Theorem halfnqq 7543

Description: One-half of any positive fraction is a fraction. (Contributed by Jim Kingdon, 23-Sep-2019.)

Assertion

Ref	Expression
halfnqq	⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem halfnqq

Step	Hyp	Ref	Expression
1		1nq 7499	. . . . . . . . 9 ⊢ 1Q ∈ Q
2		addclnq 7508	. . . . . . . . 9 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) ∈ Q)
3	1, 1, 2	mp2an 426	. . . . . . . 8 ⊢ (1Q +Q 1Q) ∈ Q
4		recclnq 7525	. . . . . . . . 9 ⊢ ((1Q +Q 1Q) ∈ Q → (*Q‘(1Q +Q 1Q)) ∈ Q)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (*Q‘(1Q +Q 1Q)) ∈ Q
6		distrnqg 7520	. . . . . . . 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)))))
7	3, 5, 5, 6	mp3an 1350	. . . . . . 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 7526	. . . . . . . . 9 ⊢ ((1Q +Q 1Q) ∈ Q → ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q)
9	3, 8	ax-mp 5	. . . . . . . 8 ⊢ ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) = 1Q
10	9, 9	oveq12i 5969	. . . . . . 7 ⊢ (((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q))) +Q ((1Q +Q 1Q) ·Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
11	7, 10	eqtri 2227	. . . . . 6 ⊢ ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
12	11	oveq1i 5967	. . . . 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)))
13	9	oveq2i 5968	. . . . . 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 7508	. . . . . . . . 9 ⊢ (((*Q‘(1Q +Q 1Q)) ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
15	5, 5, 14	mp2an 426	. . . . . . . 8 ⊢ ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q
16		mulassnqg 7517	. . . . . . . 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)))))
17	15, 3, 5, 16	mp3an 1350	. . . . . . 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 7516	. . . . . . . . 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)))))
19	15, 3, 18	mp2an 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))))
20	19	oveq1i 5967	. . . . . . 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)))
21	17, 20	eqtr3i 2229	. . . . . 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)))
22	4, 4, 14	syl2anc 411	. . . . . . 7 ⊢ ((1Q +Q 1Q) ∈ Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
23		mulidnq 7522	. . . . . . 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))))
24	3, 22, 23	mp2b 8	. . . . . 6 ⊢ (((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ·Q 1Q) = ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))
25	13, 21, 24	3eqtr3i 2235	. . . . 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)))
26	12, 25, 9	3eqtr3i 2235	. . . 4 ⊢ ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
27	26	oveq2i 5968	. . 3 ⊢ (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
28		distrnqg 7520	. . . 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)))))
29	5, 5, 28	mp3an23 1342	. . 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 7522	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
31	27, 29, 30	3eqtr3a 2263	. 2 ⊢ (𝐴 ∈ Q → ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴)
32		mulclnq 7509	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∈ Q)
33	5, 32	mpan2 425	. . 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))))
35	34, 34	oveq12d 5975	. . . . 5 ⊢ (𝑥 = (𝐴 ·Q (*Q‘(1Q +Q 1Q))) → (𝑥 +Q 𝑥) = ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))))
36	35	eqeq1d 2215	. . . 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))
37	36	adantl 277	. . 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))
38	33, 37	rspcedv 2885	. 2 ⊢ (𝐴 ∈ Q → (((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴 → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝐴))
39	31, 38	mpd 13	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ Q (𝑥 +Q 𝑥) = 𝐴)

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  ‘cfv 5280  (class class class)co 5957  Qcnq 7413  1_Qc1q 7414   +_Q cplq 7415   ·_Q cmq 7416  *_Qcrq 7417

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-coll 4167  ax-sep 4170  ax-nul 4178  ax-pow 4226  ax-pr 4261  ax-un 4488  ax-setind 4593  ax-iinf 4644

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-csb 3098  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3623  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-int 3892  df-iun 3935  df-br 4052  df-opab 4114  df-mpt 4115  df-tr 4151  df-id 4348  df-iord 4421  df-on 4423  df-suc 4426  df-iom 4647  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-rn 4694  df-res 4695  df-ima 4696  df-iota 5241  df-fun 5282  df-fn 5283  df-f 5284  df-f1 5285  df-fo 5286  df-f1o 5287  df-fv 5288  df-ov 5960  df-oprab 5961  df-mpo 5962  df-1st 6239  df-2nd 6240  df-recs 6404  df-irdg 6469  df-1o 6515  df-oadd 6519  df-omul 6520  df-er 6633  df-ec 6635  df-qs 6639  df-ni 7437  df-pli 7438  df-mi 7439  df-plpq 7477  df-mpq 7478  df-enq 7480  df-nqqs 7481  df-plqqs 7482  df-mqqs 7483  df-1nqqs 7484  df-rq 7485

This theorem is referenced by:  halfnq  7544  nsmallnqq  7545  subhalfnqq  7547  addlocpr  7669  addcanprleml  7747  addcanprlemu  7748  cauappcvgprlemm  7778  cauappcvgprlem1  7792  caucvgprlemm  7801