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Theorem pitoregt0 7769
Description: Embedding from N to yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
Assertion
Ref Expression
pitoregt0 (𝑁N → 0 < ⟨[⟨(⟨{𝑙𝑙 <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
StepHypRef Expression
1 1pr 7474 . . . . . 6 1PP
2 addclpr 7457 . . . . . 6 ((1PP ∧ 1PP) → (1P +P 1P) ∈ P)
31, 1, 2mp2an 423 . . . . 5 (1P +P 1P) ∈ P
4 nnprlu 7473 . . . . 5 (𝑁N → ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P)
5 ltaddpr 7517 . . . . 5 (((1P +P 1P) ∈ P ∧ ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (1P +P 1P)<P ((1P +P 1P) +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩))
63, 4, 5sylancr 411 . . . 4 (𝑁N → (1P +P 1P)<P ((1P +P 1P) +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩))
71a1i 9 . . . . . 6 (𝑁N → 1PP)
8 addassprg 7499 . . . . . 6 ((1PP ∧ 1PP ∧ ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → ((1P +P 1P) +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (1P +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)))
97, 7, 4, 8syl3anc 1220 . . . . 5 (𝑁N → ((1P +P 1P) +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (1P +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)))
10 addcomprg 7498 . . . . . . 7 ((1PP ∧ ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P) → (1P +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
111, 4, 10sylancr 411 . . . . . 6 (𝑁N → (1P +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))
1211oveq2d 5840 . . . . 5 (𝑁N → (1P +P (1P +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩)) = (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
139, 12eqtrd 2190 . . . 4 (𝑁N → ((1P +P 1P) +P ⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩) = (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
146, 13breqtrd 3990 . . 3 (𝑁N → (1P +P 1P)<P (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P)))
15 addclpr 7457 . . . . 5 ((⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ ∈ P ∧ 1PP) → (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
164, 1, 15sylancl 410 . . . 4 (𝑁N → (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P)
17 ltsrprg 7667 . . . . 5 (((1PP ∧ 1PP) ∧ ((⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP)) → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
181, 1, 17mpanl12 433 . . . 4 (((⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P) ∈ P ∧ 1PP) → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
1916, 1, 18sylancl 410 . . 3 (𝑁N → ([⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ (1P +P 1P)<P (1P +P (⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P))))
2014, 19mpbird 166 . 2 (𝑁N → [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
21 df-0 7739 . . . 4 0 = ⟨0R, 0R
2221breq1i 3972 . . 3 (0 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ ⟨0R, 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
23 ltresr 7759 . . 3 (⟨0R, 0R⟩ < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ 0R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
24 df-0r 7651 . . . 4 0R = [⟨1P, 1P⟩] ~R
2524breq1i 3972 . . 3 (0R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R ↔ [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
2622, 23, 253bitri 205 . 2 (0 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ [⟨1P, 1P⟩] ~R <R [⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
2720, 26sylibr 133 1 (𝑁N → 0 < ⟨[⟨(⟨{𝑙𝑙 <Q [⟨𝑁, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑁, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1335  wcel 2128  {cab 2143  cop 3563   class class class wbr 3965  (class class class)co 5824  1oc1o 6356  [cec 6478  Ncnpi 7192   ~Q ceq 7199   <Q cltq 7205  Pcnp 7211  1Pc1p 7212   +P cpp 7213  <P cltp 7215   ~R cer 7216  0Rc0r 7218   <R cltr 7223  0cc0 7732   < cltrr 7736
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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-coll 4079  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-reu 2442  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-tr 4063  df-eprel 4249  df-id 4253  df-po 4256  df-iso 4257  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-ov 5827  df-oprab 5828  df-mpo 5829  df-1st 6088  df-2nd 6089  df-recs 6252  df-irdg 6317  df-1o 6363  df-2o 6364  df-oadd 6367  df-omul 6368  df-er 6480  df-ec 6482  df-qs 6486  df-ni 7224  df-pli 7225  df-mi 7226  df-lti 7227  df-plpq 7264  df-mpq 7265  df-enq 7267  df-nqqs 7268  df-plqqs 7269  df-mqqs 7270  df-1nqqs 7271  df-rq 7272  df-ltnqqs 7273  df-enq0 7344  df-nq0 7345  df-0nq0 7346  df-plq0 7347  df-mq0 7348  df-inp 7386  df-i1p 7387  df-iplp 7388  df-iltp 7390  df-enr 7646  df-nr 7647  df-ltr 7650  df-0r 7651  df-0 7739  df-r 7742  df-lt 7745
This theorem is referenced by:  recriota  7810
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