Theorem apdifflemf 16676

Description: Lemma for apdiff 16678. 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 8208	. . . . . 6 |- ( ph -> A e. CC )
3		apdifflemf.r	. . . . . . 7 |- ( ph -> R e. QQ )
4		qcn 9868	. . . . . . 7 |- ( R e. QQ -> R e. CC )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> R e. CC )
6	2, 5	subcld 8490	. . . . 5 |- ( ph -> ( A - R ) e. CC )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( A - R ) e. CC )
8	7	abscld 11746	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) e. RR )
9		apdifflemf.q	. . . . . . 7 |- ( ph -> Q e. QQ )
10		qcn 9868	. . . . . . 7 |- ( Q e. QQ -> Q e. CC )
11	9, 10	syl 14	. . . . . 6 |- ( ph -> Q e. CC )
12	2, 11	subcld 8490	. . . . 5 |- ( ph -> ( A - Q ) e. CC )
13	12	abscld 11746	. . . 4 |- ( ph -> ( abs ` ( A - Q ) ) e. RR )
14	13	adantr 276	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) e. RR )
15		qre 9859	. . . . . . . . . 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 9869	. . . . . . . . . . . . . 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 9859	. . . . . . . . . . . . 13 |- ( ( Q + R ) e. QQ -> ( Q + R ) e. RR )
22	20, 21	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( Q + R ) e. RR )
23	22	rehalfcld 9391	. . . . . . . . . . 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 9859	. . . . . . . . . . . . . 14 |- ( R e. QQ -> R e. RR )
27	3, 26	syl 14	. . . . . . . . . . . . 13 |- ( ph -> R e. RR )
28		avglt1 9383	. . . . . . . . . . . . 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 8305	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < A )
34	17, 18, 33	ltled 8298	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q <_ A )
35	17, 18, 34	abssubge0d 11741	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) = ( A - Q ) )
36	35	oveq2d 6034	. . . . . 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 8520	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( A - Q ) ) = ( ( R + Q ) - A ) )
41	37, 39	addcomd 8330	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + Q ) = ( Q + R ) )
42	41	oveq1d 6033	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( R + Q ) - A ) = ( ( Q + R ) - A ) )
43	36, 40, 42	3eqtrd 2268	. . . . 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 9893	. . . . . . . . . 10 |- 2 e. RR+
46	45	a1i 9	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> 2 e. RR+ )
47	44, 18, 46	ltdivmuld 9983	. . . . . . . 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 9387	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( 2 x. A ) = ( A + A ) )
50	48, 49	breqtrd 4114	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) < ( A + A ) )
51	44, 18, 18	ltsubaddd 8721	. . . . . 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 4110	. . . 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 9927	. . . . . . . 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 9965	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( A + ( R - Q ) ) )
60	35	oveq2d 6034	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( R + ( A - Q ) ) )
61	37, 38, 39	addsub12d 8513	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( A - Q ) ) = ( A + ( R - Q ) ) )
62	60, 61	eqtrd 2264	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( A + ( R - Q ) ) )
63	59, 62	breqtrrd 4116	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( R + ( abs ` ( A - Q ) ) ) )
64	18, 55, 14	absdifltd 11743	. . . 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 952	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) < ( abs ` ( A - Q ) ) )
66	8, 14, 65	gtapd 8817	. 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 11746	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) e. RR )
70	11, 5, 2	subsubd 8518	. . . . . . 7 |- ( ph -> ( Q - ( R - A ) ) = ( ( Q - R ) + A ) )
71	16, 27	sublt0d 8750	. . . . . . . . 9 |- ( ph -> ( ( Q - R ) < 0 <-> Q < R ) )
72	25, 71	mpbird 167	. . . . . . . 8 |- ( ph -> ( Q - R ) < 0 )
73	16, 27	resubcld 8560	. . . . . . . . 9 |- ( ph -> ( Q - R ) e. RR )
74		ltaddnegr 8605	. . . . . . . . 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 4110	. . . . . 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 9392	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( A + A ) < ( Q + R ) )
83	79, 79, 80	ltaddsub2d 8726	. . . . . . 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 8510	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) - A ) = ( Q + ( R - A ) ) )
89	84, 88	breqtrd 4114	. . . . 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 8560	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( R - A ) e. RR )
93	79, 90, 92	absdifltd 11743	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( abs ` ( A - Q ) ) < ( R - A ) <-> ( ( Q - ( R - A ) ) < A /\ A < ( Q + ( R - A ) ) ) ) )
94	78, 89, 93	mpbir2and 952	. . . 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 9384	. . . . . . . . . 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 8305	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < R )
101	79, 91, 100	ltled 8298	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A <_ R )
102	79, 91, 101	abssuble0d 11742	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) = ( R - A ) )
103	94, 102	breqtrrd 4116	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) < ( abs ` ( A - R ) ) )
104	67, 69, 103	ltapd 8818	. 2 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
105		apdifflemf.ap	. . 3 |- ( ph -> ( ( Q + R ) / 2 ) # A )
106		reaplt 8768	. . . 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 805	1 |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   + caddc 8035   x. cmul 8037   < clt 8214   - cmin 8350   # cap 8761   / cdiv 8852   2c2 9194   QQcq 9853   RR+crp 9888   abscabs 11562

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150  ax-arch 8151  ax-caucvg 8152

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-n0 9403  df-z 9480  df-uz 9756  df-q 9854  df-rp 9889  df-seqfrec 10711  df-exp 10802  df-cj 11407  df-re 11408  df-im 11409  df-rsqrt 11563  df-abs 11564

This theorem is referenced by:  apdiff  16678