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Related theorems
Unicode version

Theorem 2wsms 15008
Description: Two ways to state the midpoint of a segment.
Assertion
Ref Expression
2wsms |- ((A e. RR /\ B e. RR /\ A < B) -> ((A + B) / 2) = (B - ((abs` (A - B)) / 2)))

Proof of Theorem 2wsms
StepHypRef Expression
1 3simpa 872 . . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A < B) -> (A e. RR /\ B e. RR))
2 recn 6466 . . . . . . . . . . 11 |- (A e. RR -> A e. CC)
3 recn 6466 . . . . . . . . . . 11 |- (B e. RR -> B e. CC)
42, 3anim12i 360 . . . . . . . . . 10 |- ((A e. RR /\ B e. RR) -> (A e. CC /\ B e. CC))
5 subcl 6524 . . . . . . . . . 10 |- ((A e. CC /\ B e. CC) -> (A - B) e. CC)
61, 4, 53syl 24 . . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A < B) -> (A - B) e. CC)
7 abscl 8084 . . . . . . . . 9 |- ((A - B) e. CC -> (abs` (A - B)) e. RR)
8 recn 6466 . . . . . . . . 9 |- ((abs` (A - B)) e. RR -> (abs` (A - B)) e. CC)
96, 7, 83syl 24 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) e. CC)
10 2cn 7164 . . . . . . . . 9 |- 2 e. CC
1110a1i 8 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> 2 e. CC)
12 2ne0 7174 . . . . . . . . 9 |- 2 =/= 0
1312a1i 8 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> 2 =/= 0)
14 divcan2 6910 . . . . . . . 8 |- (((abs` (A - B)) e. CC /\ 2 e. CC /\ 2 =/= 0) -> (2 x. ((abs`
(A - B)) / 2)) = (abs` (A - B)))
159, 11, 13, 14syl111anc 1100 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((abs`
(A - B)) / 2)) = (abs` (A - B)))
16 resubcl 6601 . . . . . . . . 9 |- ((A e. RR /\ B e. RR) -> (A - B) e. RR)
17 recn 6466 . . . . . . . . 9 |- ((A - B) e. RR -> (A - B) e. CC)
181, 16, 173syl 24 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (A - B) e. CC)
1918, 7, 83syl 24 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) e. CC)
2015, 19eqeltrd 1971 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((abs`
(A - B)) / 2)) e. CC)
2123ad2ant1 897 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> A e. CC)
2233ad2ant2 898 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> B e. CC)
23 addass 6460 . . . . . . 7 |- (((2 x. ((abs` (A - B)) / 2)) e. CC /\ A e. CC /\ B e. CC) -> (((2 x. ((abs` (A - B)) / 2)) + A) + B) = ((2 x. ((abs` (A - B)) / 2)) + (A + B)))
2423eqcomd 1889 . . . . . 6 |- (((2 x. ((abs` (A - B)) / 2)) e. CC /\ A e. CC /\ B e. CC) -> ((2 x. ((abs` (A - B)) / 2)) + (A + B)) = (((2 x. ((abs`
(A - B)) / 2)) + A) + B))
2520, 21, 22, 24syl111anc 1100 . . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. ((abs` (A - B)) / 2)) + (A + B)) = (((2 x. ((abs`
(A - B)) / 2)) + A) + B))
26 addcl 6454 . . . . . . 7 |- (((2 x. ((abs` (A - B)) / 2)) e. CC /\ A e. CC) -> ((2 x. ((abs` (A - B)) / 2)) + A) e. CC)
2720, 21, 26syl11anc 524 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. ((abs` (A - B)) / 2)) + A) e. CC)
28 addcom 6458 . . . . . 6 |- ((((2 x. ((abs`
(A - B)) / 2)) + A) e. CC /\ B e. CC) -> (((2 x. ((abs` (A - B)) / 2)) + A) + B) = (B + ((2 x. ((abs` (A - B)) / 2)) + A)))
2927, 22, 28syl11anc 524 . . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> (((2 x. ((abs` (A - B)) / 2)) + A) + B) = (B + ((2 x. ((abs` (A - B)) / 2)) + A)))
