Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ogrpinvOLD | Structured version Visualization version GIF version |
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 30-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
ogrpsub.0 | ⊢ 𝐵 = (Base‘𝐺) |
ogrpsub.1 | ⊢ ≤ = (le‘𝐺) |
ogrpinv.2 | ⊢ 𝐼 = (invg‘𝐺) |
ogrpinv.3 | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
ogrpinvOLD | ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isogrp 29702 | . . . . 5 ⊢ (𝐺 ∈ oGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ oMnd)) | |
2 | 1 | simprbi 480 | . . . 4 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ oMnd) |
3 | 2 | 3ad2ant1 1082 | . . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝐺 ∈ oMnd) |
4 | 1 | simplbi 476 | . . . . 5 ⊢ (𝐺 ∈ oGrp → 𝐺 ∈ Grp) |
5 | 4 | 3ad2ant1 1082 | . . . 4 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝐺 ∈ Grp) |
6 | ogrpsub.0 | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
7 | ogrpinv.3 | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
8 | 6, 7 | grpidcl 17450 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
9 | 5, 8 | syl 17 | . . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 0 ∈ 𝐵) |
10 | simp2 1062 | . . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 𝑋 ∈ 𝐵) | |
11 | ogrpinv.2 | . . . . 5 ⊢ 𝐼 = (invg‘𝐺) | |
12 | 6, 11 | grpinvcl 17467 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝐼‘𝑋) ∈ 𝐵) |
13 | 5, 10, 12 | syl2anc 693 | . . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ∈ 𝐵) |
14 | simp3 1063 | . . 3 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → 0 ≤ 𝑋) | |
15 | ogrpsub.1 | . . . 4 ⊢ ≤ = (le‘𝐺) | |
16 | eqid 2622 | . . . 4 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
17 | 6, 15, 16 | omndadd 29706 | . . 3 ⊢ ((𝐺 ∈ oMnd ∧ ( 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ (𝐼‘𝑋) ∈ 𝐵) ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋))) |
18 | 3, 9, 10, 13, 14, 17 | syl131anc 1339 | . 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) ≤ (𝑋(+g‘𝐺)(𝐼‘𝑋))) |
19 | 6, 16, 7 | grplid 17452 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝐼‘𝑋) ∈ 𝐵) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋)) |
20 | 5, 13, 19 | syl2anc 693 | . 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → ( 0 (+g‘𝐺)(𝐼‘𝑋)) = (𝐼‘𝑋)) |
21 | 6, 16, 7, 11 | grprinv 17469 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 ) |
22 | 5, 10, 21 | syl2anc 693 | . 2 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝑋(+g‘𝐺)(𝐼‘𝑋)) = 0 ) |
23 | 18, 20, 22 | 3brtr3d 4684 | 1 ⊢ ((𝐺 ∈ oGrp ∧ 𝑋 ∈ 𝐵 ∧ 0 ≤ 𝑋) → (𝐼‘𝑋) ≤ 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 lecple 15948 0gc0g 16100 Grpcgrp 17422 invgcminusg 17423 oMndcomnd 29697 oGrpcogrp 29698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-omnd 29699 df-ogrp 29700 |
This theorem is referenced by: (None) |
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