Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orngrmulle | 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 |
---|---|
orngrmulle | ⊢ (𝜑 → (𝑋 · 𝑍) ≤ (𝑌 · 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ornglmullt.1 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ oRing) | |
2 | orngogrp 31497 | . . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ oGrp) |
4 | isogrp 31325 | . . . . 5 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd)) | |
5 | 4 | simprbi 497 | . . . 4 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ oMnd) |
7 | orngring 31496 | . . . . . 6 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | ringgrp 19786 | . . . . 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 18605 | . . . 4 ⊢ (𝑅 ∈ Grp → 0 ∈ 𝐵) |
14 | 10, 13 | syl 17 | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) |
15 | ornglmullt.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
16 | ornglmullt.4 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
17 | ornglmullt.t | . . . . . 6 ⊢ · = (.r‘𝑅) | |
18 | 11, 17 | ringcl 19798 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 · 𝑍) ∈ 𝐵) |
19 | 8, 15, 16, 18 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑌 · 𝑍) ∈ 𝐵) |
20 | ornglmullt.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | 11, 17 | ringcl 19798 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
22 | 8, 20, 16, 21 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
23 | eqid 2738 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
24 | 11, 23 | grpsubcl 18653 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑌 · 𝑍) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) → ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍)) ∈ 𝐵) |
25 | 10, 19, 22, 24 | syl3anc 1370 | . . 3 ⊢ (𝜑 → ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍)) ∈ 𝐵) |
26 | 11, 23 | grpsubcl 18653 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
27 | 10, 15, 20, 26 | syl3anc 1370 | . . . . 5 ⊢ (𝜑 → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
28 | 11, 12, 23 | grpsubid 18657 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(-g‘𝑅)𝑋) = 0 ) |
29 | 10, 20, 28 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → (𝑋(-g‘𝑅)𝑋) = 0 ) |
30 | orngmulle.5 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ≤ 𝑌) | |
31 | orngmulle.l | . . . . . . . 8 ⊢ ≤ = (le‘𝑅) | |
32 | 11, 31, 23 | ogrpsub 31339 | . . . . . . 7 ⊢ ((𝑅 ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋(-g‘𝑅)𝑋) ≤ (𝑌(-g‘𝑅)𝑋)) |
33 | 3, 20, 15, 20, 30, 32 | syl131anc 1382 | . . . . . 6 ⊢ (𝜑 → (𝑋(-g‘𝑅)𝑋) ≤ (𝑌(-g‘𝑅)𝑋)) |
34 | 29, 33 | eqbrtrrd 5100 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝑌(-g‘𝑅)𝑋)) |
35 | orngmulle.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑍) | |
36 | 11, 31, 12, 17 | orngmul 31499 | . . . . 5 ⊢ ((𝑅 ∈ oRing ∧ ((𝑌(-g‘𝑅)𝑋) ∈ 𝐵 ∧ 0 ≤ (𝑌(-g‘𝑅)𝑋)) ∧ (𝑍 ∈ 𝐵 ∧ 0 ≤ 𝑍)) → 0 ≤ ((𝑌(-g‘𝑅)𝑋) · 𝑍)) |
37 | 1, 27, 34, 16, 35, 36 | syl122anc 1378 | . . . 4 ⊢ (𝜑 → 0 ≤ ((𝑌(-g‘𝑅)𝑋) · 𝑍)) |
38 | 11, 17, 23, 8, 15, 20, 16 | rngsubdir 19837 | . . . 4 ⊢ (𝜑 → ((𝑌(-g‘𝑅)𝑋) · 𝑍) = ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))) |
39 | 37, 38 | breqtrd 5102 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))) |
