Theorem pitoregt0 7650

Description: Embedding from N to ℝ yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)

Assertion

Ref	Expression
pitoregt0	⊢ (𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Distinct variable group:   𝑁,𝑙,𝑢

Proof of Theorem pitoregt0

Step	Hyp	Ref	Expression
1		1pr 7355	. . . . . 6 ⊢ 1P ∈ P
2		addclpr 7338	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P) → (1P +P 1P) ∈ P)
3	1, 1, 2	mp2an 422	. . . . 5 ⊢ (1P +P 1P) ∈ P
4		nnprlu 7354	. . . . 5 ⊢ (𝑁 ∈ N → ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
5		ltaddpr 7398	. . . . 5 ⊢ (((1P +P 1P) ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩))
6	3, 4, 5	sylancr 410	. . . 4 ⊢ (𝑁 ∈ N → (1P +P 1P)<P ((1P +P 1P) +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩))
7	1	a1i 9	. . . . . 6 ⊢ (𝑁 ∈ N → 1P ∈ P)
8		addassprg 7380	. . . . . 6 ⊢ ((1P ∈ P ∧ 1P ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → ((1P +P 1P) +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (1P +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)))
9	7, 7, 4, 8	syl3anc 1216	. . . . 5 ⊢ (𝑁 ∈ N → ((1P +P 1P) +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (1P +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)))
10		addcomprg 7379	. . . . . . 7 ⊢ ((1P ∈ P ∧ ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (1P +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
11	1, 4, 10	sylancr 410	. . . . . 6 ⊢ (𝑁 ∈ N → (1P +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
12	11	oveq2d 5783	. . . . 5 ⊢ (𝑁 ∈ N → (1P +P (1P +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)) = (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
13	9, 12	eqtrd 2170	. . . 4 ⊢ (𝑁 ∈ N → ((1P +P 1P) +P ⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
14	6, 13	breqtrd 3949	. . 3 ⊢ (𝑁 ∈ N → (1P +P 1P)<P (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
15		addclpr 7338	. . . . 5 ⊢ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1P ∈ P) → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
16	4, 1, 15	sylancl 409	. . . 4 ⊢ (𝑁 ∈ N → (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
17		ltsrprg 7548	. . . . 5 ⊢ (((1P ∈ P ∧ 1P ∈ P) ∧ ((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P)) → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
18	1, 1, 17	mpanl12 432	. . . 4 ⊢ (((⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1P ∈ P) → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
19	16, 1, 18	sylancl 409	. . 3 ⊢ (𝑁 ∈ N → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
20	14, 19	mpbird 166	. 2 ⊢ (𝑁 ∈ N → [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
21		df-0 7620	. . . 4 ⊢ 0 = ⟨0R, 0R⟩
22	21	breq1i 3931	. . 3 ⊢ (0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ ⟨0R, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
23		ltresr 7640	. . 3 ⊢ (⟨0R, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ 0R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
24		df-0r 7532	. . . 4 ⊢ 0R = [⟨1P, 1P⟩] ~R
25	24	breq1i 3931	. . 3 ⊢ (0R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
26	22, 23, 25	3bitri 205	. 2 ⊢ (0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
27	20, 26	sylibr 133	1 ⊢ (𝑁 ∈ N → 0 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  {cab 2123  ⟨cop 3525   class class class wbr 3924  (class class class)co 5767  1_oc1o 6299  [cec 6420  Ncnpi 7073   ~_Q ceq 7080   <_Q cltq 7086  Pcnp 7092  1_Pc1p 7093   +_P cpp 7094  <_P cltp 7096   ~_R cer 7097  0_Rc0r 7099   <_R cltr 7104  0cc0 7613   <_ℝ cltrr 7617

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-coll 4038  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497

This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-reu 2421  df-rab 2423  df-v 2683  df-sbc 2905  df-csb 2999  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-iun 3810  df-br 3925  df-opab 3985  df-mpt 3986  df-tr 4022  df-eprel 4206  df-id 4210  df-po 4213  df-iso 4214  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-1st 6031  df-2nd 6032  df-recs 6195  df-irdg 6260  df-1o 6306  df-2o 6307  df-oadd 6310  df-omul 6311  df-er 6422  df-ec 6424  df-qs 6428  df-ni 7105  df-pli 7106  df-mi 7107  df-lti 7108  df-plpq 7145  df-mpq 7146  df-enq 7148  df-nqqs 7149  df-plqqs 7150  df-mqqs 7151  df-1nqqs 7152  df-rq 7153  df-ltnqqs 7154  df-enq0 7225  df-nq0 7226  df-0nq0 7227  df-plq0 7228  df-mq0 7229  df-inp 7267  df-i1p 7268  df-iplp 7269  df-iltp 7271  df-enr 7527  df-nr 7528  df-ltr 7531  df-0r 7532  df-0 7620  df-r 7623  df-lt 7626

This theorem is referenced by:  recriota  7691