Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiabllem2 | Structured version Visualization version GIF version |
Description: Archimedean ordered groups with no minimal positive value are abelian. (Contributed by Thierry Arnoux, 1-May-2018.) |
Ref | Expression |
---|---|
archiabllem.b | ⊢ 𝐵 = (Base‘𝑊) |
archiabllem.0 | ⊢ 0 = (0g‘𝑊) |
archiabllem.e | ⊢ ≤ = (le‘𝑊) |
archiabllem.t | ⊢ < = (lt‘𝑊) |
archiabllem.m | ⊢ · = (.g‘𝑊) |
archiabllem.g | ⊢ (𝜑 → 𝑊 ∈ oGrp) |
archiabllem.a | ⊢ (𝜑 → 𝑊 ∈ Archi) |
archiabllem2.1 | ⊢ + = (+g‘𝑊) |
archiabllem2.2 | ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) |
archiabllem2.3 | ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) |
Ref | Expression |
---|---|
archiabllem2 | ⊢ (𝜑 → 𝑊 ∈ Abel) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | archiabllem.g | . . 3 ⊢ (𝜑 → 𝑊 ∈ oGrp) | |
2 | ogrpgrp 31020 | . . 3 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝜑 → 𝑊 ∈ Grp) |
4 | archiabllem.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑊) | |
5 | archiabllem.0 | . . . . 5 ⊢ 0 = (0g‘𝑊) | |
6 | archiabllem.e | . . . . 5 ⊢ ≤ = (le‘𝑊) | |
7 | archiabllem.t | . . . . 5 ⊢ < = (lt‘𝑊) | |
8 | archiabllem.m | . . . . 5 ⊢ · = (.g‘𝑊) | |
9 | 1 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ oGrp) |
10 | archiabllem.a | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Archi) | |
11 | 10 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ Archi) |
12 | archiabllem2.1 | . . . . 5 ⊢ + = (+g‘𝑊) | |
13 | archiabllem2.2 | . . . . . 6 ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) | |
14 | 13 | 3ad2ant1 1135 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (oppg‘𝑊) ∈ oGrp) |
15 | simp1 1138 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝜑) | |
16 | archiabllem2.3 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) | |
17 | 15, 16 | syl3an1 1165 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) |
18 | simp2 1139 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
19 | simp3 1140 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
20 | 4, 5, 6, 7, 8, 9, 11, 12, 14, 17, 18, 19 | archiabllem2b 31141 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
21 | 20 | 3expb 1122 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
22 | 21 | ralrimivva 3105 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
23 | 4, 12 | isabl2 19151 | . 2 ⊢ (𝑊 ∈ Abel ↔ (𝑊 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
24 | 3, 22, 23 | sylanbrc 586 | 1 ⊢ (𝜑 → 𝑊 ∈ Abel) |
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 Basecbs 16684 +gcplusg 16767 lecple 16774 0gc0g 16916 ltcplt 17787 Grpcgrp 18337 .gcmg 18460 oppgcoppg 18709 Abelcabl 19143 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-tpos 7957 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-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-z 12160 df-dec 12277 df-uz 12422 df-fz 13079 df-seq 13558 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-plusg 16780 df-ple 16787 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-sbg 18342 df-mulg 18461 df-oppg 18710 df-cmn 19144 df-abl 19145 df-omnd 31016 df-ogrp 31017 df-inftm 31123 df-archi 31124 |
This theorem is referenced by: archiabl 31143 |
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