Theorem apdifflemr 15691

Description: Lemma for apdiff 15692. (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 9063	. . . . 5 |- ( ph -> 2 e. CC )
2		apdifflemr.a	. . . . . 6 |- ( ph -> A e. RR )
3	2	recnd 8055	. . . . 5 |- ( ph -> A e. CC )
4	3	2timesd 9234	. . . . . 6 |- ( ph -> ( 2 x. A ) = ( A + A ) )
5		apdifflemr.1	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )
6		1cnd 8042	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. CC )
7	3, 6	subnegd 8344	. . . . . . . . . . . . . 14 |- ( ph -> ( A - -u 1 ) = ( A + 1 ) )
8	3, 6, 7	comraddd 8183	. . . . . . . . . . . . 13 |- ( ph -> ( A - -u 1 ) = ( 1 + A ) )
9	8	fveq2d 5562	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) = ( abs ` ( 1 + A ) ) )
10	3, 6	abssubd 11358	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - 1 ) ) = ( abs ` ( 1 - A ) ) )
11	5, 9, 10	3brtr3d 4064	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) )
12	6, 3	addcld 8046	. . . . . . . . . . . 12 |- ( ph -> ( 1 + A ) e. CC )
13	6, 3	subcld 8337	. . . . . . . . . . . 12 |- ( ph -> ( 1 - A ) e. CC )
14		absext 11228	. . . . . . . . . . . 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 8343	. . . . . . . . . 10 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
18	16, 17	breqtrrd 4061	. . . . . . . . 9 |- ( ph -> ( 1 + A ) # ( 1 + -u A ) )
19	3	negcld 8324	. . . . . . . . . 10 |- ( ph -> -u A e. CC )
20		apadd2 8636	. . . . . . . . . 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 1249	. . . . . . . . 9 |- ( ph -> ( A # -u A <-> ( 1 + A ) # ( 1 + -u A ) ) )
22	18, 21	mpbird 167	. . . . . . . 8 |- ( ph -> A # -u A )
23		apadd2 8636	. . . . . . . . 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 1249	. . . . . . . 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 8327	. . . . . . 7 |- ( ph -> ( A + -u A ) = 0 )
27	25, 26	breqtrd 4059	. . . . . 6 |- ( ph -> ( A + A ) # 0 )
28	4, 27	eqbrtrd 4055	. . . . 5 |- ( ph -> ( 2 x. A ) # 0 )
29	1, 3, 28	mulap0bbd 8687	. . . 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 4061	. 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 8326	. . . . . . . . . . 11 |- ( ph -> ( A - 0 ) = A )
36	35	fveq2d 5562	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
37		2z 9354	. . . . . . . . . . . . . . 15 |- 2 e. ZZ
38		zq 9700	. . . . . . . . . . . . . . 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 9711	. . . . . . . . . . . . 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 9708	. . . . . . . . . . . 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 11358	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - ( 2 x. S ) ) ) = ( abs ` ( ( 2 x. S ) - A ) ) )
47	36, 46	breq12d 4046	. . . . . . . . 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 8337	. . . . . . . . 9 |- ( ph -> ( ( 2 x. S ) - A ) e. CC )
52	51	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( ( 2 x. S ) - A ) e. CC )
53		absext 11228	. . . . . . . 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 8636	. . . . . . 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 1249	. . . . . 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 8340	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( A + ( ( 2 x. S ) - A ) ) = ( 2 x. S ) )
61	58, 60	breqtrd 4059	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( A + A ) # ( 2 x. S ) )
62	33, 61	eqbrtrd 4055	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. A ) # ( 2 x. S ) )
63		qcn 9708	. . . . . 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 9063	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 e. CC )
67		2ap0 9083	. . . . 5 |- 2 # 0
68	67	a1i 9	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 # 0 )
69		apmul2 8816	. . . 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 1253	. . 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 9337	. . . . . 6 |- 0 e. ZZ
73		zq 9700	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
74	72, 73	ax-mp 5	. . . . 5 |- 0 e. QQ
75		qdceq 10334	. . . . 5 |- ( ( S e. QQ /\ 0 e. QQ ) -> DECID S = 0 )
76	41, 74, 75	sylancl 413	. . . 4 |- ( ph -> DECID S = 0 )
77		exmiddc 837	. . . 4 |- (DECID S = 0 -> ( S = 0 \/ -. S = 0 ) )
78	76, 77	syl 14	. . 3 |- ( ph -> ( S = 0 \/ -. S = 0 ) )
79		df-ne 2368	. . . 4 |- ( S =/= 0 <-> -. S = 0 )
80	79	orbi2i 763	. . 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 799	1 |- ( ph -> A # S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709  DECID wdc 835   = wceq 1364   e. wcel 2167   =/= wne 2367   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   + caddc 7882   x. cmul 7884   - cmin 8197   -ucneg 8198   # cap 8608   2c2 9041   ZZcz 9326   QQcq 9693   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