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Theorem apreap 7371
Description: Complex apartness and real apartness agree on the real numbers. (Contributed by Jim Kingdon, 31-Jan-2020.)
Assertion
Ref Expression
apreap  RR  RR # #

Proof of Theorem apreap
Dummy variables  r  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqeq1 2043 . . . . . . . 8  r  +  _i  x.  s  r  +  _i  x.  s
21anbi1d 438 . . . . . . 7 
r  +  _i  x.  s 
t  +  _i  x.  r  +  _i  x.  s 
t  +  _i  x.
32anbi1d 438 . . . . . 6  r  +  _i  x.  s 
t  +  _i  x.  r #  t  s #  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #
432rexbidv 2343 . . . . 5  t  RR  RR 
r  +  _i  x.  s 
t  +  _i  x.  r #  t  s #  t  RR  RR  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #
542rexbidv 2343 . . . 4  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #
6 eqeq1 2043 . . . . . . . 8  t  +  _i  x.  t  +  _i  x.
76anbi2d 437 . . . . . . 7 
r  +  _i  x.  s 
t  +  _i  x.  r  +  _i  x.  s  t  +  _i  x.
87anbi1d 438 . . . . . 6  r  +  _i  x.  s 
t  +  _i  x.  r #  t  s #  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #
982rexbidv 2343 . . . . 5  t  RR  RR 
r  +  _i  x.  s 
t  +  _i  x.  r #  t  s #  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
1092rexbidv 2343 . . . 4  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
11 df-ap 7366 . . . 4 #  { <. ,  >.  |  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x. 
r #  t  s #  }
125, 10, 11brabg 3997 . . 3  RR  RR #  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
13 simplll 485 . . . . . . . . . . . . 13  RR  RR  r  RR  s  RR  t  RR  RR  RR
1413adantr 261 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  RR
15 simplrl 487 . . . . . . . . . . . . 13  RR  RR  r  RR  s  RR  t  RR  RR  r  RR
1615adantr 261 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  r  RR
17 simplrr 488 . . . . . . . . . . . . 13  RR  RR  r  RR  s  RR  t  RR  RR  s  RR
1817adantr 261 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  s  RR
19 simprll 489 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  r  +  _i  x.  s
20 rereim 7370 . . . . . . . . . . . 12  RR  r  RR  s  RR  r  +  _i  x.  s  r  s  0
2114, 16, 18, 19, 20syl22anc 1135 . . . . . . . . . . 11  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # 
r  s  0
2221simprd 107 . . . . . . . . . 10  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  s  0
23 simpllr 486 . . . . . . . . . . . . 13  RR  RR  r  RR  s  RR  t  RR  RR  RR
2423adantr 261 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  RR
25 simplrl 487 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  t  RR
26 simplrr 488 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  RR
27 simprlr 490 . . . . . . . . . . . 12  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  t  +  _i  x.
28 rereim 7370 . . . . . . . . . . . 12  RR  t  RR  RR  t  +  _i  x.  t  0
2924, 25, 26, 27, 28syl22anc 1135 . . . . . . . . . . 11  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # 
t  0
3029simprd 107 . . . . . . . . . 10  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  0
3122, 30eqtr4d 2072 . . . . . . . . 9  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  s
32 reapti 7363 . . . . . . . . . 10  s  RR  RR  s  s #
3318, 26, 32syl2anc 391 . . . . . . . . 9  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # 
s  s #
3431, 33mpbid 135 . . . . . . . 8  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  s #
35 simprr 484 . . . . . . . 8  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # 
r #  t  s #
