Theorem axarch 8019

Description: Archimedean axiom. The Archimedean property is more naturally stated once we have defined ℕ. Unless we find another way to state it, we'll just use the right hand side of dfnn2 9053 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 8059. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

Assertion

Ref	Expression
axarch	⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)

Distinct variable group:   𝐴,𝑛,𝑥,𝑦

Proof of Theorem axarch

Dummy variables 𝑙 𝑢 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elreal 7956	. . 3 ⊢ (𝐴 ∈ ℝ ↔ ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
2	1	biimpi 120	. 2 ⊢ (𝐴 ∈ ℝ → ∃𝑧 ∈ R ⟨𝑧, 0R⟩ = 𝐴)
3		archsr 7910	. . . 4 ⊢ (𝑧 ∈ R → ∃𝑤 ∈ N 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
4	3	ad2antrl 490	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑤 ∈ N 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
5		simplrr 536	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ = 𝐴)
6		simprr 531	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
7		ltresr 7967	. . . . . 6 ⊢ (⟨𝑧, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ↔ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )
8	6, 7	sylibr 134	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨𝑧, 0R⟩ <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
9	5, 8	eqbrtrrd 4074	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → 𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
10		pitonn 7976	. . . . . 6 ⊢ (𝑤 ∈ N → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
11	10	ad2antrl 490	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)})
12		simpr 110	. . . . . 6 ⊢ ((((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩)
13	12	breq2d 4062	. . . . 5 ⊢ ((((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) ∧ 𝑛 = ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩) → (𝐴 <ℝ 𝑛 ↔ 𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩))
14	11, 13	rspcedv 2885	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → (𝐴 <ℝ ⟨[⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R , 0R⟩ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛))
15	9, 14	mpd 13	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) ∧ (𝑤 ∈ N ∧ 𝑧 <R [⟨(⟨{𝑙 ∣ 𝑙 <Q [⟨𝑤, 1o⟩] ~Q }, {𝑢 ∣ [⟨𝑤, 1o⟩] ~Q <Q 𝑢}⟩ +P 1P), 1P⟩] ~R )) → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
16	4, 15	rexlimddv 2629	. 2 ⊢ ((𝐴 ∈ ℝ ∧ (𝑧 ∈ R ∧ ⟨𝑧, 0R⟩ = 𝐴)) → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)
17	2, 16	rexlimddv 2629	1 ⊢ (𝐴 ∈ ℝ → ∃𝑛 ∈ ∩ {𝑥 ∣ (1 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑦 + 1) ∈ 𝑥)}𝐴 <ℝ 𝑛)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486  ⟨cop 3640  ∩ cint 3890   class class class wbr 4050  (class class class)co 5956  1_oc1o 6507  [cec 6630  Ncnpi 7400   ~_Q ceq 7407   <_Q cltq 7413  1_Pc1p 7420   +_P cpp 7421   ~_R cer 7424  Rcnr 7425  0_Rc0r 7426   <_R cltr 7431  ℝcr 7939  1c1 7941   + caddc 7943   <_ℝ cltrr 7944

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 4166  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

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 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-iun 3934  df-br 4051  df-opab 4113  df-mpt 4114  df-tr 4150  df-eprel 4343  df-id 4347  df-po 4350  df-iso 4351  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-ov 5959  df-oprab 5960  df-mpo 5961  df-1st 6238  df-2nd 6239  df-recs 6403  df-irdg 6468  df-1o 6514  df-2o 6515  df-oadd 6518  df-omul 6519  df-er 6632  df-ec 6634  df-qs 6638  df-ni 7432  df-pli 7433  df-mi 7434  df-lti 7435  df-plpq 7472  df-mpq 7473  df-enq 7475  df-nqqs 7476  df-plqqs 7477  df-mqqs 7478  df-1nqqs 7479  df-rq 7480  df-ltnqqs 7481  df-enq0 7552  df-nq0 7553  df-0nq0 7554  df-plq0 7555  df-mq0 7556  df-inp 7594  df-i1p 7595  df-iplp 7596  df-iltp 7598  df-enr 7854  df-nr 7855  df-plr 7856  df-ltr 7858  df-0r 7859  df-1r 7860  df-c 7946  df-1 7948  df-r 7950  df-add 7951  df-lt 7953

This theorem is referenced by: (None)