ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  irraddap Unicode version
Theorem irraddap 10056
Description: The sum of an irrational number and a rational number is irrational. (Contributed by Jim Kingdon, 20-Aug-2026.)
Assertion
Ref Expression
irraddap |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> ( ( A + B ) e. RR /\ A. q e. QQ ( A + B ) # q ) )
Distinct variable groups:   A, q   B, q
Proof of Theorem irraddap
Dummy variable r is distinct from all other variables.
StepHypRef Expression
1 simpll 531 . . 3 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> A e. RR )
2 qre 10034 . . . 4 |- ( B e. QQ -> B e. RR )
32adantl 277 . . 3 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> B e. RR )
41, 3readdcld 8355 . 2 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> ( A + B ) e. RR )
5 breq2 4134 . . . . . . 7 |- ( q = ( r - B ) -> ( A # q <-> A # ( r - B ) ) )
6 simpllr 540 . . . . . . 7 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> A. q e. QQ A # q )
7 simpr 110 . . . . . . . 8 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> r e. QQ )
8 simplr 533 . . . . . . . 8 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> B e. QQ )
9 qsubcl 10047 . . . . . . . 8 |- ( ( r e. QQ /\ B e. QQ ) -> ( r - B ) e. QQ )
107, 8, 9syl2anc 415 . . . . . . 7 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( r - B ) e. QQ )
115, 6, 10rspcdva 2934 . . . . . 6 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> A # ( r - B ) )
12 simplll 539 . . . . . . 7 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> A e. RR )
13 qre 10034 . . . . . . . 8 |- ( ( r - B ) e. QQ -> ( r - B ) e. RR )
1410, 13syl 14 . . . . . . 7 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( r - B ) e. RR )
158, 2syl 14 . . . . . . 7 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> B e. RR )
16 reapadd1 8926 . . . . . . 7 |- ( ( A e. RR /\ ( r - B ) e. RR /\ B e. RR ) -> ( A # ( r - B ) <-> ( A + B ) # ( ( r - B ) + B ) ) )
1712, 14, 15, 16syl3anc 1278 . . . . . 6 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( A # ( r - B ) <-> ( A + B ) # ( ( r - B ) + B ) ) )
1811, 17mpbid 147 . . . . 5 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( A + B ) # ( ( r - B ) + B ) )
19 qcn 10043 . . . . . . 7 |- ( r e. QQ -> r e. CC )
2019adantl 277 . . . . . 6 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> r e. CC )
2115recnd 8354 . . . . . 6 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> B e. CC )
2220, 21npcand 8642 . . . . 5 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( ( r - B ) + B ) = r )
2318, 22breqtrd 4156 . . . 4 |- ( ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) /\ r e. QQ ) -> ( A + B ) # r )
2423ralrimiva 2623 . . 3 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> A. r e. QQ ( A + B ) # r )
25 breq2 4134 . . . 4 |- ( r = q -> ( ( A + B ) # r <-> ( A + B ) # q ) )
2625cbvralv 2786 . . 3 |- ( A. r e. QQ ( A + B ) # r <-> A. q e. QQ ( A + B ) # q )
2724, 26sylib 122 . 2 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> A. q e. QQ ( A + B ) # q )
284, 27jca 306 1 |- ( ( ( A e. RR /\ A. q e. QQ A # q ) /\ B e. QQ ) -> ( ( A + B ) e. RR /\ A. q e. QQ ( A + B ) # q ) )
Colors of variables:    wff set class
This proof depends on syntax axioms:   -> wi 4   /\ wa 104   <-> wb 105   e. wcel 2209   A.wral 2528   class class class wbr 4130  (class class class)co 6085   CCcc 8177   RRcr 8178   + caddc 8182   - cmin 8498   # cap 8911   QQcq 10028
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297
This proof depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-po 4441  df-iso 4442  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-n0 9568  df-z 9649  df-q 10029
This theorem is used by:  zprmlogbaplem2  16135
  Copyright terms: Public domain W3C validator