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Theorem archpr 6741
 Description: For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer 𝑥 is embedded into the reals as described at nnprlu 6651. (Contributed by Jim Kingdon, 22-Apr-2020.)
Assertion
Ref Expression
archpr (𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
Distinct variable group:   𝐴,𝑙,𝑢,𝑥

Proof of Theorem archpr
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 prop 6573 . . 3 (𝐴P → ⟨(1st𝐴), (2nd𝐴)⟩ ∈ P)
2 prmu 6576 . . 3 (⟨(1st𝐴), (2nd𝐴)⟩ ∈ P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
31, 2syl 14 . 2 (𝐴P → ∃𝑧Q 𝑧 ∈ (2nd𝐴))
4 archnqq 6515 . . . 4 (𝑧Q → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
54ad2antrl 459 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
6 simprl 483 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧Q)
76ad2antrr 457 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧Q)
8 simprr 484 . . . . . . . 8 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → 𝑧 ∈ (2nd𝐴))
98ad2antrr 457 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ (2nd𝐴))
10 simpr 103 . . . . . . . 8 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
11 vex 2560 . . . . . . . . 9 𝑧 ∈ V
12 breq1 3767 . . . . . . . . 9 (𝑙 = 𝑧 → (𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ))
13 ltnqex 6647 . . . . . . . . . 10 {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q } ∈ V
14 gtnqex 6648 . . . . . . . . . 10 {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢} ∈ V
1513, 14op1st 5773 . . . . . . . . 9 (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) = {𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }
1611, 12, 15elab2 2690 . . . . . . . 8 (𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q )
1710, 16sylibr 137 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))
18 eleq1 2100 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (2nd𝐴) ↔ 𝑧 ∈ (2nd𝐴)))
19 eleq1 2100 . . . . . . . . 9 (𝑤 = 𝑧 → (𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩) ↔ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)))
2018, 19anbi12d 442 . . . . . . . 8 (𝑤 = 𝑧 → ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)) ↔ (𝑧 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))))
2120rspcev 2656 . . . . . . 7 ((𝑧Q ∧ (𝑧 ∈ (2nd𝐴) ∧ 𝑧 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))) → ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)))
227, 9, 17, 21syl12anc 1133 . . . . . 6 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)))
23 simplll 485 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴P)
24 nnprlu 6651 . . . . . . . 8 (𝑥N → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
2524ad2antlr 458 . . . . . . 7 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P)
26 ltdfpr 6604 . . . . . . 7 ((𝐴P ∧ ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ∈ P) → (𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))))
2723, 25, 26syl2anc 391 . . . . . 6 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → (𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩ ↔ ∃𝑤Q (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (1st ‘⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))))
2822, 27mpbird 156 . . . . 5 ((((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) ∧ 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q ) → 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
2928ex 108 . . . 4 (((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) ∧ 𝑥N) → (𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))
3029reximdva 2421 . . 3 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → (∃𝑥N 𝑧 <Q [⟨𝑥, 1𝑜⟩] ~Q → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩))
315, 30mpd 13 . 2 ((𝐴P ∧ (𝑧Q𝑧 ∈ (2nd𝐴))) → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
323, 31rexlimddv 2437 1 (𝐴P → ∃𝑥N 𝐴<P ⟨{𝑙𝑙 <Q [⟨𝑥, 1𝑜⟩] ~Q }, {𝑢 ∣ [⟨𝑥, 1𝑜⟩] ~Q <Q 𝑢}⟩)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∈ wcel 1393  {cab 2026  ∃wrex 2307  ⟨cop 3378   class class class wbr 3764  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766  1𝑜c1o 5994  [cec 6104  Ncnpi 6370   ~Q ceq 6377  Qcnq 6378
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