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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 33011 | . . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ oGrp) |
4 | isogrp 32777 | . . . . 5 ⊢ (𝑅 ∈ oGrp ↔ (𝑅 ∈ Grp ∧ 𝑅 ∈ oMnd)) | |
5 | 4 | simprbi 496 | . . . 4 ⊢ (𝑅 ∈ oGrp → 𝑅 ∈ oMnd) |
6 | 3, 5 | syl 17 | . . 3 ⊢ (𝜑 → 𝑅 ∈ oMnd) |
7 | orngring 33010 | . . . . . 6 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | ringgrp 20172 | . . . . 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 18916 | . . . 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 20184 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑍 · 𝑌) ∈ 𝐵) |
19 | 8, 15, 16, 18 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝑍 · 𝑌) ∈ 𝐵) |
20 | ornglmullt.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
21 | 11, 17 | ringcl 20184 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑍 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑍 · 𝑋) ∈ 𝐵) |
22 | 8, 15, 20, 21 | syl3anc 1369 | . . . 4 ⊢ (𝜑 → (𝑍 · 𝑋) ∈ 𝐵) |
23 | eqid 2728 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
24 | 11, 23 | grpsubcl 18970 | . . . 4 ⊢ ((𝑅 ∈ Grp ∧ (𝑍 · 𝑌) ∈ 𝐵 ∧ (𝑍 · 𝑋) ∈ 𝐵) → ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋)) ∈ 𝐵) |
25 | 10, 19, 22, 24 | syl3anc 1369 | . . 3 ⊢ (𝜑 → ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋)) ∈ 𝐵) |
26 | orngmulle.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑍) | |
27 | 11, 23 | grpsubcl 18970 | . . . . . 6 ⊢ ((𝑅 ∈ Grp ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
28 | 10, 16, 20, 27 | syl3anc 1369 | . . . . 5 ⊢ (𝜑 → (𝑌(-g‘𝑅)𝑋) ∈ 𝐵) |
29 | 11, 12, 23 | grpsubid 18974 | . . . . . . 7 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(-g‘𝑅)𝑋) = 0 ) |
30 | 10, 20, 29 | syl2anc 583 | . . . . . 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 32791 | . . . . . . 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 1381 | . . . . . 6 ⊢ (𝜑 → (𝑋(-g‘𝑅)𝑋) ≤ (𝑌(-g‘𝑅)𝑋)) |
35 | 30, 34 | eqbrtrrd 5167 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝑌(-g‘𝑅)𝑋)) |
36 | 11, 32, 12, 17 | orngmul 33013 | . . . . 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 1377 | . . . 4 ⊢ (𝜑 → 0 ≤ (𝑍 · (𝑌(-g‘𝑅)𝑋))) |
38 | 11, 17, 23, 8, 15, 16, 20 | ringsubdi 20237 | . . . 4 ⊢ (𝜑 → (𝑍 · (𝑌(-g‘𝑅)𝑋)) = ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))) |
39 | 37, 38 | breqtrd 5169 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))) |
40 | eqid 2728 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
41 | 11, 32, 40 | omndadd 32781 | . . 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 1381 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) ≤ (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋))) |
43 | 11, 40, 12 | grplid 18918 | . . 3 ⊢ ((𝑅 ∈ Grp ∧ (𝑍 · 𝑋) ∈ 𝐵) → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑋)) |
44 | 10, 22, 43 | syl2anc 583 | . 2 ⊢ (𝜑 → ( 0 (+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑋)) |
45 | 11, 40, 23 | grpnpcan 18982 | . . 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 1369 | . 2 ⊢ (𝜑 → (((𝑍 · 𝑌)(-g‘𝑅)(𝑍 · 𝑋))(+g‘𝑅)(𝑍 · 𝑋)) = (𝑍 · 𝑌)) |
47 | 42, 44, 46 | 3brtr3d 5174 | 1 ⊢ (𝜑 → (𝑍 · 𝑋) ≤ (𝑍 · 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 class class class wbr 5143 ‘cfv 6543 (class class class)co 7415 Basecbs 17174 +gcplusg 17227 .rcmulr 17228 lecple 17234 0gc0g 17415 Grpcgrp 18884 -gcsg 18886 Ringcrg 20167 oMndcomnd 32772 oGrpcogrp 32773 oRingcorng 33005 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 ax-cnex 11189 ax-resscn 11190 ax-1cn 11191 ax-icn 11192 ax-addcl 11193 ax-addrcl 11194 ax-mulcl 11195 ax-mulrcl 11196 ax-mulcom 11197 ax-addass 11198 ax-mulass 11199 ax-distr 11200 ax-i2m1 11201 ax-1ne0 11202 ax-1rid 11203 ax-rnegex 11204 ax-rrecex 11205 ax-cnre 11206 ax-pre-lttri 11207 ax-pre-lttrn 11208 ax-pre-ltadd 11209 ax-pre-mulgt0 11210 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-nel 3043 df-ral 3058 df-rex 3067 df-rmo 3372 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5571 df-eprel 5577 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7866 df-1st 7988 df-2nd 7989 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-er 8719 df-en 8959 df-dom 8960 df-sdom 8961 df-pnf 11275 df-mnf 11276 df-xr 11277 df-ltxr 11278 df-le 11279 df-sub 11471 df-neg 11472 df-nn 12238 df-2 12300 df-sets 17127 df-slot 17145 df-ndx 17157 df-base 17175 df-plusg 17240 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18887 df-minusg 18888 df-sbg 18889 df-cmn 19731 df-abl 19732 df-mgp 20069 df-rng 20087 df-ur 20116 df-ring 20169 df-omnd 32774 df-ogrp 32775 df-orng 33007 |
This theorem is referenced by: ornglmullt 33017 |
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