Theorem apdifflemr 16601

Description: Lemma for apdiff 16602. (Contributed by Jim Kingdon, 19-May-2024.)

Hypotheses

Ref	Expression
apdifflemr.a	|- ( ph -> A e. RR )
apdifflemr.s	|- ( ph -> S e. QQ )
apdifflemr.1	|- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )
apdifflemr.as	|- ( ( ph /\ S =/= 0 ) -> ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) )

Assertion

Ref	Expression
apdifflemr	|- ( ph -> A # S )

Proof of Theorem apdifflemr

Step	Hyp	Ref	Expression
1		2cnd 9209	. . . . 5 |- ( ph -> 2 e. CC )
2		apdifflemr.a	. . . . . 6 |- ( ph -> A e. RR )
3	2	recnd 8201	. . . . 5 |- ( ph -> A e. CC )
4	3	2timesd 9380	. . . . . 6 |- ( ph -> ( 2 x. A ) = ( A + A ) )
5		apdifflemr.1	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )
6		1cnd 8188	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. CC )
7	3, 6	subnegd 8490	. . . . . . . . . . . . . 14 |- ( ph -> ( A - -u 1 ) = ( A + 1 ) )
8	3, 6, 7	comraddd 8329	. . . . . . . . . . . . 13 |- ( ph -> ( A - -u 1 ) = ( 1 + A ) )
9	8	fveq2d 5639	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) = ( abs ` ( 1 + A ) ) )
10	3, 6	abssubd 11747	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - 1 ) ) = ( abs ` ( 1 - A ) ) )
11	5, 9, 10	3brtr3d 4117	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) )
12	6, 3	addcld 8192	. . . . . . . . . . . 12 |- ( ph -> ( 1 + A ) e. CC )
13	6, 3	subcld 8483	. . . . . . . . . . . 12 |- ( ph -> ( 1 - A ) e. CC )
14		absext 11617	. . . . . . . . . . . 12 |- ( ( ( 1 + A ) e. CC /\ ( 1 - A ) e. CC ) -> ( ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) -> ( 1 + A ) # ( 1 - A ) ) )
15	12, 13, 14	syl2anc 411	. . . . . . . . . . 11 |- ( ph -> ( ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) -> ( 1 + A ) # ( 1 - A ) ) )
16	11, 15	mpd 13	. . . . . . . . . 10 |- ( ph -> ( 1 + A ) # ( 1 - A ) )
17	6, 3	negsubd 8489	. . . . . . . . . 10 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
18	16, 17	breqtrrd 4114	. . . . . . . . 9 |- ( ph -> ( 1 + A ) # ( 1 + -u A ) )
19	3	negcld 8470	. . . . . . . . . 10 |- ( ph -> -u A e. CC )
20		apadd2 8782	. . . . . . . . . 10 |- ( ( A e. CC /\ -u A e. CC /\ 1 e. CC ) -> ( A # -u A <-> ( 1 + A ) # ( 1 + -u A ) ) )
21	3, 19, 6, 20	syl3anc 1271	. . . . . . . . 9 |- ( ph -> ( A # -u A <-> ( 1 + A ) # ( 1 + -u A ) ) )
22	18, 21	mpbird 167	. . . . . . . 8 |- ( ph -> A # -u A )
23		apadd2 8782	. . . . . . . . 9 |- ( ( A e. CC /\ -u A e. CC /\ A e. CC ) -> ( A # -u A <-> ( A + A ) # ( A + -u A ) ) )
24	3, 19, 3, 23	syl3anc 1271	. . . . . . . 8 |- ( ph -> ( A # -u A <-> ( A + A ) # ( A + -u A ) ) )
25	22, 24	mpbid 147	. . . . . . 7 |- ( ph -> ( A + A ) # ( A + -u A ) )
26	3	negidd 8473	. . . . . . 7 |- ( ph -> ( A + -u A ) = 0 )
27	25, 26	breqtrd 4112	. . . . . 6 |- ( ph -> ( A + A ) # 0 )
28	4, 27	eqbrtrd 4108	. . . . 5 |- ( ph -> ( 2 x. A ) # 0 )
29	1, 3, 28	mulap0bbd 8833	. . . 4 |- ( ph -> A # 0 )
30	29	adantr 276	. . 3 |- ( ( ph /\ S = 0 ) -> A # 0 )
31		simpr 110	. . 3 |- ( ( ph /\ S = 0 ) -> S = 0 )
32	30, 31	breqtrrd 4114	. 2 |- ( ( ph /\ S = 0 ) -> A # S )
33	4	adantr 276	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. A ) = ( A + A ) )
34		apdifflemr.as	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) )
35	3	subid1d 8472	. . . . . . . . . . 11 |- ( ph -> ( A - 0 ) = A )
36	35	fveq2d 5639	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
37		2z 9500	. . . . . . . . . . . . . . 15 |- 2 e. ZZ
38		zq 9853	. . . . . . . . . . . . . . 15 |- ( 2 e. ZZ -> 2 e. QQ )
39	37, 38	ax-mp 5	. . . . . . . . . . . . . 14 |- 2 e. QQ
40	39	a1i 9	. . . . . . . . . . . . 13 |- ( ph -> 2 e. QQ )
41		apdifflemr.s	. . . . . . . . . . . . 13 |- ( ph -> S e. QQ )
42		qmulcl 9864	. . . . . . . . . . . . 13 |- ( ( 2 e. QQ /\ S e. QQ ) -> ( 2 x. S ) e. QQ )
43	40, 41, 42	syl2anc 411	. . . . . . . . . . . 12 |- ( ph -> ( 2 x. S ) e. QQ )
