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Mathbox for Thierry Arnoux |
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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 2798 | . . . . . . . 8 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
8 | ogrpaddlt.1 | . . . . . . . 8 ⊢ < = (lt‘𝐺) | |
9 | 7, 8 | oppglt 30667 | . . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → < = (lt‘(oppg‘𝐺))) |
10 | 6, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → < = (lt‘(oppg‘𝐺))) |
11 | 10 | breqd 5041 | . . . . 5 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 𝑋(lt‘(oppg‘𝐺))𝑌)) |
12 | 5, 11 | mpbid 235 | . . . 4 ⊢ (𝜑 → 𝑋(lt‘(oppg‘𝐺))𝑌) |
13 | ogrpaddlt.0 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | 7, 13 | oppgbas 18471 | . . . . 5 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
15 | eqid 2798 | . . . . 5 ⊢ (lt‘(oppg‘𝐺)) = (lt‘(oppg‘𝐺)) | |
16 | eqid 2798 | . . . . 5 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
17 | 14, 15, 16 | ogrpaddlt 30768 | . . . 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 1380 | . . 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 18469 | . . 3 ⊢ (𝑋(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑋) |
21 | 19, 7, 16 | oppgplus 18469 | . . 3 ⊢ (𝑌(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑌) |
22 | 18, 20, 21 | 3brtr3g 5063 | . 2 ⊢ (𝜑 → (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌)) |
23 | 10 | breqd 5041 | . 2 ⊢ (𝜑 → ((𝑍 + 𝑋) < (𝑍 + 𝑌) ↔ (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌))) |
24 | 22, 23 | mpbird 260 | 1 ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 ltcplt 17543 oppgcoppg 18465 oGrpcogrp 30749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-dec 12087 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-ple 16577 df-0g 16707 df-plt 17560 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-oppg 18466 df-omnd 30750 df-ogrp 30751 |
This theorem is referenced by: ogrpaddltrbid 30771 archiabllem2a 30873 archiabllem2c 30874 |
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