Theorem archnq 9802

Description: For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)

Assertion

Ref	Expression
archnq	⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1𝑜⟩)

Distinct variable group:   𝑥,𝐴

Proof of Theorem archnq

Step	Hyp	Ref	Expression
1		elpqn 9747	. . . 4 ⊢ (𝐴 ∈ Q → 𝐴 ∈ (N × N))
2		xp1st 7198	. . . 4 ⊢ (𝐴 ∈ (N × N) → (1st ‘𝐴) ∈ N)
3	1, 2	syl 17	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) ∈ N)
4		1pi 9705	. . 3 ⊢ 1𝑜 ∈ N
5		addclpi 9714	. . 3 ⊢ (((1st ‘𝐴) ∈ N ∧ 1𝑜 ∈ N) → ((1st ‘𝐴) +N 1𝑜) ∈ N)
6	3, 4, 5	sylancl 694	. 2 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) +N 1𝑜) ∈ N)
7		xp2nd 7199	. . . . . 6 ⊢ (𝐴 ∈ (N × N) → (2nd ‘𝐴) ∈ N)
8	1, 7	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → (2nd ‘𝐴) ∈ N)
9		mulclpi 9715	. . . . 5 ⊢ ((((1st ‘𝐴) +N 1𝑜) ∈ N ∧ (2nd ‘𝐴) ∈ N) → (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) ∈ N)
10	6, 8, 9	syl2anc 693	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) ∈ N)
11		eqid 2622	. . . . . . 7 ⊢ ((1st ‘𝐴) +N 1𝑜) = ((1st ‘𝐴) +N 1𝑜)
12		oveq2 6658	. . . . . . . . 9 ⊢ (𝑥 = 1𝑜 → ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜))
13	12	eqeq1d 2624	. . . . . . . 8 ⊢ (𝑥 = 1𝑜 → (((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜) ↔ ((1st ‘𝐴) +N 1𝑜) = ((1st ‘𝐴) +N 1𝑜)))
14	13	rspcev 3309	. . . . . . 7 ⊢ ((1𝑜 ∈ N ∧ ((1st ‘𝐴) +N 1𝑜) = ((1st ‘𝐴) +N 1𝑜)) → ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜))
15	4, 11, 14	mp2an 708	. . . . . 6 ⊢ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜)
16		ltexpi 9724	. . . . . 6 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1𝑜) ∈ N) → ((1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜) ↔ ∃𝑥 ∈ N ((1st ‘𝐴) +N 𝑥) = ((1st ‘𝐴) +N 1𝑜)))
17	15, 16	mpbiri 248	. . . . 5 ⊢ (((1st ‘𝐴) ∈ N ∧ ((1st ‘𝐴) +N 1𝑜) ∈ N) → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜))
18	3, 6, 17	syl2anc 693	. . . 4 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜))
19		nlt1pi 9728	. . . . 5 ⊢ ¬ (2nd ‘𝐴) <N 1𝑜
20		ltmpi 9726	. . . . . . 7 ⊢ (((1st ‘𝐴) +N 1𝑜) ∈ N → ((2nd ‘𝐴) <N 1𝑜 ↔ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1𝑜) ·N 1𝑜)))
21	6, 20	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1𝑜 ↔ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1𝑜) ·N 1𝑜)))
22		mulidpi 9708	. . . . . . . 8 ⊢ (((1st ‘𝐴) +N 1𝑜) ∈ N → (((1st ‘𝐴) +N 1𝑜) ·N 1𝑜) = ((1st ‘𝐴) +N 1𝑜))
23	6, 22	syl 17	. . . . . . 7 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) +N 1𝑜) ·N 1𝑜) = ((1st ‘𝐴) +N 1𝑜))
24	23	breq2d 4665	. . . . . 6 ⊢ (𝐴 ∈ Q → ((((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N (((1st ‘𝐴) +N 1𝑜) ·N 1𝑜) ↔ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜)))
25	21, 24	bitrd 268	. . . . 5 ⊢ (𝐴 ∈ Q → ((2nd ‘𝐴) <N 1𝑜 ↔ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜)))
26	19, 25	mtbii 316	. . . 4 ⊢ (𝐴 ∈ Q → ¬ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜))
27		ltsopi 9710	. . . . 5 ⊢ <N Or N
28		ltrelpi 9711	. . . . 5 ⊢ <N ⊆ (N × N)
29	27, 28	sotri3 5526	. . . 4 ⊢ (((((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) ∈ N ∧ (1st ‘𝐴) <N ((1st ‘𝐴) +N 1𝑜) ∧ ¬ (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)) <N ((1st ‘𝐴) +N 1𝑜)) → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)))