3634, 35ecased 1238 . . . . . . 7  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  r #  t
3721simpld 105 . . . . . . 7  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  r
3829simpld 105 . . . . . . 7  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  t
3936, 37, 383brtr3d 3784 . . . . . 6  RR  RR 
r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # #
4039ex 108 . . . . 5  RR  RR  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # #
4140rexlimdvva 2434 . . . 4  RR  RR  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # #
4241rexlimdvva 2434 . . 3  RR  RR  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s # #
4312, 42sylbid 139 . 2  RR  RR # #
44 ax-icn 6778 . . . . . . . 8  _i  CC
4544mul01i 7184 . . . . . . 7  _i  x.  0  0
4645oveq2i 5466 . . . . . 6  +  _i  x.  0  +  0
47 simp1 903 . . . . . . . 8  RR  RR #  RR
4847recnd 6851 . . . . . . 7  RR  RR #  CC
4948addid1d 6959 . . . . . 6  RR  RR #  + 
0
5046, 49syl5req 2082 . . . . 5  RR  RR #  +  _i  x.  0
5145oveq2i 5466 . . . . . 6  +  _i  x.  0  +  0
52 simp2 904 . . . . . . . 8  RR  RR #  RR
5352recnd 6851 . . . . . . 7  RR  RR #  CC
5453addid1d 6959 . . . . . 6  RR  RR #  + 
0
5551, 54syl5req 2082 . . . . 5  RR  RR #  +  _i  x.  0
56 olc 631 . . . . . . 7 #  0 #  0 #
57563ad2ant3 926 . . . . . 6  RR  RR #  0 #  0 #
5857orcomd 647 . . . . 5  RR  RR # #  0 #  0
5950, 55, 58jca31 292 . . . 4  RR  RR #  +  _i  x.  0  +  _i  x.  0 #  0 #  0
60 0red 6826 . . . . . . . 8  RR  RR #  0  RR
61 simpr 103 . . . . . . . . . . . . 13  RR  RR #  0  0
6261oveq2d 5471 . . . . . . . . . . . 12  RR  RR #  0  _i  x.  _i  x.  0
6362oveq2d 5471 . . . . . . . . . . 11  RR  RR #  0  +  _i  x.  +  _i  x.  0
6463eqeq2d 2048 . . . . . . . . . 10  RR  RR #  0  +  _i  x.  +  _i  x.  0
6564anbi2d 437 . . . . . . . . 9  RR  RR #  0  +  _i  x.  0  +  _i  x.  +  _i  x.  0  +  _i  x.  0
6661breq2d 3767 . . . . . . . . . 10  RR  RR #  0  0 #  0 #  0
6766orbi2d 703 . . . . . . . . 9  RR  RR #  0 #  0 # #  0 #  0
6865, 67anbi12d 442 . . . . . . . 8  RR  RR #  0  +  _i  x.  0  +  _i  x. #  0 #  +  _i  x.  0  +  _i  x.  0 #  0 #  0
6960, 68rspcedv 2654 . . . . . . 7  RR  RR #  +  _i  x.  0  +  _i  x.  0 #  0 #  0  RR  +  _i  x.  0  +  _i  x. #  0 #
70 simpr 103 . . . . . . . . . . . . 13  RR  RR #  t  t
7170oveq1d 5470 . . . . . . . . . . . 12  RR  RR #  t 
t  +  _i  x.  +  _i  x.
7271eqeq2d 2048 . . . . . . . . . . 11  RR  RR #  t  t  +  _i  x.  +  _i  x.
7372anbi2d 437 . . . . . . . . . 10  RR  RR #  t  +  _i  x.  0  t  +  _i  x.  +  _i  x.  0  +  _i  x.
7470breq2d 3767 . . . . . . . . . . 11  RR  RR #  t #  t #
7574orbi1d 704 . . . . . . . . . 10  RR  RR #  t #  t  0 # #  0 #
7673, 75anbi12d 442 . . . . . . . . 9  RR  RR #  t  +  _i  x.  0  t  +  _i  x. #  t  0 #  +  _i  x.  0  +  _i  x. #  0 #
7776rexbidv 2321 . . . . . . . 8  RR  RR #  t  RR  +  _i  x.  0  t  +  _i  x. #  t  0 #  RR  +  _i  x.  0  +  _i  x. #  0 #
7852, 77rspcedv 2654 . . . . . . 7  RR  RR #  RR  +  _i  x.  0  +  _i  x. #  0 #  t  RR  RR  +  _i  x.  0  t  +  _i  x. #  t  0 #
7969, 78syld 40 . . . . . 6  RR  RR #  +  _i  x.  0  +  _i  x.  0 #  0 #  0  t  RR  RR  +  _i  x.  0  t  +  _i  x. #  t  0 #
80 simpr 103 . . . . . . . . . . . . 13  RR  RR #  s  0  s  0
8180oveq2d 5471 . . . . . . . . . . . 12  RR  RR #  s  0  _i  x.  s  _i  x.  0
8281oveq2d 5471 . . . . . . . . . . 11  RR  RR #  s  0  +  _i  x.  s  +  _i  x.  0
8382eqeq2d 2048 . . . . . . . . . 10  RR  RR #  s  0  +  _i  x.  s  +  _i  x.  0
8483anbi1d 438 . . . . . . . . 9  RR  RR #  s  0  +  _i  x.  s  t  +  _i  x.  +  _i  x.  0  t  +  _i  x.
