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Mirrors > Home > MPE Home > Th. List > Mathboxes > archiabllem2b | Structured version Visualization version GIF version |
Description: Lemma for archiabl 29752. (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)) |
archiabllem2b.4 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
archiabllem2b.5 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
Ref | Expression |
---|---|
archiabllem2b | ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | archiabllem.b | . . 3 ⊢ 𝐵 = (Base‘𝑊) | |
2 | archiabllem.0 | . . 3 ⊢ 0 = (0g‘𝑊) | |
3 | archiabllem.e | . . 3 ⊢ ≤ = (le‘𝑊) | |
4 | archiabllem.t | . . 3 ⊢ < = (lt‘𝑊) | |
5 | archiabllem.m | . . 3 ⊢ · = (.g‘𝑊) | |
6 | archiabllem.g | . . 3 ⊢ (𝜑 → 𝑊 ∈ oGrp) | |
7 | archiabllem.a | . . 3 ⊢ (𝜑 → 𝑊 ∈ Archi) | |
8 | archiabllem2.1 | . . 3 ⊢ + = (+g‘𝑊) | |
9 | archiabllem2.2 | . . 3 ⊢ (𝜑 → (oppg‘𝑊) ∈ oGrp) | |
10 | archiabllem2.3 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵 ∧ 0 < 𝑎) → ∃𝑏 ∈ 𝐵 ( 0 < 𝑏 ∧ 𝑏 < 𝑎)) | |
11 | archiabllem2b.4 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
12 | archiabllem2b.5 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 | archiabllem2c 29749 | . 2 ⊢ (𝜑 → ¬ (𝑋 + 𝑌) < (𝑌 + 𝑋)) |
14 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 11 | archiabllem2c 29749 | . 2 ⊢ (𝜑 → ¬ (𝑌 + 𝑋) < (𝑋 + 𝑌)) |
15 | isogrp 29702 | . . . . 5 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd)) | |
16 | 15 | simprbi 480 | . . . 4 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd) |
17 | omndtos 29705 | . . . 4 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset) | |
18 | 6, 16, 17 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝑊 ∈ Toset) |
19 | ogrpgrp 29703 | . . . . 5 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ Grp) | |
20 | 6, 19 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑊 ∈ Grp) |
21 | 1, 8 | grpcl 17430 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
22 | 20, 11, 12, 21 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑋 + 𝑌) ∈ 𝐵) |
23 | 1, 8 | grpcl 17430 | . . . 4 ⊢ ((𝑊 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 + 𝑋) ∈ 𝐵) |
24 | 20, 12, 11, 23 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝑌 + 𝑋) ∈ 𝐵) |
25 | 1, 4 | tlt3 29665 | . . 3 ⊢ ((𝑊 ∈ Toset ∧ (𝑋 + 𝑌) ∈ 𝐵 ∧ (𝑌 + 𝑋) ∈ 𝐵) → ((𝑋 + 𝑌) = (𝑌 + 𝑋) ∨ (𝑋 + 𝑌) < (𝑌 + 𝑋) ∨ (𝑌 + 𝑋) < (𝑋 + 𝑌))) |
26 | 18, 22, 24, 25 | syl3anc 1326 | . 2 ⊢ (𝜑 → ((𝑋 + 𝑌) = (𝑌 + 𝑋) ∨ (𝑋 + 𝑌) < (𝑌 + 𝑋) ∨ (𝑌 + 𝑋) < (𝑋 + 𝑌))) |
27 | 13, 14, 26 | ecase23d 1436 | 1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∨ w3o 1036 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 lecple 15948 0gc0g 16100 ltcplt 16941 Tosetctos 17033 Grpcgrp 17422 .gcmg 17540 oppgcoppg 17775 oMndcomnd 29697 oGrpcogrp 29698 Archicarchi 29731 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-ple 15961 df-0g 16102 df-preset 16928 df-poset 16946 df-plt 16958 df-toset 17034 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-oppg 17776 df-omnd 29699 df-ogrp 29700 df-inftm 29732 df-archi 29733 |
This theorem is referenced by: archiabllem2 29751 |
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