30	10, 18, 26, 29	syl3anc 1326	. . 3 ⊢ (𝐴 ∈ Q → (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)))
31		pinq 9749	. . . . . 6 ⊢ (((1st ‘𝐴) +N 1𝑜) ∈ N → ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
32	6, 31	syl 17	. . . . 5 ⊢ (𝐴 ∈ Q → ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q)
33		ordpinq 9765	. . . . 5 ⊢ ((𝐴 ∈ Q ∧ ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ∈ Q) → (𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd ‘𝐴))))
34	32, 33	mpdan 702	. . . 4 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ↔ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd ‘𝐴))))
35		ovex 6678	. . . . . . . 8 ⊢ ((1st ‘𝐴) +N 1𝑜) ∈ V
36	4	elexi 3213	. . . . . . . 8 ⊢ 1𝑜 ∈ V
37	35, 36	op2nd 7177	. . . . . . 7 ⊢ (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) = 1𝑜
38	37	oveq2i 6661	. . . . . 6 ⊢ ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) = ((1st ‘𝐴) ·N 1𝑜)
39		mulidpi 9708	. . . . . . 7 ⊢ ((1st ‘𝐴) ∈ N → ((1st ‘𝐴) ·N 1𝑜) = (1st ‘𝐴))
40	3, 39	syl 17	. . . . . 6 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N 1𝑜) = (1st ‘𝐴))
41	38, 40	syl5eq 2668	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) = (1st ‘𝐴))
42	35, 36	op1st 7176	. . . . . . 7 ⊢ (1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) = ((1st ‘𝐴) +N 1𝑜)
43	42	oveq1i 6660	. . . . . 6 ⊢ ((1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd ‘𝐴)) = (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴))
44	43	a1i 11	. . . . 5 ⊢ (𝐴 ∈ Q → ((1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd ‘𝐴)) = (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴)))
45	41, 44	breq12d 4666	. . . 4 ⊢ (𝐴 ∈ Q → (((1st ‘𝐴) ·N (2nd ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)) <N ((1st ‘⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) ·N (2nd ‘𝐴)) ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴))))
46	34, 45	bitrd 268	. . 3 ⊢ (𝐴 ∈ Q → (𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩ ↔ (1st ‘𝐴) <N (((1st ‘𝐴) +N 1𝑜) ·N (2nd ‘𝐴))))
47	30, 46	mpbird 247	. 2 ⊢ (𝐴 ∈ Q → 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)
48		opeq1 4402	. . . 4 ⊢ (𝑥 = ((1st ‘𝐴) +N 1𝑜) → ⟨𝑥, 1𝑜⟩ = ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩)
49	48	breq2d 4665	. . 3 ⊢ (𝑥 = ((1st ‘𝐴) +N 1𝑜) → (𝐴 <Q ⟨𝑥, 1𝑜⟩ ↔ 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩))
50	49	rspcev 3309	. 2 ⊢ ((((1st ‘𝐴) +N 1𝑜) ∈ N ∧ 𝐴 <Q ⟨((1st ‘𝐴) +N 1𝑜), 1𝑜⟩) → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1𝑜⟩)
51	6, 47, 50	syl2anc 693	1 ⊢ (𝐴 ∈ Q → ∃𝑥 ∈ N 𝐴 <Q ⟨𝑥, 1𝑜⟩)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913  ⟨cop 4183   class class class wbr 4653   × cxp 5112  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  1_𝑜c1o 7553  Ncnpi 9666   +_N cpli 9667   ·_N cmi 9668   <_N clti 9669  Qcnq 9674   <_Q cltq 9680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-ni 9694  df-pli 9695  df-mi 9696  df-lti 9697  df-ltpq 9732  df-nq 9734  df-ltnq 9740

This theorem is referenced by:  prlem934  9855