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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 33311 | . . . . 5 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ oGrp) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑅 ∈ oGrp) |
4 | isogrp 33062 | . . . . 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 33310 | . . . . . 6 ⊢ (𝑅 ∈ oRing → 𝑅 ∈ Ring) | |
8 | 1, 7 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑅 ∈ Ring) |
9 | ringgrp 20256 | . . . . 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 18996 | . . . 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 20268 | . . . . 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 20268 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 · 𝑍) ∈ 𝐵) |
22 | 8, 20, 16, 21 | syl3anc 1370 | . . . 4 ⊢ (𝜑 → (𝑋 · 𝑍) ∈ 𝐵) |
23 | eqid 2735 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
24 | 11, 23 | grpsubcl 19051 | . . . 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 19051 | . . . . . 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 19055 | . . . . . . 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 33076 | . . . . . . 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 5172 | . . . . 5 ⊢ (𝜑 → 0 ≤ (𝑌(-g‘𝑅)𝑋)) |
35 | orngmulle.6 | . . . . 5 ⊢ (𝜑 → 0 ≤ 𝑍) | |
36 | 11, 31, 12, 17 | orngmul 33313 | . . . . 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 | ringsubdir 20322 | . . . 4 ⊢ (𝜑 → ((𝑌(-g‘𝑅)𝑋) · 𝑍) = ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))) |
39 | 37, 38 | breqtrd 5174 | . . 3 ⊢ (𝜑 → 0 ≤ ((𝑌 · 𝑍)(-g‘𝑅)(𝑋 · 𝑍))) |
40 | eqid 2735 | . . . 4 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
41 | 11, 31, 40 | omndadd 33066 | . . 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 18998 | . . 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 19063 | . . 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 5179 | 1 ⊢ (𝜑 → (𝑋 · 𝑍) ≤ (𝑌 · 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 class class class wbr 5148 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 +gcplusg 17298 .rcmulr 17299 lecple 17305 0gc0g 17486 Grpcgrp 18964 -gcsg 18966 Ringcrg 20251 oMndcomnd 33057 oGrpcogrp 33058 oRingcorng 33305 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-omnd 33059 df-ogrp 33060 df-orng 33307 |
This theorem is referenced by: orngrmullt 33318 |
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