Theorem apdifflemr 16823

Description: Lemma for apdiff 16824. (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 9309	. . . . 5 |- ( ph -> 2 e. CC )
2		apdifflemr.a	. . . . . 6 |- ( ph -> A e. RR )
3	2	recnd 8301	. . . . 5 |- ( ph -> A e. CC )
4	3	2timesd 9480	. . . . . 6 |- ( ph -> ( 2 x. A ) = ( A + A ) )
5		apdifflemr.1	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) # ( abs ` ( A - 1 ) ) )
6		1cnd 8289	. . . . . . . . . . . . . 14 |- ( ph -> 1 e. CC )
7	3, 6	subnegd 8590	. . . . . . . . . . . . . 14 |- ( ph -> ( A - -u 1 ) = ( A + 1 ) )
8	3, 6, 7	comraddd 8429	. . . . . . . . . . . . 13 |- ( ph -> ( A - -u 1 ) = ( 1 + A ) )
9	8	fveq2d 5673	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - -u 1 ) ) = ( abs ` ( 1 + A ) ) )
10	3, 6	abssubd 11874	. . . . . . . . . . . 12 |- ( ph -> ( abs ` ( A - 1 ) ) = ( abs ` ( 1 - A ) ) )
11	5, 9, 10	3brtr3d 4139	. . . . . . . . . . 11 |- ( ph -> ( abs ` ( 1 + A ) ) # ( abs ` ( 1 - A ) ) )
12	6, 3	addcld 8292	. . . . . . . . . . . 12 |- ( ph -> ( 1 + A ) e. CC )
13	6, 3	subcld 8583	. . . . . . . . . . . 12 |- ( ph -> ( 1 - A ) e. CC )
14		absext 11744	. . . . . . . . . . . 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 8589	. . . . . . . . . 10 |- ( ph -> ( 1 + -u A ) = ( 1 - A ) )
18	16, 17	breqtrrd 4136	. . . . . . . . 9 |- ( ph -> ( 1 + A ) # ( 1 + -u A ) )
19	3	negcld 8570	. . . . . . . . . 10 |- ( ph -> -u A e. CC )
20		apadd2 8882	. . . . . . . . . 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 1274	. . . . . . . . 9 |- ( ph -> ( A # -u A <-> ( 1 + A ) # ( 1 + -u A ) ) )
22	18, 21	mpbird 167	. . . . . . . 8 |- ( ph -> A # -u A )
23		apadd2 8882	. . . . . . . . 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 1274	. . . . . . . 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 8573	. . . . . . 7 |- ( ph -> ( A + -u A ) = 0 )
27	25, 26	breqtrd 4134	. . . . . 6 |- ( ph -> ( A + A ) # 0 )
28	4, 27	eqbrtrd 4130	. . . . 5 |- ( ph -> ( 2 x. A ) # 0 )
29	1, 3, 28	mulap0bbd 8933	. . . 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 4136	. 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 8572	. . . . . . . . . . 11 |- ( ph -> ( A - 0 ) = A )
36	35	fveq2d 5673	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - 0 ) ) = ( abs ` A ) )
37		2z 9604	. . . . . . . . . . . . . . 15 |- 2 e. ZZ
38		zq 9957	. . . . . . . . . . . . . . 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 9968	. . . . . . . . . . . . 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 9965	. . . . . . . . . . . 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 11874	. . . . . . . . . 10 |- ( ph -> ( abs ` ( A - ( 2 x. S ) ) ) = ( abs ` ( ( 2 x. S ) - A ) ) )
47	36, 46	breq12d 4121	. . . . . . . . 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 8583	. . . . . . . . 9 |- ( ph -> ( ( 2 x. S ) - A ) e. CC )
52	51	adantr 276	. . . . . . . 8 |- ( ( ph /\ S =/= 0 ) -> ( ( 2 x. S ) - A ) e. CC )
53		absext 11744	. . . . . . . 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 8882	. . . . . . 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 1274	. . . . . 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 8586	. . . . 5 |- ( ( ph /\ S =/= 0 ) -> ( A + ( ( 2 x. S ) - A ) ) = ( 2 x. S ) )
61	58, 60	breqtrd 4134	. . . 4 |- ( ( ph /\ S =/= 0 ) -> ( A + A ) # ( 2 x. S ) )
62	33, 61	eqbrtrd 4130	. . 3 |- ( ( ph /\ S =/= 0 ) -> ( 2 x. A ) # ( 2 x. S ) )
63		qcn 9965	. . . . . 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 9309	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 e. CC )
67		2ap0 9329	. . . . 5 |- 2 # 0
68	67	a1i 9	. . . 4 |- ( ( ph /\ S =/= 0 ) -> 2 # 0 )
69		apmul2 9062	. . . 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 1278	. . 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 9587	. . . . . 6 |- 0 e. ZZ
73		zq 9957	. . . . . 6 |- ( 0 e. ZZ -> 0 e. QQ )
74	72, 73	ax-mp 5	. . . . 5 |- 0 e. QQ
75		qdceq 10603	. . . . 5 |- ( ( S e. QQ /\ 0 e. QQ ) -> DECID S = 0 )
76	41, 74, 75	sylancl 413	. . . 4 |- ( ph -> DECID S = 0 )
77		exmiddc 844	. . . 4 |- (DECID S = 0 -> ( S = 0 \/ -. S = 0 ) )
78	76, 77	syl 14	. . 3 |- ( ph -> ( S = 0 \/ -. S = 0 ) )
79		df-ne 2413	. . . 4 |- ( S =/= 0 <-> -. S = 0 )
80	79	orbi2i 770	. . 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 806	1 |- ( ph -> A # S )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   e. wcel 2203   =/= wne 2412   class class class wbr 4108   ` cfv 5351  (class class class)co 6049   CCcc 8124   RRcr 8125   0cc0 8126   1c1 8127   + caddc 8129   x. cmul 8131   - cmin 8443   -ucneg 8444   # cap 8854   2c2 9287   ZZcz 9576   QQcq 9950   abscabs 11678

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8217  ax-resscn 8218  ax-1cn 8219  ax-1re 8220  ax-icn 8221  ax-addcl 8222  ax-addrcl 8223  ax-mulcl 8224  ax-mulrcl 8225  ax-addcom 8226  ax-mulcom 8227  ax-addass 8228  ax-mulass 8229  ax-distr 8230  ax-i2m1 8231  ax-0lt1 8232  ax-1rid 8233  ax-0id 8234  ax-rnegex 8235  ax-precex 8236  ax-cnre 8237  ax-pre-ltirr 8238  ax-pre-ltwlin 8239  ax-pre-lttrn 8240  ax-pre-apti 8241  ax-pre-ltadd 8242  ax-pre-mulgt0 8243  ax-pre-mulext 8244  ax-arch 8245  ax-caucvg 8246

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8309  df-mnf 8310  df-xr 8311  df-ltxr 8312  df-le 8313  df-sub 8445  df-neg 8446  df-reap 8848  df-ap 8855  df-div 8946  df-inn 9237  df-2 9295  df-3 9296  df-4 9297  df-n0 9496  df-z 9577  df-uz 9853  df-q 9951  df-rp 9986  df-seqfrec 10809  df-exp 10900  df-cj 11523  df-re 11524  df-im 11525  df-rsqrt 11679  df-abs 11680

This theorem is referenced by:  apdiff  16824