(D e. RR, D, 0) -> ((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2) <-> (abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (if(D e. RR, D, 0) / 2)))
2725breq2d 3350 . . . 4 |- (D = if(D e. RR, D, 0) -> ((abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2) <-> (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (if(D e. RR, D, 0) / 2)))
2826, 27anbi12d 690 . . 3 |- (D = if(D e. RR, D, 0) -> (((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2)) <-> ((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (if(D e. RR, D, 0) / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (if(D e. RR, D, 0) / 2))))
29 breq2 3342 . . 3 |- (D = if(D e. RR, D, 0) -> ((abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D <-> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < if(D e. RR, D, 0)))
3028, 29imbi12d 688 . 2 |- (D = if(D e. RR, D, 0) -> ((((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (D / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (D / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < D) <-> (((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (if(D e. RR, D, 0) / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (if(D e. RR, D, 0) / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < if(D e. RR, D, 0))))
31 0cn 6481 . . . 4 |- 0 e. CC
3231elimel 3025 . . 3 |- if(A e. CC, A, 0) e. CC
3331elimel 3025 . . 3 |- if(B e. CC, B, 0) e. CC
3431elimel 3025 . . 3 |- if(C e. CC, C, 0) e. CC
35 0re 6603 . . . 4 |- 0 e. RR
3635elimel 3025 . . 3 |- if(D e. RR, D, 0) e. RR
3732, 33, 34, 36abs3lemi 8153 . 2 |- (((abs` (if(A e. CC, A, 0) - if(C e. CC, C, 0))) < (if(D e. RR, D, 0) / 2) /\ (abs` (if(C e. CC, C, 0) - if(B e. CC, B, 0))) < (if(D e. RR, D, 0) / 2)) -> (abs` (if(A e. CC, A, 0) - if(B e. CC, B, 0))) < if(D e. RR, D, 0))
388, 16, 24, 30, 37dedth4h 3017 1 |- (((A e. CC /\ B e. CC) /\ (C e. CC /\ D e. RR)) -> (((abs` (A - C)) < (D / 2) /\ (abs` (C - B)) < (D / 2)) -> (abs` (A - B)) < D))
Colors of variables: wff set class
Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  ifcif 2982   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   - cmin 6445   / cdiv 6447   < clt 6653  2c2 7145  abscabs 8000
This theorem is referenced by:  climunii 8358
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-inf2 5731
This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-nel 2020  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-mpt 5006  df-1st 5020  df-2nd 5021  df-iota 5089  df-rdg 5140  df-1o 5177  df-oadd 5179  df-omul 5180  df-er 5318  df-ec 5320  df-qs 5323  df-en 5427  df-dom 5428  df-sdom 5429  df-undef 5556  df-riota 5560  df-sup 5664  df-ni 6152  df-pli 6153  df-mi 6154  df-lti 6155  df-plpq 6187  df-mpq 6188  df-enq 6189  df-nq 6190  df-plq 6191  df-mq 6192  df-rq 6193  df-ltq 6194  df-1q 6195  df-np 6238  df-1p 6239  df-plp 6240  df-mp 6241  df-ltp 6242  df-plpr 6316  df-mpr 6317  df-enr 6318  df-nr 6319  df-plr 6320  df-mr 6321  df-ltr 6322  df-0r 6323  df-1r 6324  df-m1r 6325  df-c 6392  df-0 6393  df-1 6394  df-i 6395  df-r 6396  df-plus 6397  df-mul 6398  df-lt 6399  df-sub 6511  df-neg 6513  df-pnf 6654  df-mnf 6655  df-xr 6656  df-ltxr 6657  df-le 6658  df-div 6892  df-n 7108  df-2 7154  df-n0 7309  df-z 7345  df-seq1 7721  df-exp 7812  df-sqr 7920  df-re 8001  df-im 8002  df-cj 8003  df-abs 8004
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