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Theorem halfnq 11014
Description: One-half of any positive fraction exists. Lemma for Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 16-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
Assertion
Ref Expression
halfnq (𝐴Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem halfnq
StepHypRef Expression
1 distrnq 10999 . . . 4 (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q))))
2 distrnq 10999 . . . . . . . 8 ((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))))
3 1nq 10966 . . . . . . . . . . 11 1QQ
4 addclnq 10983 . . . . . . . . . . 11 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) ∈ Q)
53, 3, 4mp2an 692 . . . . . . . . . 10 (1Q +Q 1Q) ∈ Q
6 recidnq 11003 . . . . . . . . . 10 ((1Q +Q 1Q) ∈ Q → ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q)
75, 6ax-mp 5 . . . . . . . . 9 ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q
87, 7oveq12i 7443 . . . . . . . 8 (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
92, 8eqtri 2763 . . . . . . 7 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
109oveq1i 7441 . . . . . 6 (((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)))
117oveq2i 7442 . . . . . . 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)
12 mulassnq 10997 . . . . . . . 8 ((((*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))))
13 mulcomnq 10991 . . . . . . . . 9 (((*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))))
1413oveq1i 7441 . . . . . . . 8 ((((*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)))
1512, 14eqtr3i 2765 . . . . . . 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)))
16 recclnq 11004 . . . . . . . . 9 ((1Q +Q 1Q) ∈ Q → (*Q‘(1Q +Q 1Q)) ∈ Q)
17 addclnq 10983 . . . . . . . . 9 (((*Q‘(1Q +Q 1Q)) ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
1816, 16, 17syl2anc 584 . . . . . . . 8 ((1Q +Q 1Q) ∈ Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
19 mulidnq 11001 . . . . . . . 8 (((*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))))
205, 18, 19mp2b 10 . . . . . . 7 (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q 1Q) = ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))
2111, 15, 203eqtr3i 2771 . . . . . 6 (((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)))
2210, 21, 73eqtr3i 2771 . . . . 5 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
2322oveq2i 7442 . . . 4 (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
241, 23eqtr3i 2765 . . 3 ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
25 mulidnq 11001 . . 3 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
2624, 25eqtrid 2787 . 2 (𝐴Q → ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴)
27 ovex 7464 . . 3 (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∈ V
28 oveq12 7440 . . . . 5 ((𝑥 = (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∧ 𝑥 = (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) → (𝑥 +Q 𝑥) = ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))))
2928anidms 566 . . . 4 (𝑥 = (𝐴 ·Q (*Q‘(1Q +Q 1Q))) → (𝑥 +Q 𝑥) = ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))))
3029eqeq1d 2737 . . 3 (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))
3127, 30spcev 3606 . 2 (((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴 → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
3226, 31syl 17 1 (𝐴Q → ∃𝑥(𝑥 +Q 𝑥) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wex 1776  wcel 2106  cfv 6563  (class class class)co 7431  Qcnq 10890  1Qc1q 10891   +Q cplq 10893   ·Q cmq 10894  *Qcrq 10895
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-oadd 8509  df-omul 8510  df-er 8744  df-ni 10910  df-pli 10911  df-mi 10912  df-lti 10913  df-plpq 10946  df-mpq 10947  df-enq 10949  df-nq 10950  df-erq 10951  df-plq 10952  df-mq 10953  df-1nq 10954  df-rq 10955
This theorem is referenced by:  nsmallnq  11015
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