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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 2733 | . . . . . . . 8 ⊢ (oppg‘𝐺) = (oppg‘𝐺) | |
8 | ogrpaddlt.1 | . . . . . . . 8 ⊢ < = (lt‘𝐺) | |
9 | 7, 8 | oppglt 31871 | . . . . . . 7 ⊢ (𝐺 ∈ 𝑉 → < = (lt‘(oppg‘𝐺))) |
10 | 6, 9 | syl 17 | . . . . . 6 ⊢ (𝜑 → < = (lt‘(oppg‘𝐺))) |
11 | 10 | breqd 5117 | . . . . 5 ⊢ (𝜑 → (𝑋 < 𝑌 ↔ 𝑋(lt‘(oppg‘𝐺))𝑌)) |
12 | 5, 11 | mpbid 231 | . . . 4 ⊢ (𝜑 → 𝑋(lt‘(oppg‘𝐺))𝑌) |
13 | ogrpaddlt.0 | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
14 | 7, 13 | oppgbas 19135 | . . . . 5 ⊢ 𝐵 = (Base‘(oppg‘𝐺)) |
15 | eqid 2733 | . . . . 5 ⊢ (lt‘(oppg‘𝐺)) = (lt‘(oppg‘𝐺)) | |
16 | eqid 2733 | . . . . 5 ⊢ (+g‘(oppg‘𝐺)) = (+g‘(oppg‘𝐺)) | |
17 | 14, 15, 16 | ogrpaddlt 31974 | . . . 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 1384 | . . 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 19132 | . . 3 ⊢ (𝑋(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑋) |
21 | 19, 7, 16 | oppgplus 19132 | . . 3 ⊢ (𝑌(+g‘(oppg‘𝐺))𝑍) = (𝑍 + 𝑌) |
22 | 18, 20, 21 | 3brtr3g 5139 | . 2 ⊢ (𝜑 → (𝑍 + 𝑋)(lt‘(oppg‘𝐺))(𝑍 + 𝑌)) |
23 | 10 | breqd 5117 | . 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 1542 ∈ wcel 2107 class class class wbr 5106 ‘cfv 6497 (class class class)co 7358 Basecbs 17088 +gcplusg 17138 ltcplt 18202 oppgcoppg 19128 oGrpcogrp 31955 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-er 8651 df-en 8887 df-dom 8888 df-sdom 8889 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-dec 12624 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-plusg 17151 df-ple 17158 df-0g 17328 df-plt 18224 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-grp 18756 df-oppg 19129 df-omnd 31956 df-ogrp 31957 |
This theorem is referenced by: ogrpaddltrbid 31977 archiabllem2a 32079 archiabllem2c 32080 |
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