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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 33053 | . . 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 33176 | . . . 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 3208 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) |
23 | 4, 12 | isabl2 19832 | . 2 ⊢ (𝑊 ∈ Abel ↔ (𝑊 ∈ Grp ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))) |
24 | 3, 22, 23 | sylanbrc 582 | 1 ⊢ (𝜑 → 𝑊 ∈ Abel) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1537 ∈ wcel 2108 ∀wral 3067 ∃wrex 3076 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 Basecbs 17258 +gcplusg 17311 lecple 17318 0gc0g 17499 ltcplt 18378 Grpcgrp 18973 .gcmg 19107 oppgcoppg 19385 Abelcabl 19823 oGrpcogrp 33048 Archicarchi 33157 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-tpos 8267 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-nn 12294 df-2 12356 df-3 12357 df-4 12358 df-5 12359 df-6 12360 df-7 12361 df-8 12362 df-9 12363 df-n0 12554 df-z 12640 df-dec 12759 df-uz 12904 df-fz 13568 df-seq 14053 df-sets 17211 df-slot 17229 df-ndx 17241 df-base 17259 df-plusg 17324 df-ple 17331 df-0g 17501 df-proset 18365 df-poset 18383 df-plt 18400 df-toset 18487 df-mgm 18678 df-sgrp 18757 df-mnd 18773 df-grp 18976 df-minusg 18977 df-sbg 18978 df-mulg 19108 df-oppg 19386 df-cmn 19824 df-abl 19825 df-omnd 33049 df-ogrp 33050 df-inftm 33158 df-archi 33159 |
This theorem is referenced by: archiabl 33178 |
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