40 | eqid 2738 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
41 | 11, 31, 40 | omndadd 31329 | . . 3 ⊢ ((𝑅 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍)) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) ∧ 0 ≤ ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))) → ( 0 (+g‘𝑅)(𝑋 · 𝑍)) ≤ (((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))(+g‘𝑅)(𝑋 · 𝑍))) |
42 | 6, 14, 25, 22, 39, 41 | syl131anc 1382 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑋 · 𝑍)) ≤ (((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))(+g‘𝑅)(𝑋 · 𝑍))) |
43 | 11, 40, 12 | grplid 18607 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑋 · 𝑍) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑋 · 𝑍)) = (𝑋 · 𝑍)) |
44 | 10, 22, 43 | syl2anc 584 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑋 · 𝑍)) = (𝑋 · 𝑍)) |
45 | 11, 40, 23 | grpnpcan 18665 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑌 · 𝑍) ∈ 𝐵 ∧ (𝑋 · 𝑍) ∈ 𝐵) → (((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))(+g‘𝑅)(𝑋 · 𝑍)) = (𝑌 · 𝑍)) |
46 | 10, 19, 22, 45 | syl3anc 1370 | . 2 ⊢ (𝜑 → (((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))(+g‘𝑅)(𝑋 · 𝑍)) = (𝑌 · 𝑍)) |
47 | 42, 44, 46 | 3brtr3d 5107 | 1 ⊢ (𝜑 → (𝑋 · 𝑍) ≤ (𝑌 · 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2106 class class class wbr 5076 ‘cfv 6435 (class class class)co 7277 Basecbs 16910 +gcplusg 16960 .rcmulr 16961 lecple 16967 0gc0g 17148 Grpcgrp 18575 -gcsg 18577 Ringcrg 19781 oMndcomnd 31320 oGrpcogrp 31321 oRingcorng 31491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5225 ax-nul 5232 ax-pow 5290 ax-pr 5354 ax-un 7588 ax-cnex 10925 ax-resscn 10926 ax-1cn 10927 ax-icn 10928 ax-addcl 10929 ax-addrcl 10930 ax-mulcl 10931 ax-mulrcl 10932 ax-mulcom 10933 ax-addass 10934 ax-mulass 10935 ax-distr 10936 ax-i2m1 10937 ax-1ne0 10938 ax-1rid 10939 ax-rnegex 10940 ax-rrecex 10941 ax-cnre 10942 ax-pre-lttri 10943 ax-pre-lttrn 10944 ax-pre-ltadd 10945 ax-pre-mulgt0 10946 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3433 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4259 df-if 4462 df-pw 4537 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-iun 4928 df-br 5077 df-opab 5139 df-mpt 5160 df-tr 5194 df-id 5491 df-eprel 5497 df-po 5505 df-so 5506 df-fr 5546 df-we 5548 df-xp 5597 df-rel 5598 df-cnv 5599 df-co 5600 df-dm 5601 df-rn 5602 df-res 5603 df-ima 5604 df-pred 6204 df-ord 6271 df-on 6272 df-lim 6273 df-suc 6274 df-iota 6393 df-fun 6437 df-fn 6438 df-f 6439 df-f1 6440 df-fo 6441 df-f1o 6442 df-fv 6443 df-riota 7234 df-ov 7280 df-oprab 7281 df-mpo 7282 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8095 df-wrecs 8126 df-recs 8200 df-rdg 8239 df-er 8496 df-en 8732 df-dom 8733 df-sdom 8734 df-pnf 11009 df-mnf 11010 df-xr 11011 df-ltxr 11012 df-le 11013 df-sub 11205 df-neg 11206 df-nn 11972 df-2 12034 df-sets 16863 df-slot 16881 df-ndx 16893 df-base 16911 df-plusg 16973 df-0g 17150 df-mgm 18324 df-sgrp 18373 df-mnd 18384 df-grp 18578 df-minusg 18579 df-sbg 18580 df-mgp 19719 df-ur 19736 df-ring 19783 df-omnd 31322 df-ogrp 31323 df-orng 31493 |
This theorem is referenced by: orngrmullt 31504 |
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