44		qcn 9861	. . . . . . . . . . . 12 |- ( ( 2 x. S ) e. QQ -> ( 2 x. S ) e. CC )
45	43, 44	syl 14	. . . . . . . . . . 11 |- ( ph -> ( 2 x. S ) e. CC )
46	3, 45	abssubd 11747	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - ( 2 x. S ) ) ) = ( abs ` ( ( 2 x. S ) - A ) ) )
47	36, 46	breq12d 4099	. . . . . . . . 9 |- ( ph -> ( ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) <-> ( abs ` A ) # ( abs ` ( ( 2 x. S ) - A ) ) ) )
48	47	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( ( abs ` ( A - 0 ) ) # ( abs ` ( A - ( 2 x. S ) ) ) <-> ( abs ` A ) # ( abs ` ( ( 2 x. S ) - A ) ) ) )
49	34, 48	mpbid 147	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> ( abs ` A ) # ( abs ` ( ( 2 x. S ) - A ) ) )
50	3	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> A e. CC )
51	45, 3	subcld 8483	. . . . . . . . 9 |- ( ph -> ( ( 2 x. S ) - A ) e. CC )
52	51	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( ( 2 x. S ) - A ) e. CC )
53		absext 11617	. . . . . . . 8 |- ( ( A e. CC /\ ( ( 2 x. S ) - A ) e. CC ) -> ( ( abs ` A ) # ( abs ` ( ( 2 x. S ) - A ) ) -> A # ( ( 2 x. S ) - A ) ) )
54	50, 52, 53	syl2anc 411	. . . . . . 7 |- ( ( ph /\ S =/= 0 ) -> ( ( abs ` A ) # ( abs ` ( ( 2 x. S ) - A ) ) -> A # ( ( 2 x. S ) - A ) ) )
55	49, 54	mpd 13	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> A # ( ( 2 x. S ) - A ) )
56		apadd2 8782	. . . . . . 7 |- ( ( A e. CC /\ ( ( 2 x. S ) - A ) e. CC /\ A e. CC ) -> ( A # ( ( 2 x. S ) - A ) <-> ( A + A ) # ( A + ( ( 2 x. S ) - A ) ) ) )
57	50, 52, 50, 56	syl3anc 1271	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> ( A # ( ( 2 x. S ) - A ) <-> ( A + A ) # ( A + ( ( 2 x. S ) - A ) ) ) )
58	55, 57	mpbid 147	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( A + A ) # ( A + ( ( 2 x. S ) - A ) ) )
59	45	adantr 276	. . . . . 6 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. S ) e. CC )
60	50, 59	pncan3d 8486	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( A + ( ( 2 x. S ) - A ) ) = ( 2 x. S ) )
61	58, 60	breqtrd 4112	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( A + A ) # ( 2 x. S ) )
62	33, 61	eqbrtrd 4108	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. A ) # ( 2 x. S ) )
63		qcn 9861	. . . . . 6 |- ( S e. QQ -> S e. CC )
64	41, 63	syl 14	. . . . 5 |- ( ph -> S e. CC )
65	64	adantr 276	. . . 4 |- ( ( ph /\ S =/= 0 ) -> S e. CC )
66		2cnd 9209	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 e. CC )
67		2ap0 9229	. . . . 5 |- 2 # 0
68	67	a1i 9	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 # 0 )
69		apmul2 8962	. . . 4 |- ( ( A e. CC /\ S e. CC /\ ( 2 e. CC /\ 2 # 0 ) ) -> ( A # S <-> ( 2 x. A ) # ( 2 x. S ) ) )
70	50, 65, 66, 68, 69	syl112anc 1275	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( A # S <-> ( 2 x. A ) # ( 2 x. S ) ) )
71	62, 70	mpbird 167	. 2 |- ( ( ph /\ S =/= 0 ) -> A # S )
72		0z 9483	. . . . . 6 |- 0 e. ZZ
73		zq 9853	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
74	72, 73	ax-mp 5	. . . . 5 |- 0 e. QQ
75		qdceq 10497	. . . . 5 |- ( ( S e. QQ /\ 0 e. QQ ) -> DECID S = 0 )
76	41, 74, 75	sylancl 413	. . . 4 |- ( ph -> DECID S = 0 )
77		exmiddc 841	. . . 4 |- (DECID S = 0 -> ( S = 0 \/ -. S = 0 ) )
78	76, 77	syl 14	. . 3 |- ( ph -> ( S = 0 \/ -. S = 0 ) )
79		df-ne 2401	. . . 4 |- ( S =/= 0 <-> -. S = 0 )
80	79	orbi2i 767	. . 3 |- ( ( S = 0 \/ S =/= 0 ) <-> ( S = 0 \/ -. S = 0 ) )
81	78, 80	sylibr 134	. 2 |- ( ph -> ( S = 0 \/ S =/= 0 ) )
82	32, 71, 81	mpjaodan 803	1 |- ( ph -> A # S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839   = wceq 1395   e. wcel 2200   =/= wne 2400   class class class wbr 4086   ` cfv 5324  (class class class)co 6013   CCcc 8023   RRcr 8024   0cc0 8025   1c1 8026   + caddc 8028   x. cmul 8030   - cmin 8343   -ucneg 8344   # cap 8754   2c2 9187   ZZcz 9472   QQcq 9846   abscabs 11551

