Theorem apdifflemr 14957

Description: Lemma for apdiff 14958. (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 8995	. . . . 5 |- ( ph -> 2 e. CC )
2		apdifflemr.a	. . . . . 6 |- ( ph -> A e. RR )
3	2	recnd 7989	. . . . 5 |- ( ph -> A e. CC )
4	3	2timesd 9164	. . . . . 6 |- ( ph -> ( 2 x. A ) = ( A + A ) )
5		apdifflemr.1	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )
6		1cnd 7976	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. CC )
7	3, 6	subnegd 8278	. . . . . . . . . . . . . 14 |- ( ph -> ( A - -u 1 ) = ( A + 1 ) )
8	3, 6, 7	comraddd 8117	. . . . . . . . . . . . 13 |- ( ph -> ( A - -u 1 ) = ( 1 + A ) )
9	8	fveq2d 5521	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) = ( abs ` ( 1 + A ) ) )
10	3, 6	abssubd 11205	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - 1 ) ) = ( abs ` ( 1 - A ) ) )
11	5, 9, 10	3brtr3d 4036	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) )
12	6, 3	addcld 7980	. . . . . . . . . . . 12 |- ( ph -> ( 1 + A ) e. CC )
13	6, 3	subcld 8271	. . . . . . . . . . . 12 |- ( ph -> ( 1 - A ) e. CC )
14		absext 11075	. . . . . . . . . . . 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 8277	. . . . . . . . . 10 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
18	16, 17	breqtrrd 4033	. . . . . . . . 9 |- ( ph -> ( 1 + A ) # ( 1 + -u A ) )
19	3	negcld 8258	. . . . . . . . . 10 |- ( ph -> -u A e. CC )
20		apadd2 8569	. . . . . . . . . 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 1238	. . . . . . . . 9 |- ( ph -> ( A # -u A <-> ( 1 + A ) # ( 1 + -u A ) ) )
22	18, 21	mpbird 167	. . . . . . . 8 |- ( ph -> A # -u A )
23		apadd2 8569	. . . . . . . . 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 1238	. . . . . . . 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 8261	. . . . . . 7 |- ( ph -> ( A + -u A ) = 0 )
27	25, 26	breqtrd 4031	. . . . . 6 |- ( ph -> ( A + A ) # 0 )
28	4, 27	eqbrtrd 4027	. . . . 5 |- ( ph -> ( 2 x. A ) # 0 )
29	1, 3, 28	mulap0bbd 8620	. . . 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 4033	. 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 8260	. . . . . . . . . . 11 |- ( ph -> ( A - 0 ) = A )
36	35	fveq2d 5521	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
37		2z 9284	. . . . . . . . . . . . . . 15 |- 2 e. ZZ
38		zq 9629	. . . . . . . . . . . . . . 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 9640	. . . . . . . . . . . . 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 9637	. . . . . . . . . . . 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 11205	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - ( 2 x. S ) ) ) = ( abs ` ( ( 2 x. S ) - A ) ) )
47	36, 46	breq12d 4018	. . . . . . . . 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 8271	. . . . . . . . 9 |- ( ph -> ( ( 2 x. S ) - A ) e. CC )
52	51	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( ( 2 x. S ) - A ) e. CC )
53		absext 11075	. . . . . . . 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 8569	. . . . . . 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 1238	. . . . . 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 8274	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( A + ( ( 2 x. S ) - A ) ) = ( 2 x. S ) )
61	58, 60	breqtrd 4031	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( A + A ) # ( 2 x. S ) )
62	33, 61	eqbrtrd 4027	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. A ) # ( 2 x. S ) )
63		qcn 9637	. . . . . 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 8995	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 e. CC )
67		2ap0 9015	. . . . 5 |- 2 # 0
68	67	a1i 9	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 # 0 )
69		apmul2 8749	. . . 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 1242	. . 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 9267	. . . . . 6 |- 0 e. ZZ
73		zq 9629	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
74	72, 73	ax-mp 5	. . . . 5 |- 0 e. QQ
75		qdceq 10250	. . . . 5 |- ( ( S e. QQ /\ 0 e. QQ ) -> DECID S = 0 )
76	41, 74, 75	sylancl 413	. . . 4 |- ( ph -> DECID S = 0 )
77		exmiddc 836	. . . 4 |- (DECID S = 0 -> ( S = 0 \/ -. S = 0 ) )
78	76, 77	syl 14	. . 3 |- ( ph -> ( S = 0 \/ -. S = 0 ) )
79		df-ne 2348	. . . 4 |- ( S =/= 0 <-> -. S = 0 )
80	79	orbi2i 762	. . 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 798	1 |- ( ph -> A # S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 708  DECID wdc 834   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   0cc0 7814   1c1 7815   + caddc 7817   x. cmul 7819   - cmin 8131   -ucneg 8132   # cap 8541   2c2 8973   ZZcz 9256   QQcq 9622   abscabs 11009

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-1st 6144  df-2nd 6145  df-recs 6309  df-frec 6395  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-seqfrec 10449  df-exp 10523  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011

This theorem is referenced by:  apdiff  14958