8580breq1d 3765 . . . . . . . . . 10  RR  RR #  s  0  s #  0 #
8685orbi2d 703 . . . . . . . . 9  RR  RR #  s  0 #  t  s # #  t  0 #
8784, 86anbi12d 442 . . . . . . . 8  RR  RR #  s  0  +  _i  x.  s  t  +  _i  x. #  t  s #  +  _i  x.  0  t  +  _i  x. #  t  0 #
88872rexbidv 2343 . . . . . . 7  RR  RR #  s  0  t  RR  RR  +  _i  x.  s  t  +  _i  x. #  t  s #  t  RR  RR  +  _i  x.  0  t  +  _i  x. #  t  0 #
8960, 88rspcedv 2654 . . . . . 6  RR  RR #  t  RR  RR  +  _i  x.  0  t  +  _i  x. #  t  0 #  s  RR  t  RR  RR  +  _i  x.  s  t  +  _i  x. #  t  s #
90 simpr 103 . . . . . . . . . . . . 13  RR  RR #  r  r
9190oveq1d 5470 . . . . . . . . . . . 12  RR  RR #  r 
r  +  _i  x.  s  +  _i  x.  s
9291eqeq2d 2048 . . . . . . . . . . 11  RR  RR #  r  r  +  _i  x.  s  +  _i  x.  s
9392anbi1d 438 . . . . . . . . . 10  RR  RR #  r 
r  +  _i  x.  s  t  +  _i  x.  +  _i  x.  s  t  +  _i  x.
9490breq1d 3765 . . . . . . . . . . 11  RR  RR #  r 
r #  t #  t
9594orbi1d 704 . . . . . . . . . 10  RR  RR #  r  r #  t  s # #  t  s #
9693, 95anbi12d 442 . . . . . . . . 9  RR  RR #  r  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  +  _i  x.  s  t  +  _i  x. #  t  s #
9796rexbidv 2321 . . . . . . . 8  RR  RR #  r  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  RR  +  _i  x.  s  t  +  _i  x. #  t  s #
98972rexbidv 2343 . . . . . . 7  RR  RR #  r  s  RR  t  RR  RR 
r  +  _i  x.  s  t  +  _i  x.  r #  t  s #  s  RR  t  RR  RR  +  _i  x.  s  t  +  _i  x. #  t  s #
9947, 98rspcedv 2654 . . . . . 6  RR  RR #  s  RR  t  RR  RR  +  _i  x.  s  t  +  _i  x. #  t  s #  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
10079, 89, 993syld 51 . . . . 5  RR  RR #  +  _i  x.  0  +  _i  x.  0 #  0 #  0  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
101123adant3 923 . . . . 5  RR  RR # #  r  RR  s  RR  t  RR  RR  r  +  _i  x.  s  t  +  _i  x.  r #  t  s #
102100, 101sylibrd 158 . . . 4  RR  RR #  +  _i  x.  0  +  _i  x.  0 #  0 #  0 #
10359, 102mpd 13 . . 3  RR  RR # #
1041033expia 1105 . 2  RR  RR # #
10543, 104impbid 120 1  RR  RR # #
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97   wb 98   wo 628   w3a 884   wceq 1242   wcel 1390  wrex 2301   class class class wbr 3755  (class class class)co 5455   RRcr 6710   0cc0 6711   _ici 6713    + caddc 6714    x. cmul 6716   # creap 7358   # cap 7365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-coll 3863  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-iinf 4254  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-mulrcl 6782  ax-addcom 6783  ax-mulcom 6784  ax-addass 6785  ax-mulass 6786  ax-distr 6787  ax-i2m1 6788  ax-1rid 6790  ax-0id 6791  ax-rnegex 6792  ax-precex 6793  ax-cnre 6794  ax-pre-ltirr 6795  ax-pre-lttrn 6797  ax-pre-apti 6798  ax-pre-ltadd 6799  ax-pre-mulgt0 6800
This theorem depends on definitions:  df-bi 110  df-dc 742  df-3or 885  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-eprel 4017  df-id 4021  df-po 4024  df-iso 4025  df-iord 4069  df-on 4071  df-suc 4074  df-iom 4257  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  df-2o 5941  df-oadd 5944  df-omul 5945  df-er 6042  df-ec 6044  df-qs 6048  df-ni 6288  df-pli 6289  df-mi 6290  df-lti 6291  df-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336  df-ltnqqs 6337  df-enq0 6407  df-nq0 6408  df-0nq0 6409  df-plq0 6410  df-mq0 6411  df-inp 6449  df-i1p 6450  df-iplp 6451  df-iltp 6453  df-enr 6654  df-nr 6655  df-ltr 6658  df-0r 6659  df-1r 6660  df-0 6718  df-1 6719  df-r 6721  df-lt 6724  df-pnf 6859  df-mnf 6860  df-ltxr 6862  df-sub 6981  df-neg 6982  df-reap 7359  df-ap 7366
This theorem is referenced by:  reaplt  7372  apreim  7387  apirr  7389  apti  7406  recexap  7416  rerecclap  7488
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