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Theorem apdifflemr 13301
Description: Lemma for apdiff 13302. (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
StepHypRef Expression
1 2cnd 8805 . . . . 5  |-  ( ph  ->  2  e.  CC )
2 apdifflemr.a . . . . . 6  |-  ( ph  ->  A  e.  RR )
32recnd 7806 . . . . 5  |-  ( ph  ->  A  e.  CC )
432timesd 8974 . . . . . 6  |-  ( ph  ->  ( 2  x.  A
)  =  ( A  +  A ) )
5 apdifflemr.1 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A  -  -u 1 ) ) #  ( abs `  ( A  -  1 ) ) )
6 1cnd 7794 . . . . . . . . . . . . . 14  |-  ( ph  ->  1  e.  CC )
73, 6subnegd 8092 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( A  -  -u 1
)  =  ( A  +  1 ) )
83, 6, 7comraddd 7931 . . . . . . . . . . . . 13  |-  ( ph  ->  ( A  -  -u 1
)  =  ( 1  +  A ) )
98fveq2d 5425 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A  -  -u 1 ) )  =  ( abs `  ( 1  +  A
) ) )
103, 6abssubd 10977 . . . . . . . . . . . 12  |-  ( ph  ->  ( abs `  ( A  -  1 ) )  =  ( abs `  ( 1  -  A
) ) )
115, 9, 103brtr3d 3959 . . . . . . . . . . 11  |-  ( ph  ->  ( abs `  (
1  +  A ) ) #  ( abs `  (
1  -  A ) ) )
126, 3addcld 7797 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  +  A
)  e.  CC )
136, 3subcld 8085 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1  -  A
)  e.  CC )
14 absext 10847 . . . . . . . . . . . 12  |-  ( ( ( 1  +  A
)  e.  CC  /\  ( 1  -  A
)  e.  CC )  ->  ( ( abs `  ( 1  +  A
) ) #  ( abs `  ( 1  -  A
) )  ->  (
1  +  A ) #  ( 1  -  A
) ) )
1512, 13, 14syl2anc 408 . . . . . . . . . . 11  |-  ( ph  ->  ( ( abs `  (
1  +  A ) ) #  ( abs `  (
1  -  A ) )  ->  ( 1  +  A ) #  ( 1  -  A ) ) )
1611, 15mpd 13 . . . . . . . . . 10  |-  ( ph  ->  ( 1  +  A
) #  ( 1  -  A ) )
176, 3negsubd 8091 . . . . . . . . . 10  |-  ( ph  ->  ( 1  +  -u A )  =  ( 1  -  A ) )
1816, 17breqtrrd 3956 . . . . . . . . 9  |-  ( ph  ->  ( 1  +  A
) #  ( 1  + 
-u A ) )
193negcld 8072 . . . . . . . . . 10  |-  ( ph  -> 
-u A  e.  CC )
20 apadd2 8383 . . . . . . . . . 10  |-  ( ( A  e.  CC  /\  -u A  e.  CC  /\  1  e.  CC )  ->  ( A #  -u A  <->  ( 1  +  A ) #  ( 1  +  -u A ) ) )
213, 19, 6, 20syl3anc 1216 . . . . . . . . 9  |-  ( ph  ->  ( A #  -u A  <->  ( 1  +  A ) #  ( 1  +  -u A ) ) )
2218, 21mpbird 166 . . . . . . . 8  |-  ( ph  ->  A #  -u A )
23 apadd2 8383 . . . . . . . . 9  |-  ( ( A  e.  CC  /\  -u A  e.  CC  /\  A  e.  CC )  ->  ( A #  -u A  <->  ( A  +  A ) #  ( A  +  -u A ) ) )
243, 19, 3, 23syl3anc 1216 . . . . . . . 8  |-  ( ph  ->  ( A #  -u A  <->  ( A  +  A ) #  ( A  +  -u A ) ) )
2522, 24mpbid 146 . . . . . . 7  |-  ( ph  ->  ( A  +  A
) #  ( A  +  -u A ) )
263negidd 8075 . . . . . . 7  |-  ( ph  ->  ( A  +  -u A )  =  0 )
2725, 26breqtrd 3954 . . . . . 6  |-  ( ph  ->  ( A  +  A
) #  0 )
284, 27eqbrtrd 3950 . . . . 5  |-  ( ph  ->  ( 2  x.  A
) #  0 )
291, 3, 28mulap0bbd 8433 . . . 4  |-  ( ph  ->  A #  0 )
3029adantr 274 . . 3  |-  ( (
ph  /\  S  = 
0 )  ->  A #  0 )
31 simpr 109 . . 3  |-  ( (
ph  /\  S  = 
