Theorem apdifflemf 17314

Description: Lemma for apdiff 17316. 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 8361	. . . . . 6 |- ( ph -> A e. CC )
3		apdifflemf.r	. . . . . . 7 |- ( ph -> R e. QQ )
4		qcn 10051	. . . . . . 7 |- ( R e. QQ -> R e. CC )
5	3, 4	syl 14	. . . . . 6 |- ( ph -> R e. CC )
6	2, 5	subcld 8645	. . . . 5 |- ( ph -> ( A - R ) e. CC )
7	6	adantr 276	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( A - R ) e. CC )
8	7	abscld 11971	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) e. RR )
9		apdifflemf.q	. . . . . . 7 |- ( ph -> Q e. QQ )
10		qcn 10051	. . . . . . 7 |- ( Q e. QQ -> Q e. CC )
11	9, 10	syl 14	. . . . . 6 |- ( ph -> Q e. CC )
12	2, 11	subcld 8645	. . . . 5 |- ( ph -> ( A - Q ) e. CC )
13	12	abscld 11971	. . . 4 |- ( ph -> ( abs ` ( A - Q ) ) e. RR )
14	13	adantr 276	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) e. RR )
15		qre 10042	. . . . . . . . . 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 10052	. . . . . . . . . . . . . 14 |- ( ( Q e. QQ /\ R e. QQ ) -> ( Q + R ) e. QQ )
20	9, 3, 19	syl2anc 415	. . . . . . . . . . . . 13 |- ( ph -> ( Q + R ) e. QQ )
21		qre 10042	. . . . . . . . . . . . 13 |- ( ( Q + R ) e. QQ -> ( Q + R ) e. RR )
22	20, 21	syl 14	. . . . . . . . . . . 12 |- ( ph -> ( Q + R ) e. RR )
23	22	rehalfcld 9564	. . . . . . . . . . 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 10042	. . . . . . . . . . . . . 14 |- ( R e. QQ -> R e. RR )
27	3, 26	syl 14	. . . . . . . . . . . . 13 |- ( ph -> R e. RR )
28		avglt1 9556	. . . . . . . . . . . . 13 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> Q < ( ( Q + R ) / 2 ) ) )
29	16, 27, 28	syl2anc 415	. . . . . . . . . . . 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 8460	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q < A )
34	17, 18, 33	ltled 8453	. . . . . . . 8 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> Q <_ A )
35	17, 18, 34	abssubge0d 11966	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - Q ) ) = ( A - Q ) )
36	35	oveq2d 6104	. . . . . 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 8675	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R - ( A - Q ) ) = ( ( R + Q ) - A ) )
41	37, 39	addcomd 8485	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + Q ) = ( Q + R ) )
42	41	oveq1d 6103	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( ( R + Q ) - A ) = ( ( Q + R ) - A ) )
43	36, 40, 42	3eqtrd 2275	. . . . 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 10077	. . . . . . . . . 10 |- 2 e. RR+
46	45	a1i 9	. . . . . . . . 9 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> 2 e. RR+ )
47	44, 18, 46	ltdivmuld 10167	. . . . . . . 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 9560	. . . . . . 7 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( 2 x. A ) = ( A + A ) )
50	48, 49	breqtrd 4157	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( Q + R ) < ( A + A ) )
51	44, 18, 18	ltsubaddd 8877	. . . . . 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 4153	. . . 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 10111	. . . . . . . 8 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> ( R - Q ) e. RR+ ) )
57	17, 55, 56	syl2anc 415	. . . . . . 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 10149	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( A + ( R - Q ) ) )
60	35	oveq2d 6104	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( R + ( A - Q ) ) )
61	37, 38, 39	addsub12d 8668	. . . . . 6 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( A - Q ) ) = ( A + ( R - Q ) ) )
62	60, 61	eqtrd 2271	. . . . 5 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( R + ( abs ` ( A - Q ) ) ) = ( A + ( R - Q ) ) )
63	59, 62	breqtrrd 4159	. . . 4 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> A < ( R + ( abs ` ( A - Q ) ) ) )
64	18, 55, 14	absdifltd 11968	. . . 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 957	. . 3 |- ( ( ph /\ ( ( Q + R ) / 2 ) < A ) -> ( abs ` ( A - R ) ) < ( abs ` ( A - Q ) ) )
66	8, 14, 65	gtapd 8974	. 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 11971	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) e. RR )
70	11, 5, 2	subsubd 8673	. . . . . . 7 |- ( ph -> ( Q - ( R - A ) ) = ( ( Q - R ) + A ) )
71	16, 27	sublt0d 8907	. . . . . . . . 9 |- ( ph -> ( ( Q - R ) < 0 <-> Q < R ) )
72	25, 71	mpbird 167	. . . . . . . 8 |- ( ph -> ( Q - R ) < 0 )
73	16, 27	resubcld 8716	. . . . . . . . 9 |- ( ph -> ( Q - R ) e. RR )
74		ltaddnegr 8761	. . . . . . . . 9 |- ( ( ( Q - R ) e. RR /\ A e. RR ) -> ( ( Q - R ) < 0 <-> ( ( Q - R ) + A ) < A ) )
75	73, 1, 74	syl2anc 415	. . . . . . . 8 |- ( ph -> ( ( Q - R ) < 0 <-> ( ( Q - R ) + A ) < A ) )
76	72, 75	mpbid 147	. . . . . . 7 |- ( ph -> ( ( Q - R ) + A ) < A )
77	70, 76	eqbrtrd 4153	. . . . . 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 9565	. . . . . . 7 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( A + A ) < ( Q + R ) )
83	79, 79, 80	ltaddsub2d 8882	. . . . . . 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 8665	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( Q + R ) - A ) = ( Q + ( R - A ) ) )
89	84, 88	breqtrd 4157	. . . . 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 8716	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( R - A ) e. RR )
93	79, 90, 92	absdifltd 11968	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( ( abs ` ( A - Q ) ) < ( R - A ) <-> ( ( Q - ( R - A ) ) < A /\ A < ( Q + ( R - A ) ) ) ) )
94	78, 89, 93	mpbir2and 957	. . . 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 9557	. . . . . . . . . 10 |- ( ( Q e. RR /\ R e. RR ) -> ( Q < R <-> ( ( Q + R ) / 2 ) < R ) )
97	16, 27, 96	syl2anc 415	. . . . . . . . 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 8460	. . . . . 6 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A < R )
101	79, 91, 100	ltled 8453	. . . . 5 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> A <_ R )
102	79, 91, 101	abssuble0d 11967	. . . 4 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - R ) ) = ( R - A ) )
103	94, 102	breqtrrd 4159	. . 3 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) < ( abs ` ( A - R ) ) )
104	67, 69, 103	ltapd 8975	. 2 |- ( ( ph /\ A < ( ( Q + R ) / 2 ) ) -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )
105		apdifflemf.ap	. . 3 |- ( ph -> ( ( Q + R ) / 2 ) # A )
106		reaplt 8925	. . . 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 415	. . 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 810	1 |- ( ph -> ( abs ` ( A - Q ) ) # ( abs ` ( A - R ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   e. wcel 2209   class class class wbr 4131   ` cfv 5380  (class class class)co 6088   CCcc 8184   RRcr 8185   0cc0 8186   + caddc 8189   x. cmul 8191   < clt 8367   - cmin 8505   # cap 8918   / cdiv 9011   2c2 9365   QQcq 10036   RR+crp 10072   abscabs 11786

