![]() |
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 10328 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 2651 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 17514 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1100 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
5 | grpsubadd.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
6 | 1, 5 | grpcl 17477 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1411 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
8 | grpsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
9 | 1, 5, 2, 8 | grpsubval 17512 | . . 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 694 | . 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 17553 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
12 | 4, 11 | syld3an3 1411 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
13 | 1, 5, 2, 8 | grpsubval 17512 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
14 | 13 | 3adant1 1099 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
15 | 14 | eqcomd 2657 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 − 𝑌)) |
16 | 1, 2 | grpinvinv 17529 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
17 | 16 | 3adant2 1100 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
18 | 15, 17 | oveq12d 6708 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) + ((invg‘𝐺)‘((invg‘𝐺)‘𝑌))) = ((𝑋 − 𝑌) + 𝑌)) |
19 | 10, 12, 18 | 3eqtr3rd 2694 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 ‘cfv 5926 (class class class)co 6690 Basecbs 15904 +gcplusg 15988 Grpcgrp 17469 invgcminusg 17470 -gcsg 17471 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-1st 7210 df-2nd 7211 df-0g 16149 df-mgm 17289 df-sgrp 17331 df-mnd 17342 df-grp 17472 df-minusg 17473 df-sbg 17474 |
This theorem is referenced by: grpsubsub4 17555 grpnpncan 17557 grpnnncan2 17559 dfgrp3 17561 nsgconj 17674 conjghm 17738 conjnmz 17741 sylow2blem1 18081 ablpncan3 18268 lmodvnpcan 18965 coe1subfv 19684 ipsubdir 20035 ipsubdi 20036 mdetunilem9 20474 subgntr 21957 ghmcnp 21965 tgpt0 21969 r1pid 23964 archiabllem1a 29873 archiabllem2a 29876 ornglmulle 29933 orngrmulle 29934 kercvrlsm 37970 hbtlem5 38015 |
Copyright terms: Public domain | W3C validator |