Theorem apdifflemf 16822

Description: Lemma for apdiff 16824. Being apart from the point halfway between Q and R suffices for A to be a different distance from Q and from R. (Contributed by Jim Kingdon, 18-May-2024.)

Hypotheses

Ref	Expression
apdifflemf.a	|- ( ph -> A e. RR )
apdifflemf.q	|- ( ph -> Q e. QQ )
apdifflemf.r	|- ( ph -> R e. QQ )
apdifflemf.qr	|- ( ph -> Q < R )
apdifflemf.ap	|- ( ph -> ( ( Q + R ) / 2 ) # A )

Assertion

Ref	Expression
apdifflemf	|- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )

Proof of Theorem apdifflemf

Step	Hyp	Ref	Expression
1		apdifflemf.a	. . . . . . 7 |- ( ph -> A e. RR )
2	1	recnd 8301	. . . . . 6 |- ( ph -> A e. CC )
3		apdifflemf.r	. . . . . . 7 |- ( ph -> R e. QQ )
4		qcn 9965	. . . . . . 7 |- ( R e. QQ -> R e. CC )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> R e. CC )
6	2, 5	subcld 8583	. . . . 5 |- ( ph -> ( A - R ) e. CC )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( A - R ) e. CC )
8	7	abscld 11862	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) e. RR )
9		apdifflemf.q	. . . . . . 7 |- ( ph -> Q e. QQ )
10		qcn 9965	. . . . . . 7 |- ( Q e. QQ -> Q e. CC )
11	9, 10	syl 14	. . . . . 6 |- ( ph -> Q e. CC )
12	2, 11	subcld 8583	. . . . 5 |- ( ph -> ( A - Q ) e. CC )
13	12	abscld 11862	. . . 4 |- ( ph -> ( abs ` ( A - Q ) ) e. RR )
14	13	adantr 276	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) e. RR )
15		qre 9956	. . . . . . . . . 10 |- ( Q e. QQ -> Q e. RR )
16	9, 15	syl 14	. . . . . . . . 9 |- ( ph -> Q e. RR )
17	16	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q e. RR )
18	1	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A e. RR )
19		qaddcl 9966	. . . . . . . . . . . . . 14 |- ( ( Q e. QQ /\ R e. QQ ) -> ( Q + R ) e. QQ )
20	9, 3, 19	syl2anc 411	. . . . . . . . . . . . 13 |- ( ph -> ( Q + R ) e. QQ )
21		qre 9956	. . . . . . . . . . . . 13 |- ( ( Q + R ) e. QQ -> ( Q + R ) e. RR )
22	20, 21	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( Q + R ) e. RR )
23	22	rehalfcld 9484	. . . . . . . . . . 11 |- ( ph -> ( ( Q + R ) / 2 ) e. RR )
24	23	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( Q + R ) / 2 ) e. RR )
25		apdifflemf.qr	. . . . . . . . . . . 12 |- ( ph -> Q < R )
26		qre 9956	. . . . . . . . . . . . . 14 |- ( R e. QQ -> R e. RR )
27	3, 26	syl 14	. . . . . . . . . . . . 13 |- ( ph -> R e. RR )
28		avglt1 9476	. . . . . . . . . . . . 13 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> Q < ( ( Q + R ) / 2 ) ) )
29	16, 27, 28	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> ( Q < R <-> Q < ( ( Q + R ) / 2 ) ) )
30	25, 29	mpbid 147	. . . . . . . . . . 11 |- ( ph -> Q < ( ( Q + R ) / 2 ) )
31	30	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < ( ( Q + R ) / 2 ) )
32		simpr 110	. . . . . . . . . 10 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( Q + R ) / 2 ) < A )
33	17, 24, 18, 31, 32	lttrd 8398	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < A )
34	17, 18, 33	ltled 8391	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q <_ A )
35	17, 18, 34	abssubge0d 11857	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) = ( A - Q ) )
36	35	oveq2d 6065	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( abs ` ( A - Q ) ) ) = ( R - ( A - Q ) ) )
37	5	adantr 276	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> R e. CC )
38	2	adantr 276	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A e. CC )
39	11	adantr 276	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q e. CC )
