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Theorem halfnqq 6916
Description: One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)
Assertion
Ref Expression
halfnqq (𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem halfnqq
StepHypRef Expression
1 1nq 6872 . . . . . . . . 9 1QQ
2 addclnq 6881 . . . . . . . . 9 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) ∈ Q)
31, 1, 2mp2an 417 . . . . . . . 8 (1Q +Q 1Q) ∈ Q
4 recclnq 6898 . . . . . . . . 9 ((1Q +Q 1Q) ∈ Q → (*Q‘(1Q +Q 1Q)) ∈ Q)
53, 4ax-mp 7 . . . . . . . 8 (*Q‘(1Q +Q 1Q)) ∈ Q
6 distrnqg 6893 . . . . . . . 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 1271 . . . . . . 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 6899 . . . . . . . . 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 5627 . . . . . . 7 (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
117, 10eqtri 2105 . . . . . 6 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
1211oveq1i 5625 . . . . 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 5626 . . . . . 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 6881 . . . . . . . . 9 (((*Q‘(1Q +Q 1Q)) ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
155, 5, 14mp2an 417 . . . . . . . 8 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q
16 mulassnqg 6890 . . . . . . . 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 1271 . . . . . . 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 6889 . . . . . . . . 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 417 . . . . . . . 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 5625 . . . . . . 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 2107 . . . . . 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 403 . . . . . . 7 ((1Q +Q 1Q) ∈ Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
23 mulidnq 6895 . . . . . . 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 2113 . . . . 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 2113 . . . 4 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
2726oveq2i 5626 . . 3 (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
28 distrnqg 6893 . . . 4 ((𝐴Q ∧ (*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 1263 . . 3 (𝐴Q → (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 6895 . . 3 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
3127, 29, 303eqtr3a 2141 . 2 (𝐴Q → ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴)
32 mulclnq 6882 . . . 4 ((𝐴Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∈ Q)
335, 32mpan2 416 . . 3 (𝐴Q → (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 5633 . . . . 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 2093 . . . 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 271 . . 3 ((𝐴Q𝑥 = (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 2719 . 2 (𝐴Q → (((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴 → ∃𝑥Q (𝑥 +Q 𝑥) = 𝐴))
3931, 38mpd 13 1 (𝐴Q → ∃𝑥Q (𝑥 +Q 𝑥) = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1287  wcel 1436  wrex 2356  cfv 4983  (class class class)co 5615  Qcnq 6786  1Qc1q 6787   +Q cplq 6788   ·Q cmq 6789  *Qcrq 6790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-coll 3931  ax-sep 3934  ax-nul 3942  ax-pow 3986  ax-pr 4012  ax-un 4236  ax-setind 4328  ax-iinf 4378
This theorem depends on definitions:  df-bi 115  df-dc 779  df-3or 923  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-csb 2923  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3639  df-int 3674  df-iun 3717  df-br 3823  df-opab 3877  df-mpt 3878  df-tr 3914  df-id 4096  df-iord 4169  df-on 4171  df-suc 4174  df-iom 4381  df-xp 4419  df-rel 4420  df-cnv 4421  df-co 4422  df-dm 4423  df-rn 4424  df-res 4425  df-ima 4426  df-iota 4948  df-fun 4985  df-fn 4986  df-f 4987  df-f1 4988  df-fo 4989  df-f1o 4990  df-fv 4991  df-ov 5618  df-oprab 5619  df-mpt2 5620  df-1st 5870  df-2nd 5871  df-recs 6026  df-irdg 6091  df-1o 6137  df-oadd 6141  df-omul 6142  df-er 6246  df-ec 6248  df-qs 6252  df-ni 6810  df-pli 6811  df-mi 6812  df-plpq 6850  df-mpq 6851  df-enq 6853  df-nqqs 6854  df-plqqs 6855  df-mqqs 6856  df-1nqqs 6857  df-rq 6858
This theorem is referenced by:  halfnq  6917  nsmallnqq  6918  subhalfnqq  6920  addlocpr  7042  addcanprleml  7120  addcanprlemu  7121  cauappcvgprlemm  7151  cauappcvgprlem1  7165  caucvgprlemm  7174
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