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Theorem halfnqq 6267
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 6225 . . . . . . . . 9 1Q Q
2 addclnq 6234 . . . . . . . . 9 ((1Q Q 1Q Q) → (1Q +Q 1Q) Q)
31, 1, 2mp2an 404 . . . . . . . 8 (1Q +Q 1Q) Q
4 recclnq 6251 . . . . . . . . 9 ((1Q +Q 1Q) Q → (*Q‘(1Q +Q 1Q)) Q)
53, 4ax-mp 7 . . . . . . . 8 (*Q‘(1Q +Q 1Q)) Q
6 distrnqg 6246 . . . . . . . 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 1217 . . . . . . 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 6252 . . . . . . . . 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 5448 . . . . . . 7 (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
117, 10eqtri 2042 . . . . . 6 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
1211oveq1i 5446 . . . . 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 5447 . . . . . 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 6234 . . . . . . . . 9 (((*Q‘(1Q +Q 1Q)) Q (*Q‘(1Q +Q 1Q)) Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q)
155, 5, 14mp2an 404 . . . . . . . 8 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q
16 mulassnqg 6243 . . . . . . . 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 1217 . . . . . . 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 6242 . . . . . . . . 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 404 . . . . . . . 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 5446 . . . . . . 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 2044 . . . . . 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 393 . . . . . . 7 ((1Q +Q 1Q) Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) Q)
23 mulidnq 6248 . . . . . . 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 2050 . . . . 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 2050 . . . 4 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
2726oveq2i 5447 . . 3 (A ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (A ·Q 1Q)
28 distrnqg 6246 . . . 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 1209 . . 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 6248 . . 3 (A Q → (A ·Q 1Q) = A)
3127, 29, 303eqtr3a 2078 . 2 (A Q → ((A ·Q (*Q‘(1Q +Q 1Q))) +Q (A ·Q (*Q‘(1Q +Q 1Q)))) = A)
32 mulclnq 6235 . . . 4 ((A Q (*Q‘(1Q +Q 1Q)) Q) → (A ·Q (*Q‘(1Q +Q 1Q))) Q)
335, 32mpan2 403 . . 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 5454 . . . . 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 2030 . . . 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 2637 . 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 1228   wcel 1374  wrex 2285  cfv 4829  (class class class)co 5436  Qcnq 6138  1Qc1q 6139   +Q cplq 6140   ·Q cmq 6141  *Qcrq 6142
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 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-coll 3846  ax-sep 3849  ax-nul 3857  ax-pow 3901  ax-pr 3918  ax-un 4120  ax-setind 4204  ax-iinf 4238
This theorem depends on definitions:  df-bi 110  df-dc 734  df-3or 874  df-3an 875  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ne 2188  df-ral 2289  df-rex 2290  df-reu 2291  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-dif 2897  df-un 2899  df-in 2901  df-ss 2908  df-nul 3202  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-int 3590  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-tr 3829  df-id 4004  df-iord 4052  df-on 4054  df-suc 4057  df-iom 4241  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-f1 4834  df-fo 4835  df-f1o 4836  df-fv 4837  df-ov 5439  df-oprab 5440  df-mpt2 5441  df-1st 5690  df-2nd 5691  df-recs 5842  df-irdg 5878  df-1o 5916  df-oadd 5920  df-omul 5921  df-er 6017  df-ec 6019  df-qs 6023  df-ni 6164  df-pli 6165  df-mi 6166  df-plpq 6203  df-mpq 6204  df-enq 6206  df-nqqs 6207  df-plqqs 6208  df-mqqs 6209  df-1nqqs 6210  df-rq 6211
This theorem is referenced by:  halfnq  6268  nsmallnqq  6269  subhalfnqq  6271  addlocpr  6391  addcanprleml  6451  addcanprlemu  6452
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