0 )  ->  S  =  0 )
3230, 31breqtrrd 3956 . 2  |-  ( (
ph  /\  S  = 
0 )  ->  A #  S )
334adantr 274 . . . 4  |-  ( (
ph  /\  S  =/=  0 )  ->  (
2  x.  A )  =  ( A  +  A ) )
34 apdifflemr.as . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  ( abs `  ( A  - 
0 ) ) #  ( abs `  ( A  -  ( 2  x.  S ) ) ) )
353subid1d 8074 . . . . . . . . . . 11  |-  ( ph  ->  ( A  -  0 )  =  A )
3635fveq2d 5425 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( A  -  0 ) )  =  ( abs `  A ) )
37 2z 9094 . . . . . . . . . . . . . . 15  |-  2  e.  ZZ
38 zq 9430 . . . . . . . . . . . . . . 15  |-  ( 2  e.  ZZ  ->  2  e.  QQ )
3937, 38ax-mp 5 . . . . . . . . . . . . . 14  |-  2  e.  QQ
4039a1i 9 . . . . . . . . . . . . 13  |-  ( ph  ->  2  e.  QQ )
41 apdifflemr.s . . . . . . . . . . . . 13  |-  ( ph  ->  S  e.  QQ )
42 qmulcl 9441 . . . . . . . . . . . . 13  |-  ( ( 2  e.  QQ  /\  S  e.  QQ )  ->  ( 2  x.  S
)  e.  QQ )
4340, 41, 42syl2anc 408 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2  x.  S
)  e.  QQ )
44 qcn 9438 . . . . . . . . . . . 12  |-  ( ( 2  x.  S )  e.  QQ  ->  (
2  x.  S )  e.  CC )
4543, 44syl 14 . . . . . . . . . . 11  |-  ( ph  ->  ( 2  x.  S
)  e.  CC )
463, 45abssubd 10977 . . . . . . . . . 10  |-  ( ph  ->  ( abs `  ( A  -  ( 2  x.  S ) ) )  =  ( abs `  ( ( 2  x.  S )  -  A
) ) )
4736, 46breq12d 3942 . . . . . . . . 9  |-  ( ph  ->  ( ( abs `  ( A  -  0 ) ) #  ( abs `  ( A  -  ( 2  x.  S ) ) )  <->  ( abs `  A
) #  ( abs `  (
( 2  x.  S
)  -  A ) ) ) )
4847adantr 274 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  (
( abs `  ( A  -  0 ) ) #  ( abs `  ( A  -  ( 2  x.  S ) ) )  <->  ( abs `  A
) #  ( abs `  (
( 2  x.  S
)  -  A ) ) ) )
4934, 48mpbid 146 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  ( abs `  A ) #  ( abs `  ( ( 2  x.  S )  -  A ) ) )
503adantr 274 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  A  e.  CC )
5145, 3subcld 8085 . . . . . . . . 9  |-  ( ph  ->  ( ( 2  x.  S )  -  A
)  e.  CC )
5251adantr 274 . . . . . . . 8  |-  ( (
ph  /\  S  =/=  0 )  ->  (
( 2  x.  S
)  -  A )  e.  CC )
53 absext 10847 . . . . . . . 8  |-  ( ( A  e.  CC  /\  ( ( 2  x.  S )  -  A
)  e.  CC )  ->  ( ( abs `  A ) #  ( abs `  ( ( 2  x.  S )  -  A
) )  ->  A #  ( ( 2  x.  S )  -  A
) ) )
5450, 52, 53syl2anc 408 . . . . . . 7  |-  ( (
ph  /\  S  =/=  0 )  ->  (
( abs `  A
) #  ( abs `  (
( 2  x.  S
)  -  A ) )  ->  A #  (
( 2  x.  S
)  -  A ) ) )
5549, 54mpd 13 . . . . . 6  |-  ( (
ph  /\  S  =/=  0 )  ->  A #  ( ( 2  x.  S )  -  A
) )
56 apadd2 8383 . . . . . . 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 ) ) ) )
5750, 52, 50, 56syl3anc 1216 . . . . . 6  |-  ( (
ph  /\  S  =/=  0 )  ->  ( A #  ( ( 2  x.  S )  -  A
)  <->  ( A  +  A ) #  ( A  +  ( ( 2  x.  S )  -  A ) ) ) )
5855, 57mpbid 146 . . . . 5  |-  ( (
ph  /\  S  =/=  0 )  ->  ( A  +  A ) #  ( A  +  (
( 2  x.  S
)  -  A ) ) )
5945adantr 274 . . . . . 6  |-  ( (
ph  /\  S  =/=  0 )  ->  (
2  x.  S )  e.  CC )
6050, 59pncan3d 8088 . . . . 5  |-  ( (
