Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ornglmulle | Structured version Visualization version GIF version |
Description: In an ordered ring, multiplication with a positive does not change comparison. (Contributed by Thierry Arnoux, 10-Apr-2018.) |
Ref | Expression |
---|---|
ornglmullt.b | ⊢ 𝐵 = (Base‘𝑅) |
ornglmullt.t | ⊢ · = (.r‘𝑅) |
ornglmullt.0 | ⊢ 0 = (0g‘𝑅) |
ornglmullt.1 | ⊢ (𝜑 → 𝑅 ∈ oRing) |
ornglmullt.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ornglmullt.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ornglmullt.4 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
orngmulle.l | ⊢ ≤ = (le‘𝑅) |
orngmulle.5 | ⊢ (𝜑 → 𝑋 ≤ 𝑌) |
orngmulle.6 | ⊢ (𝜑 → 0 ≤ 𝑍) |
Ref | Expression |
---|---|
ornglmulle | ⊢ (𝜑 → (𝑍 · 𝑋) ≤ (𝑍 · 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ornglmullt.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ oRing) | |
2 | orngogrp 29801 | . . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ oGrp) |
4 | isogrp 29702 | . . . . 5 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd)) | |
5 | 4 | simprbi 480 | . . . 4 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ oMnd) |
7 | orngring 29800 | . . . . . 6 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | ringgrp 18552 | . . . . 5 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
10 | 8, 9 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ Grp) |
11 | ornglmullt.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
12 | ornglmullt.0 | . . . . 5 ⊢ 0 = (0g‘𝑅) | |
13 | 11, 12 | grpidcl 17450 | . . . 4 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
15 | ornglmullt.4 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
16 | ornglmullt.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
17 | ornglmullt.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
18 | 11, 17 | ringcl 18561 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 · 𝑌) ∈ 𝐵) |
19 | 8, 15, 16, 18 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ 𝐵) |
20 | ornglmullt.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | 11, 17 | ringcl 18561 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑍 · 𝑋) ∈ 𝐵) |
22 | 8, 15, 20, 21 | syl3anc 1326 | . . . 4 ⊢ (𝜑 → (𝑍 · 𝑋) ∈ 𝐵) |
23 | eqid 2622 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
24 | 11, 23 | grpsubcl 17495 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑍 · 𝑌) ∈ 𝐵 ∧ (𝑍 · 𝑋) ∈ 𝐵) → ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋)) ∈ 𝐵) |
25 | 10, 19, 22, 24 | syl3anc 1326 | . . 3 ⊢ (𝜑 → ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋)) ∈ 𝐵) |
26 | orngmulle.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑍) | |
27 | 11, 23 | grpsubcl 17495 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
28 | 10, 16, 20, 27 | syl3anc 1326 | . . . . 5 ⊢ (𝜑 → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
29 | 11, 12, 23 | grpsubid 17499 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(-g‘𝑅)𝑋) = 0 ) |
30 | 10, 20, 29 | syl2anc 693 | . . . . . 6 ⊢ (𝜑 → (𝑋(-g‘𝑅)𝑋) = 0 ) |
31 | orngmulle.5 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
32 | orngmulle.l | . . . . . . . 8 ⊢ ≤ = (le‘𝑅) | |
33 | 11, 32, 23 | ogrpsub 29717 | . . . . . . 7 ⊢ ((𝑅 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋(-g‘𝑅)𝑋) ≤ (𝑌(-g‘𝑅)𝑋)) |
34 | 3, 20, 16, 20, 31, 33 | syl131anc 1339 | . . . . . 6 ⊢ (𝜑 → (𝑋(-g‘𝑅)𝑋) ≤ (𝑌(-g‘𝑅)𝑋)) |
35 | 30, 34 | eqbrtrrd 4677 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝑌(-g‘𝑅)𝑋)) |
36 | 11, 32, 12, 17 | orngmul 29803 | . . . . 5 ⊢ ((𝑅 ∈ oRing ∧ (𝑍 ∈ 𝐵 ∧ 0 ≤ 𝑍) ∧ ((𝑌(-g‘𝑅)𝑋) ∈ 𝐵 ∧ 0 ≤ (𝑌(-g‘𝑅)𝑋))) → 0 ≤ (𝑍 · (𝑌(-g‘𝑅)𝑋))) |
37 | 1, 15, 26, 28, 35, 36 | syl122anc 1335 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝑍 · (𝑌(-g‘𝑅)𝑋))) |
38 | 11, 17, 23, 8, 15, 16, 20 | ringsubdi 18599 | . . . 4 ⊢ (𝜑 → (𝑍 · (𝑌(-g‘𝑅)𝑋)) = ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))) |
39 | 37, 38 | breqtrd 4679 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))) |
40 | eqid 2622 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
41 | 11, 32, 40 | omndadd 29706 | . . 3 ⊢ ((𝑅 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋)) ∈ 𝐵 ∧ (𝑍 · 𝑋) ∈ 𝐵) ∧ 0 ≤ ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))) → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) ≤ (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋))) |
42 | 6, 14, 25, 22, 39, 41 | syl131anc 1339 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) ≤ (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋))) |
43 | 11, 40, 12 | grplid 17452 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑍 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑋)) |
44 | 10, 22, 43 | syl2anc 693 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑋)) |
45 | 11, 40, 23 | grpnpcan 17507 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑍 · 𝑌) ∈ 𝐵 ∧ (𝑍 · 𝑋) ∈ 𝐵) → (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑌)) |
46 | 10, 19, 22, 45 | syl3anc 1326 | . 2 ⊢ (𝜑 → (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑌)) |
47 | 42, 44, 46 | 3brtr3d 4684 | 1 ⊢ (𝜑 → (𝑍 · 𝑋) ≤ (𝑍 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 lecple 15948 0gc0g 16100 Grpcgrp 17422 -gcsg 17424 Ringcrg 18547 oMndcomnd 29697 oGrpcogrp 29698 oRingcorng 29795 |
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-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-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-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-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mgp 18490 df-ur 18502 df-ring 18549 df-omnd 29699 df-ogrp 29700 df-orng 29797 |
This theorem is referenced by: ornglmullt 29807 |
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