Theorem apdifflemf 15690

Description: Lemma for apdiff 15692. 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 8055	. . . . . 6 |- ( ph -> A e. CC )
3		apdifflemf.r	. . . . . . 7 |- ( ph -> R e. QQ )
4		qcn 9708	. . . . . . 7 |- ( R e. QQ -> R e. CC )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> R e. CC )
6	2, 5	subcld 8337	. . . . 5 |- ( ph -> ( A - R ) e. CC )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( A - R ) e. CC )
8	7	abscld 11346	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) e. RR )
9		apdifflemf.q	. . . . . . 7 |- ( ph -> Q e. QQ )
10		qcn 9708	. . . . . . 7 |- ( Q e. QQ -> Q e. CC )
11	9, 10	syl 14	. . . . . 6 |- ( ph -> Q e. CC )
12	2, 11	subcld 8337	. . . . 5 |- ( ph -> ( A - Q ) e. CC )
13	12	abscld 11346	. . . 4 |- ( ph -> ( abs ` ( A - Q ) ) e. RR )
14	13	adantr 276	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) e. RR )
15		qre 9699	. . . . . . . . . 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 9709	. . . . . . . . . . . . . 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 9699	. . . . . . . . . . . . 13 |- ( ( Q + R ) e. QQ -> ( Q + R ) e. RR )
22	20, 21	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( Q + R ) e. RR )
23	22	rehalfcld 9238	. . . . . . . . . . 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 9699	. . . . . . . . . . . . . 14 |- ( R e. QQ -> R e. RR )
27	3, 26	syl 14	. . . . . . . . . . . . 13 |- ( ph -> R e. RR )
28		avglt1 9230	. . . . . . . . . . . . 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 8152	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < A )
34	17, 18, 33	ltled 8145	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q <_ A )
35	17, 18, 34	abssubge0d 11341	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) = ( A - Q ) )
36	35	oveq2d 5938	. . . . . 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 8367	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( A - Q ) ) = ( ( R + Q ) - A ) )
41	37, 39	addcomd 8177	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + Q ) = ( Q + R ) )
42	41	oveq1d 5937	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( R + Q ) - A ) = ( ( Q + R ) - A ) )
43	36, 40, 42	3eqtrd 2233	. . . . 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 9733	. . . . . . . . . 10 |- 2 e. RR+
46	45	a1i 9	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> 2 e. RR+ )
47	44, 18, 46	ltdivmuld 9823	. . . . . . . 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 9234	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( 2 x. A ) = ( A + A ) )
50	48, 49	breqtrd 4059	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) < ( A + A ) )
51	44, 18, 18	ltsubaddd 8568	. . . . . 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 4055	. . . 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 9767	. . . . . . . 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 9805	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( A + ( R - Q ) ) )
60	35	oveq2d 5938	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( R + ( A - Q ) ) )
61	37, 38, 39	addsub12d 8360	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( A - Q ) ) = ( A + ( R - Q ) ) )
62	60, 61	eqtrd 2229	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( A + ( R - Q ) ) )
63	59, 62	breqtrrd 4061	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( R + ( abs ` ( A - Q ) ) ) )
64	18, 55, 14	absdifltd 11343	. . . 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 946	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) < ( abs ` ( A - Q ) ) )
66	8, 14, 65	gtapd 8664	. 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 11346	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) e. RR )
70	11, 5, 2	subsubd 8365	. . . . . . 7 |- ( ph -> ( Q - ( R - A ) ) = ( ( Q - R ) + A ) )
71	16, 27	sublt0d 8597	. . . . . . . . 9 |- ( ph -> ( ( Q - R ) < 0 <-> Q < R ) )
72	25, 71	mpbird 167	. . . . . . . 8 |- ( ph -> ( Q - R ) < 0 )
73	16, 27	resubcld 8407	. . . . . . . . 9 |- ( ph -> ( Q - R ) e. RR )
74		ltaddnegr 8452	. . . . . . . . 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 4055	. . . . . 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 9239	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( A + A ) < ( Q + R ) )
83	79, 79, 80	ltaddsub2d 8573	. . . . . . 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 8357	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) - A ) = ( Q + ( R - A ) ) )
89	84, 88	breqtrd 4059	. . . . 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 8407	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( R - A ) e. RR )
93	79, 90, 92	absdifltd 11343	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( abs ` ( A - Q ) ) < ( R - A ) <-> ( ( Q - ( R - A ) ) < A /\ A < ( Q + ( R - A ) ) ) ) )
94	78, 89, 93	mpbir2and 946	. . . 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 9231	. . . . . . . . . 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 8152	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < R )
101	79, 91, 100	ltled 8145	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A <_ R )
102	79, 91, 101	abssuble0d 11342	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) = ( R - A ) )
103	94, 102	breqtrrd 4061	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) < ( abs ` ( A - R ) ) )
104	67, 69, 103	ltapd 8665	. 2 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
105		apdifflemf.ap	. . 3 |- ( ph -> ( ( Q + R ) / 2 ) # A )
106		reaplt 8615	. . . 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 799	1 |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   + caddc 7882   x. cmul 7884   < clt 8061   - cmin 8197   # cap 8608   / cdiv 8699   2c2 9041   QQcq 9693   RR+crp 9728   abscabs 11162

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-q 9694  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  apdiff  15692