![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > ogrpaddltrd | Structured version Visualization version GIF version |
Description: In a right ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 3-Sep-2018.) |
Ref | Expression |
---|---|
ogrpaddlt.0 | ⊢ 𝐵 = (Base‘𝐺) |
ogrpaddlt.1 | ⊢ < = (lt‘𝐺) |
ogrpaddlt.2 | ⊢ + = (+g‘𝐺) |
ogrpaddltrd.1 | ⊢ (𝜑 → 𝐺 ∈ 𝑉) |
ogrpaddltrd.2 | ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) |
ogrpaddltrd.3 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
ogrpaddltrd.4 | ⊢ (𝜑 → 𝑌 ∈ 𝐵) |
ogrpaddltrd.5 | ⊢ (𝜑 → 𝑍 ∈ 𝐵) |
ogrpaddltrd.6 | ⊢ (𝜑 → 𝑋 < 𝑌) |
Ref | Expression |
---|---|
ogrpaddltrd | ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ogrpaddltrd.2 | . . . 4 ⊢ (𝜑 → (oppg‘𝐺) ∈ oGrp) | |
2 | ogrpaddltrd.3 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | ogrpaddltrd.4 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵) | |
4 | ogrpaddltrd.5 | . . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵) | |
5 | ogrpaddltrd.6 | . . . . 5 ⊢ (𝜑 → 𝑋 < 𝑌) | |
6 | ogrpaddltrd.1 | . . . . . . 7 ⊢ (𝜑 → 𝐺 ∈ 𝑉) | |
7 | eqid 2735 | . . . . . . . 8 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
8 | ogrpaddlt.1 | . . . . . . . 8 ⊢ < = (lt‘𝐺) | |
9 | 7, 8 | oppglt 32938 | . . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → < = (lt‘(oppg‘𝐺))) |
10 | 6, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → < = (lt‘(oppg‘𝐺))) |
11 | 10 | breqd 5159 | . . . . 5 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 𝑋(lt‘(oppg‘𝐺))𝑌)) |
12 | 5, 11 | mpbid 232 | . . . 4 ⊢ (𝜑 → 𝑋(lt‘(oppg‘𝐺))𝑌) |
13 | ogrpaddlt.0 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | 7, 13 | oppgbas 19383 | . . . . 5 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
15 | eqid 2735 | . . . . 5 ⊢ (lt‘(oppg‘𝐺)) = (lt‘(oppg‘𝐺)) | |
16 | eqid 2735 | . . . . 5 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
17 | 14, 15, 16 | ogrpaddlt 33077 | . . . 4 ⊢ (((oppg‘𝐺) ∈ oGrp ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋(lt‘(oppg‘𝐺))𝑌) → (𝑋(+g‘(oppg‘𝐺))𝑍)(lt‘(oppg‘𝐺))(𝑌(+g‘(oppg‘𝐺))𝑍)) |
18 | 1, 2, 3, 4, 12, 17 | syl131anc 1382 | . . 3 ⊢ (𝜑 → (𝑋(+g‘(oppg‘𝐺))𝑍)(lt‘(oppg‘𝐺))(𝑌(+g‘(oppg‘𝐺))𝑍)) |
19 | ogrpaddlt.2 | . . . 4 ⊢ + = (+g‘𝐺) | |
20 | 19, 7, 16 | oppgplus 19380 | . . 3 ⊢ (𝑋(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑋) |
21 | 19, 7, 16 | oppgplus 19380 | . . 3 ⊢ (𝑌(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑌) |
22 | 18, 20, 21 | 3brtr3g 5181 | . 2 ⊢ (𝜑 → (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌)) |
23 | 10 | breqd 5159 | . 2 ⊢ (𝜑 → ((𝑍 + 𝑋) < (𝑍 + 𝑌) ↔ (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌))) |
24 | 22, 23 | mpbird 257 | 1 ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) |
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 ltcplt 18366 oppgcoppg 19376 oGrpcogrp 33058 |
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-2nd 8014 df-tpos 8250 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-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-dec 12732 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-ple 17318 df-0g 17488 df-plt 18388 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-oppg 19377 df-omnd 33059 df-ogrp 33060 |
This theorem is referenced by: ogrpaddltrbid 33080 archiabllem2a 33184 archiabllem2c 33185 |
Copyright terms: Public domain | W3C validator |