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 2610 | . . . . . . . 8 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
8 | ogrpaddlt.1 | . . . . . . . 8 ⊢ < = (lt‘𝐺) | |
9 | 7, 8 | oppglt 28985 | . . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → < = (lt‘(oppg‘𝐺))) |
10 | 6, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → < = (lt‘(oppg‘𝐺))) |
11 | 10 | breqd 4594 | . . . . 5 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 𝑋(lt‘(oppg‘𝐺))𝑌)) |
12 | 5, 11 | mpbid 221 | . . . 4 ⊢ (𝜑 → 𝑋(lt‘(oppg‘𝐺))𝑌) |
13 | ogrpaddlt.0 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | 7, 13 | oppgbas 17604 | . . . . 5 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
15 | eqid 2610 | . . . . 5 ⊢ (lt‘(oppg‘𝐺)) = (lt‘(oppg‘𝐺)) | |
16 | eqid 2610 | . . . . 5 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
17 | 14, 15, 16 | ogrpaddlt 29049 | . . . 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 1331 | . . 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 17602 | . . 3 ⊢ (𝑋(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑋) |
21 | 19, 7, 16 | oppgplus 17602 | . . 3 ⊢ (𝑌(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑌) |
22 | 18, 20, 21 | 3brtr3g 4616 | . 2 ⊢ (𝜑 → (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌)) |
23 | 10 | breqd 4594 | . 2 ⊢ (𝜑 → ((𝑍 + 𝑋) < (𝑍 + 𝑌) ↔ (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌))) |
24 | 22, 23 | mpbird 246 | 1 ⊢ (𝜑 → (𝑍 + 𝑋) < (𝑍 + 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 ltcplt 16764 oppgcoppg 17598 oGrpcogrp 29029 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-ple 15788 df-0g 15925 df-plt 16781 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-oppg 17599 df-omnd 29030 df-ogrp 29031 |
This theorem is referenced by: ogrpaddltrbid 29052 archiabllem2a 29079 archiabllem2c 29080 |
Copyright terms: Public domain | W3C validator |