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Theorem halfnqq 7387
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 7343 . . . . . . . . 9 1QQ
2 addclnq 7352 . . . . . . . . 9 ((1QQ ∧ 1QQ) → (1Q +Q 1Q) ∈ Q)
31, 1, 2mp2an 426 . . . . . . . 8 (1Q +Q 1Q) ∈ Q
4 recclnq 7369 . . . . . . . . 9 ((1Q +Q 1Q) ∈ Q → (*Q‘(1Q +Q 1Q)) ∈ Q)
53, 4ax-mp 5 . . . . . . . 8 (*Q‘(1Q +Q 1Q)) ∈ Q
6 distrnqg 7364 . . . . . . . 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 1337 . . . . . . 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 7370 . . . . . . . . 9 ((1Q +Q 1Q) ∈ Q → ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q)
93, 8ax-mp 5 . . . . . . . 8 ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q
109, 9oveq12i 5880 . . . . . . 7 (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
117, 10eqtri 2198 . . . . . 6 ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
1211oveq1i 5878 . . . . 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 5879 . . . . . 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 7352 . . . . . . . . 9 (((*Q‘(1Q +Q 1Q)) ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
155, 5, 14mp2an 426 . . . . . . . 8 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q
16 mulassnqg 7361 . . . . . . . 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 1337 . . . . . . 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 7360 . . . . . . . . 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 426 . . . . . . . 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 5878 . . . . . . 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 2200 . . . . . 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 411 . . . . . . 7 ((1Q +Q 1Q) ∈ Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
23 mulidnq 7366 . . . . . . 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 2206 . . . . 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 2206 . . . 4 ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
2726oveq2i 5879 . . 3 (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
28 distrnqg 7364 . . . 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 1329 . . 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 7366 . . 3 (𝐴Q → (𝐴 ·Q 1Q) = 𝐴)
3127, 29, 303eqtr3a 2234 . 2 (𝐴Q → ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴)
32 mulclnq 7353 . . . 4 ((𝐴Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∈ Q)
335, 32mpan2 425 . . 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 5886 . . . . 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 2186 . . . 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 277 . . 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 2845 . 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 105   = wceq 1353  wcel 2148  wrex 2456  cfv 5211  (class class class)co 5868  Qcnq 7257  1Qc1q 7258   +Q cplq 7259   ·Q cmq 7260  *Qcrq 7261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-iinf 4583
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4289  df-iord 4362  df-on 4364  df-suc 4367  df-iom 4586  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-recs 6299  df-irdg 6364  df-1o 6410  df-oadd 6414  df-omul 6415  df-er 6528  df-ec 6530  df-qs 6534  df-ni 7281  df-pli 7282  df-mi 7283  df-plpq 7321  df-mpq 7322  df-enq 7324  df-nqqs 7325  df-plqqs 7326  df-mqqs 7327  df-1nqqs 7328  df-rq 7329
This theorem is referenced by:  halfnq  7388  nsmallnqq  7389  subhalfnqq  7391  addlocpr  7513  addcanprleml  7591  addcanprlemu  7592  cauappcvgprlemm  7622  cauappcvgprlem1  7636  caucvgprlemm  7645
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