30 simp2 877 . . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A < B) -> B e. RR)
31 2times 7188 . . . . . . . . 9 |- (B e. CC -> (2 x. B) = (B + B))
3230, 3, 313syl 24 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. B) = (B + B))
3332opreq1d 4897 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. B) - B) = ((B + B) - B))
343, 3jca 310 . . . . . . . 8 |- (B e. RR -> (B e. CC /\ B e. CC))
35 pncan 6557 . . . . . . . 8 |- ((B e. CC /\ B e. CC) -> ((B + B) - B) = B)
3630, 34, 353syl 24 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> ((B + B) - B) = B)
37 abssuble0 8148 . . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ A <_ B) -> (abs` (A - B)) = (B - A))
38 ltle 6690 . . . . . . . . . . . 12 |- ((A e. RR /\ B e. RR) -> (A < B -> A <_ B))
39383impia 1064 . . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ A < B) -> A <_ B)
4037, 39syld3an3 1142 . . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) = (B - A))
4115, 40eqtr2d 1926 . . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A < B) -> (B - A) = (2 x. ((abs` (A - B)) / 2)))
42 subadd 6532 . . . . . . . . . 10 |- ((B e. CC /\ A e. CC /\ (2 x. ((abs` (A - B)) / 2)) e. CC) -> ((B - A) = (2 x. ((abs` (A - B)) / 2)) <-> (A + (2 x. ((abs` (A - B)) / 2))) = B))
4322, 21, 20, 42syl111anc 1100 . . . . . . . . 9 |- ((A e. RR /\ B e. RR /\ A < B) -> ((B - A) = (2 x. ((abs` (A - B)) / 2)) <-> (A + (2 x. ((abs` (A - B)) / 2))) = B))
4441, 43mpbid 212 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (A + (2 x. ((abs` (A - B)) / 2))) = B)
45 addcom 6458 . . . . . . . . 9 |- ((A e. CC /\ (2 x. ((abs` (A - B)) / 2)) e. CC) -> (A + (2 x. ((abs` (A - B)) / 2))) = ((2 x. ((abs` (A - B)) / 2)) + A))
4621, 20, 45syl11anc 524 . . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (A + (2 x. ((abs` (A - B)) / 2))) = ((2 x. ((abs` (A - B)) / 2)) + A))
4744, 46eqtr3d 1927 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> B = ((2 x. ((abs` (A - B)) / 2)) + A))
4833, 36, 473eqtrd 1929 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. B) - B) = ((2 x. ((abs` (A - B)) / 2)) + A))
49 mulcl 6456 . . . . . . . . 9 |- ((2 e. CC /\ B e. CC) -> (2 x. B) e. CC)
5010, 49mpan 759 . . . . . . . 8 |- (B e. CC -> (2 x. B) e. CC)
5130, 3, 503syl 24 . . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. B) e. CC)
52 subadd 6532 . . . . . . 7 |- (((2 x. B) e. CC /\ B e. CC /\ ((2 x. ((abs`
(A - B)) / 2)) + A) e. CC) -> (((2 x. B) - B) = ((2 x. ((abs`
(A - B)) / 2)) + A) <-> (B + ((2 x. ((abs`
(A - B)) / 2)) + A)) = (2 x. B)))
5351, 22, 27, 52syl111anc 1100 . . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> (((2 x. B) - B) = ((2 x. ((abs`
(A - B)) / 2)) + A) <-> (B + ((2 x. ((abs`
(A - B)) / 2)) + A)) = (2 x. B)))
5448, 53mpbid 212 . . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> (B + ((2 x. ((abs`
(A - B)) / 2)) + A)) = (2 x. B))
5525, 29, 543eqtrd 1929 . . . 4 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. ((abs` (A - B)) / 2)) + (A + B)) = (2 x. B))
56 addcl 6454 . . . . . 6 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
