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

Step	Hyp	Ref	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)
4	2, 3	anim12i 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)
6	1, 4, 5	3syl 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)
9	6, 7, 8	3syl 24	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) e. CC)
10		2cn 7164	. . . . . . . . 9 |- 2 e. CC
11	10	a1i 8	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> 2 e. CC)
12		2ne0 7174	. . . . . . . . 9 |- 2 =/= 0
13	12	a1i 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)))
15	9, 11, 13, 14	syl111anc 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)
18	1, 16, 17	3syl 24	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (A - B) e. CC)
19	18, 7, 8	3syl 24	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) e. CC)
20	15, 19	eqeltrd 1971	. . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((abs` (A - B)) / 2)) e. CC)
21	2	3ad2ant1 897	. . . . . 6 |- ((A e. RR /\ B e. RR /\ A < B) -> A e. CC)
22	3	3ad2ant2 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)))
24	23	eqcomd 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))
25	20, 21, 22, 24	syl111anc 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)
27	20, 21, 26	syl11anc 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)))
29	27, 22, 28	syl11anc 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))
32	30, 3, 31	3syl 24	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. B) = (B + B))
33	32	opreq1d 4897	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> ((2 x. B) - B) = ((B + B) - B))
34	3, 3	jca 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)
36	30, 34, 35	3syl 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))
39	38	3impia 1064	. . . . . . . . . . 11 |- ((A e. RR /\ B e. RR /\ A < B) -> A <_ B)
40	37, 39	syld3an3 1142	. . . . . . . . . 10 |- ((A e. RR /\ B e. RR /\ A < B) -> (abs` (A - B)) = (B - A))
41	15, 40	eqtr2d 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))
43	22, 21, 20, 42	syl111anc 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))
44	41, 43	mpbid 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))
46	21, 20, 45	syl11anc 524	. . . . . . . 8 |- ((A e. RR /\ B e. RR /\ A < B) -> (A + (2 x. ((abs` (A - B)) / 2))) = ((2 x. ((abs` (A - B)) / 2)) + A))
47	44, 46	eqtr3d 1927	. . . . . . 7 |- ((A e. RR /\ B e. RR /\ A < B) -> B = ((2 x. ((abs` (A - B)) / 2)) + A))
48	33, 36, 47	3eqtrd 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)
50	10, 49	mpan 759	. . . . . . . 8 |- (B e. CC -> (2 x. B) e. CC)
51	30, 3, 50	3syl 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)))
53	51, 22, 27, 52	syl111anc 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)))
54	48, 53	mpbid 212	. . . . 5 |- ((A e. RR /\ B e. RR /\ A < B) -> (B + ((2 x. ((abs` (A - B)) / 2)) + A)) = (2 x. B))
55	25, 29, 54	3eqtrd 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)
57	1, 4, 56	3syl 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)))
59	51, 20, 57, 58	syl111anc 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)))
60	55, 59	mpbird 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)
64	61, 62, 63	3syl 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))))
66	11, 22, 64, 65	syl111anc 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))
68	57, 11, 13, 67	syl111anc 1100	. . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> (2 x. ((A + B) / 2)) = (A + B))
69	60, 66, 68	3eqtr4rd 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))
71	70	simplld 348	. . . 4 |- ((A e. CC /\ B e. CC) -> ((A + B) / 2) e. CC)
72	1, 4, 71	3syl 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)
75	1, 73, 74	3syl 24	. . 3 |- ((A e. RR /\ B e. RR /\ A < B) -> (B - ((abs` (A - B)) / 2)) e. CC)
76	12	mulcant2i 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))))
77	72, 75, 11, 76	syl111anc 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))))
78	69, 77	mpbid 212	1 |- ((A e. RR /\ B e. RR /\ A < B) -> ((A + B) / 2) = (B - ((abs` (A - B)) / 2)))

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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