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8277  ax-resscn 8278  ax-1cn 8279  ax-1re 8280  ax-icn 8281  ax-addcl 8282  ax-addrcl 8283  ax-mulcl 8284  ax-mulrcl 8285  ax-addcom 8286  ax-mulcom 8287  ax-addass 8288  ax-mulass 8289  ax-distr 8290  ax-i2m1 8291  ax-0lt1 8292  ax-1rid 8293  ax-0id 8294  ax-rnegex 8295  ax-precex 8296  ax-cnre 8297  ax-pre-ltirr 8298  ax-pre-ltwlin 8299  ax-pre-lttrn 8300  ax-pre-apti 8301  ax-pre-ltadd 8302  ax-pre-mulgt0 8303  ax-pre-mulext 8304  ax-arch 8305  ax-caucvg 8306

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-po 4443  df-iso 4444  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-frec 6666  df-pnf 8369  df-mnf 8370  df-xr 8371  df-ltxr 8372  df-le 8373  df-sub 8507  df-neg 8508  df-reap 8912  df-ap 8919  df-div 9012  df-inn 9315  df-2 9373  df-3 9374  df-4 9375  df-n0 9576  df-z 9657  df-uz 9939  df-q 10037  df-rp 10073  df-seqfrec 10907  df-exp 10998  df-cj 11630  df-re 11631  df-im 11632  df-rsqrt 11787  df-abs 11788

This theorem is used by:  apdiff  17316