Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > grpnpcan | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction (npcan 10883 analog). (Contributed by NM, 19-Apr-2014.) |
Ref | Expression |
---|---|
grpsubadd.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubadd.p | ⊢ + = (+g‘𝐺) |
grpsubadd.m | ⊢ − = (-g‘𝐺) |
Ref | Expression |
---|---|
grpnpcan | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubadd.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐺) | |
2 | eqid 2818 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18089 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1123 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
5 | grpsubadd.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
6 | 1, 5 | grpcl 18049 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1401 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
8 | grpsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
9 | 1, 5, 2, 8 | grpsubval 18087 | . . 3 ⊢ (((𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = ((𝑋 + ((invg‘𝐺)‘𝑌)) + ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)))) |
10 | 7, 4, 9 | syl2anc 584 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = ((𝑋 + ((invg‘𝐺)‘𝑌)) + ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)))) |
11 | 1, 5, 8 | grppncan 18128 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
12 | 4, 11 | syld3an3 1401 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
13 | 1, 5, 2, 8 | grpsubval 18087 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
14 | 13 | 3adant1 1122 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
15 | 14 | eqcomd 2824 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 − 𝑌)) |
16 | 1, 2 | grpinvinv 18104 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
17 | 16 | 3adant2 1123 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
18 | 15, 17 | oveq12d 7163 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) + ((invg‘𝐺)‘((invg‘𝐺)‘𝑌))) = ((𝑋 − 𝑌) + 𝑌)) |
19 | 10, 12, 18 | 3eqtr3rd 2862 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 +gcplusg 16553 Grpcgrp 18041 invgcminusg 18042 -gcsg 18043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 |
This theorem is referenced by: grpsubsub4 18130 grpnpncan 18132 grpnnncan2 18134 dfgrp3 18136 nsgconj 18249 conjghm 18327 conjnmz 18330 sylow2blem1 18674 ablpncan3 18866 lmodvnpcan 19617 coe1subfv 20362 ipsubdir 20714 ipsubdi 20715 mdetunilem9 21157 subgntr 22642 ghmcnp 22650 tgpt0 22654 r1pid 24680 archiabllem1a 30747 archiabllem2a 30750 ornglmulle 30805 orngrmulle 30806 kercvrlsm 39561 hbtlem5 39606 |
Copyright terms: Public domain | W3C validator |