Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiexdiv | Structured version Visualization version GIF version |
Description: In an Archimedean group, given two positive elements, there exists a "divisor" 𝑛. (Contributed by Thierry Arnoux, 30-Jan-2018.) |
Ref | Expression |
---|---|
archiexdiv.b | ⊢ 𝐵 = (Base‘𝑊) |
archiexdiv.0 | ⊢ 0 = (0g‘𝑊) |
archiexdiv.i | ⊢ < = (lt‘𝑊) |
archiexdiv.x | ⊢ · = (.g‘𝑊) |
Ref | Expression |
---|---|
archiexdiv | ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | archiexdiv.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
2 | archiexdiv.0 | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
3 | archiexdiv.i | . . . . 5 ⊢ < = (lt‘𝑊) | |
4 | archiexdiv.x | . . . . 5 ⊢ · = (.g‘𝑊) | |
5 | 1, 2, 3, 4 | isarchi3 31132 | . . . 4 ⊢ (𝑊 ∈ oGrp → (𝑊 ∈ Archi ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)))) |
6 | 5 | biimpa 480 | . . 3 ⊢ ((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))) |
7 | 6 | 3ad2ant1 1135 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥))) |
8 | simp3 1140 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → 0 < 𝑋) | |
9 | breq2 5047 | . . . . 5 ⊢ (𝑥 = 𝑋 → ( 0 < 𝑥 ↔ 0 < 𝑋)) | |
10 | oveq2 7210 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑛 · 𝑥) = (𝑛 · 𝑋)) | |
11 | 10 | breq2d 5055 | . . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑦 < (𝑛 · 𝑥) ↔ 𝑦 < (𝑛 · 𝑋))) |
12 | 11 | rexbidv 3209 | . . . . 5 ⊢ (𝑥 = 𝑋 → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥) ↔ ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋))) |
13 | 9, 12 | imbi12d 348 | . . . 4 ⊢ (𝑥 = 𝑋 → (( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) ↔ ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋)))) |
14 | breq1 5046 | . . . . . 6 ⊢ (𝑦 = 𝑌 → (𝑦 < (𝑛 · 𝑋) ↔ 𝑌 < (𝑛 · 𝑋))) | |
15 | 14 | rexbidv 3209 | . . . . 5 ⊢ (𝑦 = 𝑌 → (∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋) ↔ ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋))) |
16 | 15 | imbi2d 344 | . . . 4 ⊢ (𝑦 = 𝑌 → (( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑋)) ↔ ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
17 | 13, 16 | rspc2v 3540 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) → ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
18 | 17 | 3ad2ant2 1136 | . 2 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ( 0 < 𝑥 → ∃𝑛 ∈ ℕ 𝑦 < (𝑛 · 𝑥)) → ( 0 < 𝑋 → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)))) |
19 | 7, 8, 18 | mp2d 49 | 1 ⊢ (((𝑊 ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 0 < 𝑋) → ∃𝑛 ∈ ℕ 𝑌 < (𝑛 · 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∀wral 3054 ∃wrex 3055 class class class wbr 5043 ‘cfv 6369 (class class class)co 7202 ℕcn 11813 Basecbs 16684 0gc0g 16916 ltcplt 17787 .gcmg 18460 oGrpcogrp 31015 Archicarchi 31122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-er 8380 df-en 8616 df-dom 8617 df-sdom 8618 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-seq 13558 df-0g 16918 df-proset 17774 df-poset 17792 df-plt 17808 df-toset 17895 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-grp 18340 df-minusg 18341 df-mulg 18461 df-omnd 31016 df-ogrp 31017 df-inftm 31123 df-archi 31124 |
This theorem is referenced by: (None) |
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