571, 4, 563syl 24 . . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> (A + B) e. CC)
58 subadd 6532 . . . . 5 |- (((2 x. B) e. CC /\ (2 x. ((abs` (A - B)) / 2)) e. CC /\ (A + B) e. CC) -> (((2 x. B) - (2 x. ((abs`
(A - B)) / 2))) = (A + B) <-> ((2 x. ((abs` (A - B)) / 2)) + (A + B)) = (2 x. B)))
5951, 20, 57, 58syl111anc 1100 . . . 4 |- ((A e. RR /\ B e. RR /\ A < B) -> (((2 x. B) - (2 x. ((abs`
(A - B)) / 2))) = (A + B) <-> ((2 x. ((abs` (A - B)) / 2)) + (A + B)) = (2 x. B)))
6055, 59mpbird 213 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. B) - (2 x. ((abs` (A - B)) / 2))) = (A + B))
61 dmse2 15002 . . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> ((abs` (A - B)) / 2) e. RR+)
62 rpre 7236 . . . . 5 |- (((abs` (A - B)) / 2) e. RR+ -> ((abs` (A - B)) / 2) e. RR)
63 recn 6466 . . . . 5 |- (((abs` (A - B)) / 2) e. RR -> ((abs` (A - B)) / 2) e. CC)
6461, 62, 633syl 24 . . . 4 |- ((A e. RR /\ B e. RR /\ A < B) -> ((abs` (A - B)) / 2) e. CC)
65 subdi 6590 . . . 4 |- ((2 e. CC /\ B e. CC /\ ((abs`
(A - B)) / 2) e. CC) -> (2 x. (B - ((abs` (A - B)) / 2))) = ((2 x. B) - (2 x. ((abs` (A - B)) / 2))))
6611, 22, 64, 65syl111anc 1100 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. (B - ((abs` (A - B)) / 2))) = ((2 x. B) - (2 x. ((abs` (A - B)) / 2))))
67 divcan2 6910 . . . 4 |- (((A + B) e. CC /\ 2 e. CC /\ 2 =/= 0) -> (2 x. ((A + B) / 2)) = (A + B))
6857, 11, 13, 67syl111anc 1100 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((A + B) / 2)) = (A + B))
6960, 66, 683eqtr4rd 1939 . 2 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((A + B) / 2)) = (2 x. (B - ((abs` (A - B)) / 2))))
70 halfaddsubcl 7226 . . . . 5 |- ((A e. CC /\ B e. CC) -> (((A + B) / 2) e. CC /\ ((A - B) / 2) e. CC))
7170simplld 348 . . . 4 |- ((A e. CC /\ B e. CC) -> ((A + B) / 2) e. CC)
721, 4, 713syl 24 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> ((A + B) / 2) e. CC)
73 msr3 15003 . . . 4 |- ((A e. RR /\ B e. RR) -> (B - ((abs`
(A - B)) / 2)) e. RR)
74 recn 6466 . . . 4 |- ((B - ((abs` (A - B)) / 2)) e. RR -> (B - ((abs` (A - B)) / 2)) e. CC)
751, 73, 743syl 24 . . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> (B - ((abs`
(A - B)) / 2)) e. CC)
7612mulcant2i 6879 . . 3 |- ((((A + B) / 2) e. CC /\ (B - ((abs` (A - B)) / 2)) e. CC /\ 2 e. CC) -> ((2 x. ((A + B) / 2)) = (2 x. (B - ((abs` (A - B)) / 2))) <-> ((A + B) / 2) = (B - ((abs` (A - B)) / 2))))
7772, 75, 11, 76syl111anc 1100 . 2 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. ((A + B) / 2)) = (2 x. (B - ((abs` (A - B)) / 2))) <-> ((A + B) / 2) = (B - ((abs` (A - B)) / 2))))
7869, 77mpbid 212 1 |- ((A e. RR /\ B e. RR /\ A < B) -> ((A + B) / 2) = (B - ((abs` (A - B)) / 2)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017   class class class wbr 3338  ` cfv 3998  (class class class)co 4884  CCcc 6384  RRcr 6385  0cc0 6386   + caddc 6389   x. cmul 6391   - cmin 6445   / cdiv 6447   <_ cle 6448  RR+crp 6453   < clt 6653  2c2 7145  abscabs 8000
This theorem is referenced by:  msra3 15009
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-rp 7232  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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