40	37, 38, 39	subsub3d 8613	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( A - Q ) ) = ( ( R + Q ) - A ) )
41	37, 39	addcomd 8423	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + Q ) = ( Q + R ) )
42	41	oveq1d 6064	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( R + Q ) - A ) = ( ( Q + R ) - A ) )
43	36, 40, 42	3eqtrd 2269	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( abs ` ( A - Q ) ) ) = ( ( Q + R ) - A ) )
44	22	adantr 276	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) e. RR )
45		2rp 9990	. . . . . . . . . 10 |- 2 e. RR+
46	45	a1i 9	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> 2 e. RR+ )
47	44, 18, 46	ltdivmuld 10080	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( ( Q + R ) / 2 ) < A <-> ( Q + R ) < ( 2 x. A ) ) )
48	32, 47	mpbid 147	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) < ( 2 x. A ) )
49	38	2timesd 9480	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( 2 x. A ) = ( A + A ) )
50	48, 49	breqtrd 4134	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) < ( A + A ) )
51	44, 18, 18	ltsubaddd 8814	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( ( Q + R ) - A ) < A <-> ( Q + R ) < ( A + A ) ) )
52	50, 51	mpbird 167	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( Q + R ) - A ) < A )
53	43, 52	eqbrtrd 4130	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( abs ` ( A - Q ) ) ) < A )
54	25	adantr 276	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < R )
55	27	adantr 276	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> R e. RR )
56		difrp 10024	. . . . . . . 8 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> ( R - Q ) e. RR+ ) )
57	17, 55, 56	syl2anc 411	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q < R <-> ( R - Q ) e. RR+ ) )
58	54, 57	mpbid 147	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - Q ) e. RR+ )
59	18, 58	ltaddrpd 10062	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( A + ( R - Q ) ) )
60	35	oveq2d 6065	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( R + ( A - Q ) ) )
61	37, 38, 39	addsub12d 8606	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( A - Q ) ) = ( A + ( R - Q ) ) )
62	60, 61	eqtrd 2265	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( A + ( R - Q ) ) )
63	59, 62	breqtrrd 4136	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( R + ( abs ` ( A - Q ) ) ) )
64	18, 55, 14	absdifltd 11859	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( abs ` ( A - R ) ) < ( abs ` ( A - Q ) ) <-> ( ( R - ( abs ` ( A - Q ) ) ) < A /\ A < ( R + ( abs ` ( A - Q ) ) ) ) ) )
65	53, 63, 64	mpbir2and 953	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) < ( abs ` ( A - Q ) ) )
66	8, 14, 65	gtapd 8910	. 2 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
67	13	adantr 276	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) e. RR )
68	6	adantr 276	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( A - R ) e. CC )
69	68	abscld 11862	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) e. RR )
70	11, 5, 2	subsubd 8611	. . . . . . 7 |- ( ph -> ( Q - ( R - A ) ) = ( ( Q - R ) + A ) )
71	16, 27	sublt0d 8843	. . . . . . . . 9 |- ( ph -> ( ( Q - R ) < 0 <-> Q < R ) )
72	25, 71	mpbird 167	. . . . . . . 8 |- ( ph -> ( Q - R ) < 0 )
73	16, 27	resubcld 8653	. . . . . . . . 9 |- ( ph -> ( Q - R ) e. RR )
74		ltaddnegr 8698	. . . . . . . . 9 |- ( ( ( Q - R ) e. RR /\ A e. RR ) -> ( ( Q - R ) < 0 <-> ( ( Q - R ) + A ) < A ) )
75	73, 1, 74	syl2anc 411	. . . . . . . 8 |- ( ph -> ( ( Q - R ) < 0 <-> ( ( Q - R ) + A ) < A ) )
76	72, 75	mpbid 147	. . . . . . 7 |- ( ph -> ( ( Q - R ) + A ) < A )
