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 12240. (Contributed by Thierry Arnoux, 9-Apr-2018.) |
Ref | Expression |
---|---|
rearchi | ⊢ ℝfld ∈ Archi |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reofld 31552 | . . 3 ⊢ ℝfld ∈ oField | |
2 | rebase 20821 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
3 | eqid 2738 | . . . 4 ⊢ (ℤRHom‘ℝfld) = (ℤRHom‘ℝfld) | |
4 | relt 20830 | . . . 4 ⊢ < = (lt‘ℝfld) | |
5 | 2, 3, 4 | isarchiofld 31524 | . . 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 12240 | . . 3 ⊢ (𝑥 ∈ ℝ → ∃𝑛 ∈ ℕ 𝑥 < 𝑛) | |
8 | nnz 12352 | . . . . 5 ⊢ (𝑛 ∈ ℕ → 𝑛 ∈ ℤ) | |
9 | refld 20834 | . . . . . . . . 9 ⊢ ℝfld ∈ Field | |
10 | isfld 20010 | . . . . . . . . . 10 ⊢ (ℝfld ∈ Field ↔ (ℝfld ∈ DivRing ∧ ℝfld ∈ CRing)) | |
11 | 10 | simplbi 498 | . . . . . . . . 9 ⊢ (ℝfld ∈ Field → ℝfld ∈ DivRing) |
12 | drngring 20008 | . . . . . . . . 9 ⊢ (ℝfld ∈ DivRing → ℝfld ∈ Ring) | |
13 | 9, 11, 12 | mp2b 10 | . . . . . . . 8 ⊢ ℝfld ∈ Ring |
14 | eqid 2738 | . . . . . . . . 9 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
15 | re1r 20828 | . . . . . . . . 9 ⊢ 1 = (1r‘ℝfld) | |
16 | 3, 14, 15 | zrhmulg 20721 | . . . . . . . 8 ⊢ ((ℝfld ∈ Ring ∧ 𝑛 ∈ ℤ) → ((ℤRHom‘ℝfld)‘𝑛) = (𝑛(.g‘ℝfld)1)) |
17 | 13, 16 | mpan 687 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → ((ℤRHom‘ℝfld)‘𝑛) = (𝑛(.g‘ℝfld)1)) |
18 | 1re 10985 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
19 | remulg 20822 | . . . . . . . 8 ⊢ ((𝑛 ∈ ℤ ∧ 1 ∈ ℝ) → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) | |
20 | 18, 19 | mpan2 688 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛(.g‘ℝfld)1) = (𝑛 · 1)) |
21 | zcn 12334 | . . . . . . . 8 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ ℂ) | |
22 | 21 | mulid1d 11002 | . . . . . . 7 ⊢ (𝑛 ∈ ℤ → (𝑛 · 1) = 𝑛) |
23 | 17, 20, 22 | 3eqtrd 2782 | . . . . . 6 ⊢ (𝑛 ∈ ℤ → ((ℤRHom‘ℝfld)‘𝑛) = 𝑛) |
24 | 23 | breq2d 5085 | . . . . 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 3178 | . . 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 3079 | 1 ⊢ ℝfld ∈ Archi |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 class class class wbr 5073 ‘cfv 6426 (class class class)co 7267 ℝcr 10880 1c1 10882 · cmul 10886 < clt 11019 ℕcn 11983 ℤcz 12329 .gcmg 18710 Ringcrg 19793 CRingccrg 19794 DivRingcdr 20001 Fieldcfield 20002 ℤRHomczrh 20711 ℝfldcrefld 20819 Archicarchi 31439 oFieldcofld 31503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5208 ax-sep 5221 ax-nul 5228 ax-pow 5286 ax-pr 5350 ax-un 7578 ax-cnex 10937 ax-resscn 10938 ax-1cn 10939 ax-icn 10940 ax-addcl 10941 ax-addrcl 10942 ax-mulcl 10943 ax-mulrcl 10944 ax-mulcom 10945 ax-addass 10946 ax-mulass 10947 ax-distr 10948 ax-i2m1 10949 ax-1ne0 10950 ax-1rid 10951 ax-rnegex 10952 ax-rrecex 10953 ax-cnre 10954 ax-pre-lttri 10955 ax-pre-lttrn 10956 ax-pre-ltadd 10957 ax-pre-mulgt0 10958 ax-pre-sup 10959 ax-addf 10960 ax-mulf 10961 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3071 df-rmo 3072 df-rab 3073 df-v 3431 df-sbc 3716 df-csb 3832 df-dif 3889 df-un 3891 df-in 3893 df-ss 3903 df-pss 3905 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4840 df-iun 4926 df-br 5074 df-opab 5136 df-mpt 5157 df-tr 5191 df-id 5484 df-eprel 5490 df-po 5498 df-so 5499 df-fr 5539 df-we 5541 df-xp 5590 df-rel 5591 df-cnv 5592 df-co 5593 df-dm 5594 df-rn 5595 df-res 5596 df-ima 5597 df-pred 6195 df-ord 6262 df-on 6263 df-lim 6264 df-suc 6265 df-iota 6384 df-fun 6428 df-fn 6429 df-f 6430 df-f1 6431 df-fo 6432 df-f1o 6433 df-fv 6434 df-riota 7224 df-ov 7270 df-oprab 7271 df-mpo 7272 df-om 7703 df-1st 7820 df-2nd 7821 df-tpos 8029 df-frecs 8084 df-wrecs 8115 df-recs 8189 df-rdg 8228 df-1o 8284 df-er 8485 df-map 8604 df-en 8721 df-dom 8722 df-sdom 8723 df-fin 8724 df-pnf 11021 df-mnf 11022 df-xr 11023 df-ltxr 11024 df-le 11025 df-sub 11217 df-neg 11218 df-div 11643 df-nn 11984 df-2 12046 df-3 12047 df-4 12048 df-5 12049 df-6 12050 df-7 12051 df-8 12052 df-9 12053 df-n0 12244 df-z 12330 df-dec 12448 df-uz 12593 df-fz 13250 df-seq 13732 df-struct 16858 df-sets 16875 df-slot 16893 df-ndx 16905 df-base 16923 df-ress 16952 df-plusg 16985 df-mulr 16986 df-starv 16987 df-tset 16991 df-ple 16992 df-ds 16994 df-unif 16995 df-0g 17162 df-proset 18023 df-poset 18041 df-plt 18058 df-toset 18145 df-ps 18294 df-tsr 18295 df-mgm 18336 df-sgrp 18385 df-mnd 18396 df-mhm 18440 df-grp 18590 df-minusg 18591 df-sbg 18592 df-mulg 18711 df-subg 18762 df-ghm 18842 df-cmn 19398 df-mgp 19731 df-ur 19748 df-ring 19795 df-cring 19796 df-oppr 19872 df-dvdsr 19893 df-unit 19894 df-invr 19924 df-dvr 19935 df-rnghom 19969 df-drng 20003 df-field 20004 df-subrg 20032 df-cnfld 20608 df-zring 20681 df-zrh 20715 df-refld 20820 df-omnd 31333 df-ogrp 31334 df-inftm 31440 df-archi 31441 df-orng 31504 df-ofld 31505 |
This theorem is referenced by: nn0archi 31555 |
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