ph  /\  S  =/=  0 )  ->  ( A  +  ( (
2  x.  S )  -  A ) )  =  ( 2  x.  S ) )
6158, 60breqtrd 3954 . . . 4  |-  ( (
ph  /\  S  =/=  0 )  ->  ( A  +  A ) #  ( 2  x.  S
) )
6233, 61eqbrtrd 3950 . . 3  |-  ( (
ph  /\  S  =/=  0 )  ->  (
2  x.  A ) #  ( 2  x.  S
) )
63 qcn 9438 . . . . . 6  |-  ( S  e.  QQ  ->  S  e.  CC )
6441, 63syl 14 . . . . 5  |-  ( ph  ->  S  e.  CC )
6564adantr 274 . . . 4  |-  ( (
ph  /\  S  =/=  0 )  ->  S  e.  CC )
66 2cnd 8805 . . . 4  |-  ( (
ph  /\  S  =/=  0 )  ->  2  e.  CC )
67 2ap0 8825 . . . . 5  |-  2 #  0
6867a1i 9 . . . 4  |-  ( (
ph  /\  S  =/=  0 )  ->  2 #  0 )
69 apmul2 8561 . . . 4  |-  ( ( A  e.  CC  /\  S  e.  CC  /\  (
2  e.  CC  /\  2 #  0 ) )  -> 
( A #  S  <->  ( 2  x.  A ) #  ( 2  x.  S ) ) )
7050, 65, 66, 68, 69syl112anc 1220 . . 3  |-  ( (
ph  /\  S  =/=  0 )  ->  ( A #  S  <->  ( 2  x.  A ) #  ( 2  x.  S ) ) )
7162, 70mpbird 166 . 2  |-  ( (
ph  /\  S  =/=  0 )  ->  A #  S )
72 0z 9077 . . . . . 6  |-  0  e.  ZZ
73 zq 9430 . . . . . 6  |-  ( 0  e.  ZZ  ->  0  e.  QQ )
7472, 73ax-mp 5 . . . . 5  |-  0  e.  QQ
75 qdceq 10036 . . . . 5  |-  ( ( S  e.  QQ  /\  0  e.  QQ )  -> DECID  S  =  0 )
7641, 74, 75sylancl 409 . . . 4  |-  ( ph  -> DECID  S  =  0 )
77 exmiddc 821 . . . 4  |-  (DECID  S  =  0  ->  ( S  =  0  \/  -.  S  =  0 ) )
7876, 77syl 14 . . 3  |-  ( ph  ->  ( S  =  0  \/  -.  S  =  0 ) )
79 df-ne 2309 . . . 4  |-  ( S  =/=  0  <->  -.  S  =  0 )
8079orbi2i 751 . . 3  |-  ( ( S  =  0  \/  S  =/=  0 )  <-> 
( S  =  0  \/  -.  S  =  0 ) )
8178, 80sylibr 133 . 2  |-  ( ph  ->  ( S  =  0  \/  S  =/=  0
) )
8232, 71, 81mpjaodan 787 1  |-  ( ph  ->  A #  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697  DECID wdc 819    = wceq 1331    e. wcel 1480    =/= wne 2308   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   CCcc 7630   RRcr 7631   0cc0 7632   1c1 7633    + caddc 7635    x. cmul 7637    - cmin 7945   -ucneg 7946   # cap 8355   2c2 8783   ZZcz 9066   QQcq 9423   abscabs 10781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7723  ax-resscn 7724  ax-1cn 7725  ax-1re 7726  ax-icn 7727  ax-addcl 7728  ax-addrcl 7729  ax-mulcl 7730  ax-mulrcl 7731  ax-addcom 7732  ax-mulcom 7733  ax-addass 7734  ax-mulass 7735  ax-distr 7736  ax-i2m1 7737  ax-0lt1 7738  ax-1rid 7739  ax-0id 7740  ax-rnegex 7741  ax-precex 7742  ax-cnre 7743  ax-pre-ltirr 7744  ax-pre-ltwlin 7745  ax-pre-lttrn 7746  ax-pre-apti 7747  ax-pre-ltadd 7748  ax-pre-mulgt0 7749  ax-pre-mulext 7750  ax-arch 7751  ax-caucvg 7752
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-frec 6288  df-pnf 7814  df-mnf 7815  df-xr 7816  df-ltxr 7817  df-le 7818  df-sub 7947  df-neg 7948  df-reap 8349  df-ap 8356  df-div 8445  df-inn 8733  df-2 8791  df-3 8792  df-4 8793  df-n0 8990  df-z 9067  df-uz 9339  df-q 9424  df-rp 9454  df-seqfrec 10231  df-exp 10305  df-cj 10626  df-re 10627  df-im 10628  df-rsqrt 10782  df-abs 10783
This theorem is referenced by:  apdiff  13302
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