Theorem halfnqq 7342

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 7298	. . . . . . . . 9 ⊢ 1Q ∈ Q
2		addclnq 7307	. . . . . . . . 9 ⊢ ((1Q ∈ Q ∧ 1Q ∈ Q) → (1Q +Q 1Q) ∈ Q)
3	1, 1, 2	mp2an 423	. . . . . . . 8 ⊢ (1Q +Q 1Q) ∈ Q
4		recclnq 7324	. . . . . . . . 9 ⊢ ((1Q +Q 1Q) ∈ Q → (*Q‘(1Q +Q 1Q)) ∈ Q)
5	3, 4	ax-mp 5	. . . . . . . 8 ⊢ (*Q‘(1Q +Q 1Q)) ∈ Q
6		distrnqg 7319	. . . . . . . 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 1326	. . . . . . 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 7325	. . . . . . . . 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 5848	. . . . . . 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 2185	. . . . . 6 ⊢ ((1Q +Q 1Q) ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (1Q +Q 1Q)
12	11	oveq1i 5846	. . . . 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 5847	. . . . . 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 7307	. . . . . . . . 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 423	. . . . . . . 8 ⊢ ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q
16		mulassnqg 7316	. . . . . . . 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 1326	. . . . . . 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 7315	. . . . . . . . 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 423	. . . . . . . 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 5846	. . . . . . 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 2187	. . . . . 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 409	. . . . . . 7 ⊢ ((1Q +Q 1Q) ∈ Q → ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) ∈ Q)
23		mulidnq 7321	. . . . . . 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 2193	. . . . 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 2193	. . . 4 ⊢ ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q))) = 1Q
27	26	oveq2i 5847	. . 3 ⊢ (𝐴 ·Q ((*Q‘(1Q +Q 1Q)) +Q (*Q‘(1Q +Q 1Q)))) = (𝐴 ·Q 1Q)
28		distrnqg 7319	. . . 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 1318	. . 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 7321	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 ·Q 1Q) = 𝐴)
31	27, 29, 30	3eqtr3a 2221	. 2 ⊢ (𝐴 ∈ Q → ((𝐴 ·Q (*Q‘(1Q +Q 1Q))) +Q (𝐴 ·Q (*Q‘(1Q +Q 1Q)))) = 𝐴)
32		mulclnq 7308	. . . 4 ⊢ ((𝐴 ∈ Q ∧ (*Q‘(1Q +Q 1Q)) ∈ Q) → (𝐴 ·Q (*Q‘(1Q +Q 1Q))) ∈ Q)
33	5, 32	mpan2 422	. . 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 5854	. . . . 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 2173	. . . 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 275	. . 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 2829	. 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 104   = wceq 1342   ∈ wcel 2135  ∃wrex 2443  ‘cfv 5182  (class class class)co 5836  Qcnq 7212  1_Qc1q 7213   +_Q cplq 7214   ·_Q cmq 7215  *_Qcrq 7216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-coll 4091  ax-sep 4094  ax-nul 4102  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-iinf 4559

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-rex 2448  df-reu 2449  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-int 3819  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-tr 4075  df-id 4265  df-iord 4338  df-on 4340  df-suc 4343  df-iom 4562  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-f1 5187  df-fo 5188  df-f1o 5189  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-recs 6264  df-irdg 6329  df-1o 6375  df-oadd 6379  df-omul 6380  df-er 6492  df-ec 6494  df-qs 6498  df-ni 7236  df-pli 7237  df-mi 7238  df-plpq 7276  df-mpq 7277  df-enq 7279  df-nqqs 7280  df-plqqs 7281  df-mqqs 7282  df-1nqqs 7283  df-rq 7284

This theorem is referenced by:  halfnq  7343  nsmallnqq  7344  subhalfnqq  7346  addlocpr  7468  addcanprleml  7546  addcanprlemu  7547  cauappcvgprlemm  7577  cauappcvgprlem1  7591  caucvgprlemm  7600