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684  ax-cnex 8116  ax-resscn 8117  ax-1cn 8118  ax-1re 8119  ax-icn 8120  ax-addcl 8121  ax-addrcl 8122  ax-mulcl 8123  ax-mulrcl 8124  ax-addcom 8125  ax-mulcom 8126  ax-addass 8127  ax-mulass 8128  ax-distr 8129  ax-i2m1 8130  ax-0lt1 8131  ax-1rid 8132  ax-0id 8133  ax-rnegex 8134  ax-precex 8135  ax-cnre 8136  ax-pre-ltirr 8137  ax-pre-ltwlin 8138  ax-pre-lttrn 8139  ax-pre-apti 8140  ax-pre-ltadd 8141  ax-pre-mulgt0 8142  ax-pre-mulext 8143  ax-arch 8144  ax-caucvg 8145

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-po 4391  df-iso 4392  df-iord 4461  df-on 4463  df-ilim 4464  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-riota 5966  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-recs 6466  df-frec 6552  df-pnf 8209  df-mnf 8210  df-xr 8211  df-ltxr 8212  df-le 8213  df-sub 8345  df-neg 8346  df-reap 8748  df-ap 8755  df-div 8846  df-inn 9137  df-2 9195  df-3 9196  df-4 9197  df-n0 9396  df-z 9473  df-uz 9749  df-q 9847  df-rp 9882  df-seqfrec 10703  df-exp 10794  df-cj 11396  df-re 11397  df-im 11398  df-rsqrt 11552  df-abs 11553

This theorem is referenced by:  apdiff  16602