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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rearchi | Structured version Visualization version GIF version |
Description: The field of the real numbers is Archimedean. See also arch 12491. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
Ref | Expression |
---|---|
rearchi | ⊢ ℝfld ∈ Archi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reofld 32996 | . . 3 ⊢ ℝfld ∈ oField | |
2 | rebase 21525 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
3 | eqid 2727 | . . . 4 ⊢ (ℤRHom‘ℝfld) = (ℤRHom‘ℝfld) | |
4 | relt 21534 | . . . 4 ⊢ < = (lt‘ℝfld) | |
5 | 2, 3, 4 | isarchiofld 32972 | . . 3 ⊢ (ℝfld ∈ oField → (ℝfld ∈ Archi ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ ℕ 𝑥 < ((ℤRHom‘ℝfld)‘𝑛))) |
6 | 1, 5 | ax-mp 5 | . 2 ⊢ (ℝfld ∈ Archi ↔ ∀𝑥 ∈ ℝ ∃𝑛 ∈ ℕ 𝑥 < ((ℤRHom‘ℝfld)‘𝑛)) |
7 | arch 12491 | . . 3 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛) | |
8 | nnz 12601 | . . . . 5 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
9 | refld 21538 | . . . . . . . . 9 ⊢ ℝfld ∈ Field | |
10 | isfld 20624 | . . . . . . . . . 10 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
11 | 10 | simplbi 497 | . . . . . . . . 9 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
12 | drngring 20620 | . . . . . . . . 9 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
13 | 9, 11, 12 | mp2b 10 | . . . . . . . 8 ⊢ ℝfld ∈ Ring |
14 | eqid 2727 | . . . . . . . . 9 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
15 | re1r 21532 | . . . . . . . . 9 ⊢ 1 = (1r‘ℝfld) | |
16 | 3, 14, 15 | zrhmulg 21422 | . . . . . . . 8 ⊢ ((ℝfld ∈ Ring ∧ 𝑛 ∈ ℤ) → ((ℤRHom‘ℝfld)‘𝑛) = (𝑛(.g‘ℝfld)1)) |
17 | 13, 16 | mpan 689 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → ((ℤRHom‘ℝfld)‘𝑛) = (𝑛(.g‘ℝfld)1)) |
18 | 1re 11236 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
19 | remulg 21526 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℝ) → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) | |
20 | 18, 19 | mpan2 690 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) |
21 | zcn 12585 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
22 | 21 | mulridd 11253 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
23 | 17, 20, 22 | 3eqtrd 2771 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → ((ℤRHom‘ℝfld)‘𝑛) = 𝑛) |
24 | 23 | breq2d 5154 | . . . . 5 ⊢ (𝑛 ∈ ℤ → (𝑥 < ((ℤRHom‘ℝfld)‘𝑛) ↔ 𝑥 < 𝑛)) |
25 | 8, 24 | syl 17 | . . . 4 ⊢ (𝑛 ∈ ℕ → (𝑥 < ((ℤRHom‘ℝfld)‘𝑛) ↔ 𝑥 < 𝑛)) |
26 | 25 | rexbiia 3087 | . . 3 ⊢ (∃𝑛 ∈ ℕ 𝑥 < ((ℤRHom‘ℝfld)‘𝑛) ↔ ∃𝑛 ∈ ℕ 𝑥 < 𝑛) |
27 | 7, 26 | sylibr 233 | . 2 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < ((ℤRHom‘ℝfld)‘𝑛)) |
28 | 6, 27 | mprgbir 3063 | 1 ⊢ ℝfld ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∈ wcel 2099 ∀wral 3056 ∃wrex 3065 class class class wbr 5142 ‘cfv 6542 (class class class)co 7414 ℝcr 11129 1c1 11131 · cmul 11135 < clt 11270 ℕcn 12234 ℤcz 12580 .gcmg 19014 Ringcrg 20164 CRingccrg 20165 DivRingcdr 20613 Fieldcfield 20614 ℤRHomczrh 21412 ℝfldcrefld 21523 Archicarchi 32863 oFieldcofld 32951 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208 ax-addf 11209 ax-mulf 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-tpos 8225 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-er 8718 df-map 8838 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-4 12299 df-5 12300 df-6 12301 df-7 12302 df-8 12303 df-9 12304 df-n0 12495 df-z 12581 df-dec 12700 df-uz 12845 df-fz 13509 df-seq 13991 df-struct 17107 df-sets 17124 df-slot 17142 df-ndx 17154 df-base 17172 df-ress 17201 df-plusg 17237 df-mulr 17238 df-starv 17239 df-tset 17243 df-ple 17244 df-ds 17246 df-unif 17247 df-0g 17414 df-proset 18278 df-poset 18296 df-plt 18313 df-toset 18400 df-ps 18549 df-tsr 18550 df-mgm 18591 df-sgrp 18670 df-mnd 18686 df-mhm 18731 df-grp 18884 df-minusg 18885 df-sbg 18886 df-mulg 19015 df-subg 19069 df-ghm 19159 df-cmn 19728 df-abl 19729 df-mgp 20066 df-rng 20084 df-ur 20113 df-ring 20166 df-cring 20167 df-oppr 20262 df-dvdsr 20285 df-unit 20286 df-invr 20316 df-dvr 20329 df-rhm 20400 df-subrng 20472 df-subrg 20497 df-drng 20615 df-field 20616 df-cnfld 21267 df-zring 21360 df-zrh 21416 df-refld 21524 df-omnd 32757 df-ogrp 32758 df-inftm 32864 df-archi 32865 df-orng 32952 df-ofld 32953 |
This theorem is referenced by: nn0archi 32999 |
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