77	70, 76	eqbrtrd 4130	. . . . . 6 |- ( ph -> ( Q - ( R - A ) ) < A )
78	77	adantr 276	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( Q - ( R - A ) ) < A )
79	1	adantr 276	. . . . . . . 8 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A e. RR )
80	22	adantr 276	. . . . . . . 8 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( Q + R ) e. RR )
81		simpr 110	. . . . . . . 8 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < ( ( Q + R ) / 2 ) )
82	79, 79, 80, 81, 81	lt2halvesd 9485	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( A + A ) < ( Q + R ) )
83	79, 79, 80	ltaddsub2d 8819	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( A + A ) < ( Q + R ) <-> A < ( ( Q + R ) - A ) ) )
84	82, 83	mpbid 147	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < ( ( Q + R ) - A ) )
85	11	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> Q e. CC )
86	5	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> R e. CC )
87	2	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A e. CC )
88	85, 86, 87	addsubassd 8603	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) - A ) = ( Q + ( R - A ) ) )
89	84, 88	breqtrd 4134	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < ( Q + ( R - A ) ) )
90	16	adantr 276	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> Q e. RR )
91	27	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> R e. RR )
92	91, 79	resubcld 8653	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( R - A ) e. RR )
93	79, 90, 92	absdifltd 11859	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( abs ` ( A - Q ) ) < ( R - A ) <-> ( ( Q - ( R - A ) ) < A /\ A < ( Q + ( R - A ) ) ) ) )
94	78, 89, 93	mpbir2and 953	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) < ( R - A ) )
95	23	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) / 2 ) e. RR )
96		avglt2 9477	. . . . . . . . . 10 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> ( ( Q + R ) / 2 ) < R ) )
97	16, 27, 96	syl2anc 411	. . . . . . . . 9 |- ( ph -> ( Q < R <-> ( ( Q + R ) / 2 ) < R ) )
98	25, 97	mpbid 147	. . . . . . . 8 |- ( ph -> ( ( Q + R ) / 2 ) < R )
99	98	adantr 276	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) / 2 ) < R )
100	79, 95, 91, 81, 99	lttrd 8398	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < R )
101	79, 91, 100	ltled 8391	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A <_ R )
102	79, 91, 101	abssuble0d 11858	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) = ( R - A ) )
103	94, 102	breqtrrd 4136	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) < ( abs ` ( A - R ) ) )
104	67, 69, 103	ltapd 8911	. 2 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
105		apdifflemf.ap	. . 3 |- ( ph -> ( ( Q + R ) / 2 ) # A )
106		reaplt 8861	. . . 4 |- ( ( ( ( Q + R ) / 2 ) e. RR /\ A e. RR ) -> ( ( ( Q + R ) / 2 ) # A <-> ( ( ( Q + R ) / 2 ) < A \/ A < ( ( Q + R ) / 2 ) ) ) )
107	23, 1, 106	syl2anc 411	. . 3 |- ( ph -> ( ( ( Q + R ) / 2 ) # A <-> ( ( ( Q + R ) / 2 ) < A \/ A < ( ( Q + R ) / 2 ) ) ) )
108	105, 107	mpbid 147	. 2 |- ( ph -> ( ( ( Q + R ) / 2 ) < A \/ A < ( ( Q + R ) / 2 ) ) )
109	66, 104, 108	mpjaodan 806	1 |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   e. wcel 2203   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   + caddc 8129   x. cmul 8131   < clt 8307   - cmin 8443   # cap 8854   / cdiv 8945   2c2 9287   QQcq 9950   RR+crp 9985   abscabs 11678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680

This theorem is referenced by:  apdiff  16824