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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 31794 | . . 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 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑊 ∈ oGrp) |
10 | archiabllem.a | . . . . . 6 ⊢ (𝜑 → 𝑊 ∈ Archi) | |
11 | 10 | 3ad2ant1 1133 | . . . . 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 1133 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (oppg‘𝑊) ∈ oGrp) |
15 | simp1 1136 | . . . . . 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 1163 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) |
18 | simp2 1137 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵) | |
19 | simp3 1138 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵) | |
20 | 4, 5, 6, 7, 8, 9, 11, 12, 14, 17, 18, 19 | archiabllem2b 31915 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
21 | 20 | 3expb 1120 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
22 | 21 | ralrimivva 3195 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
23 | 4, 12 | isabl2 19563 | . 2 ⊢ (𝑊 ∈ Abel ↔ (𝑊 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
24 | 3, 22, 23 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝑊 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3062 ∃wrex 3071 class class class wbr 5103 ‘cfv 6493 (class class class)co 7353 Basecbs 17075 +gcplusg 17125 lecple 17132 0gc0g 17313 ltcplt 18189 Grpcgrp 18740 .gcmg 18863 oppgcoppg 19114 Abelcabl 19554 oGrpcogrp 31789 Archicarchi 31896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 ax-cnex 11103 ax-resscn 11104 ax-1cn 11105 ax-icn 11106 ax-addcl 11107 ax-addrcl 11108 ax-mulcl 11109 ax-mulrcl 11110 ax-mulcom 11111 ax-addass 11112 ax-mulass 11113 ax-distr 11114 ax-i2m1 11115 ax-1ne0 11116 ax-1rid 11117 ax-rnegex 11118 ax-rrecex 11119 ax-cnre 11120 ax-pre-lttri 11121 ax-pre-lttrn 11122 ax-pre-ltadd 11123 ax-pre-mulgt0 11124 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7309 df-ov 7356 df-oprab 7357 df-mpo 7358 df-om 7799 df-1st 7917 df-2nd 7918 df-tpos 8153 df-frecs 8208 df-wrecs 8239 df-recs 8313 df-rdg 8352 df-er 8644 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11187 df-mnf 11188 df-xr 11189 df-ltxr 11190 df-le 11191 df-sub 11383 df-neg 11384 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12410 df-z 12496 df-dec 12615 df-uz 12760 df-fz 13417 df-seq 13899 df-sets 17028 df-slot 17046 df-ndx 17058 df-base 17076 df-plusg 17138 df-ple 17145 df-0g 17315 df-proset 18176 df-poset 18194 df-plt 18211 df-toset 18298 df-mgm 18489 df-sgrp 18538 df-mnd 18549 df-grp 18743 df-minusg 18744 df-sbg 18745 df-mulg 18864 df-oppg 19115 df-cmn 19555 df-abl 19556 df-omnd 31790 df-ogrp 31791 df-inftm 31897 df-archi 31898 |
This theorem is referenced by: archiabl 31917 |
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