ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  halfnqq Structured version   GIF version

Theorem halfnqq 6393
Description: One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
Assertion
Ref Expression
halfnqq (A Qx Q (x +Q x) = A)
Distinct variable group:   x,A

Proof of Theorem halfnqq
StepHypRef Expression
1 1nq 6350 . . . . . . . . 9 1Q Q
2 addclnq 6359 . . . . . . . . 9 ((1Q Q 1Q Q) → (1Q +Q 1Q) Q)
31, 1, 2mp2an 402 . . . . . . . 8 (1Q +Q 1Q) Q
4 recclnq 6376 . . . . . . . . 9 ((1Q +Q 1Q) Q → (*Q‘(1Q +Q 1Q)) Q)
53, 4ax-mp 7 . . . . . . . 8 (*Q‘(1Q +Q 1Q)) Q
6 distrnqg 6371 . . . . . . . 8 (((1Q +Q 1Q) Q (*Q‘(1Q +Q 1Q)) Q (*Q‘(1Q +Q 1Q)) Q) → ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))))
73, 5, 5, 6mp3an 1231 . . . . . . 7 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))))
8 recidnq 6377 . . . . . . . . 9 ((1Q +Q 1Q) Q → ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q)
93, 8ax-mp 7 . . . . . . . 8 ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q
109, 9oveq12i 5467 . . . . . . 7 (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
117, 10eqtri 2057 . . . . . 6 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
1211oveq1i 5465 . . . . 5 (((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) ·Q (*Q‘(1Q +Q 1Q))) = ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))
139oveq2i 5466 . . . . . 6 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q 1Q)
14 addclnq 6359 . . . . . . . . 9 (((*Q‘(1Q +Q 1Q)) Q (*Q‘(1Q +Q 1Q)) Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q)
155, 5, 14mp2an 402 . . . . . . . 8 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q
16 mulassnqg 6368 . . . . . . . 8 ((((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q (1Q +Q 1Q) Q (*Q‘(1Q +Q 1Q)) Q) → ((((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q (1Q +Q 1Q)) ·Q (*Q‘(1Q +Q 1Q))) = (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))))
1715, 3, 5, 16mp3an 1231 . . . . . . 7 ((((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q (1Q +Q 1Q)) ·Q (*Q‘(1Q +Q 1Q))) = (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))))
18 mulcomnqg 6367 . . . . . . . . 9 ((((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q (1Q +Q 1Q) Q) → (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q (1Q +Q 1Q)) = ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))))
1915, 3, 18mp2an 402 . . . . . . . 8 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q (1Q +Q 1Q)) = ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))))
2019oveq1i 5465 . . . . . . 7 ((((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q (1Q +Q 1Q)) ·Q (*Q‘(1Q +Q 1Q))) = (((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) ·Q (*Q‘(1Q +Q 1Q)))
2117, 20eqtr3i 2059 . . . . . 6 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) ·Q (*Q‘(1Q +Q 1Q)))
224, 4, 14syl2anc 391 . . . . . . 7 ((1Q +Q 1Q) Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q)
23 mulidnq 6373 . . . . . . 7 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q → (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q 1Q) = ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))))
243, 22, 23mp2b 8 . . . . . 6 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q 1Q) = ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))
2513, 21, 243eqtr3i 2065 . . . . 5 (((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) ·Q (*Q‘(1Q +Q 1Q))) = ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))
2612, 25, 93eqtr3i 2065 . . . 4 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
2726oveq2i 5466 . . 3 (A ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (A ·Q 1Q)
28 distrnqg 6371 . . . 4 ((A Q (*Q‘(1Q +Q 1Q)) Q (*Q‘(1Q +Q 1Q)) Q) → (A ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))))
295, 5, 28mp3an23 1223 . . 3 (A Q → (A ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))))
30 mulidnq 6373 . . 3 (A Q → (A ·Q 1Q) = A)
3127, 29, 303eqtr3a 2093 . 2 (A Q → ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))) = A)
32 mulclnq 6360 . . . 4 ((A Q (*Q‘(1Q +Q 1Q)) Q) → (A ·Q (*Q‘(1Q +Q 1Q))) Q)
335, 32mpan2 401 . . 3 (A Q → (A ·Q (*Q‘(1Q +Q 1Q))) Q)
34 id 19 . . . . . 6 (x = (A ·Q (*Q‘(1Q +Q 1Q))) → x = (A ·Q (*Q‘(1Q +Q 1Q))))
3534, 34oveq12d 5473 . . . . 5 (x = (A ·Q (*Q‘(1Q +Q 1Q))) → (x +Q x) = ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))))
3635eqeq1d 2045 . . . 4 (x = (A ·Q (*Q‘(1Q +Q 1Q))) → ((x +Q x) = A ↔ ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))) = A))
3736adantl 262 . . 3 ((A Q x = (A ·Q (*Q‘(1Q +Q 1Q)))) → ((x +Q x) = A ↔ ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))) = A))
3833, 37rspcedv 2654 . 2 (A Q → (((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))) = Ax Q (x +Q x) = A))
3931, 38mpd 13 1 (A Qx Q (x +Q x) = A)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242   wcel 1390  wrex 2301  cfv 4845  (class class class)co 5455  Qcnq 6264  1Qc1q 6265   +Q cplq 6266   ·Q cmq 6267  *Qcrq 6268
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-bnd 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
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-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-id 4021  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-ov 5458  df-oprab 5459  df-mpt2 5460  df-1st 5709  df-2nd 5710  df-recs 5861  df-irdg 5897  df-1o 5940  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-plpq 6328  df-mpq 6329  df-enq 6331  df-nqqs 6332  df-plqqs 6333  df-mqqs 6334  df-1nqqs 6335  df-rq 6336
This theorem is referenced by:  halfnq  6394  nsmallnqq  6395  subhalfnqq  6397  addlocpr  6519  addcanprleml  6586  addcanprlemu  6587
  Copyright terms: Public domain W3C validator