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Mirrors > Home > MPE Home > Th. List > grpnpcan | Structured version Visualization version GIF version |
Description: Cancellation law for subtraction (npcan 11509 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 2728 | . . . . . 6 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
3 | 1, 2 | grpinvcl 18958 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
4 | 3 | 3adant2 1128 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) ∈ 𝐵) |
5 | grpsubadd.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
6 | 1, 5 | grpcl 18912 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
7 | 4, 6 | syld3an3 1406 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) ∈ 𝐵) |
8 | grpsubadd.m | . . . 4 ⊢ − = (-g‘𝐺) | |
9 | 1, 5, 2, 8 | grpsubval 18956 | . . 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 582 | . 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 19001 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑌) ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
12 | 4, 11 | syld3an3 1406 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) − ((invg‘𝐺)‘𝑌)) = 𝑋) |
13 | 1, 5, 2, 8 | grpsubval 18956 | . . . . 5 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
14 | 13 | 3adant1 1127 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
15 | 14 | eqcomd 2734 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 − 𝑌)) |
16 | 1, 2 | grpinvinv 18976 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
17 | 16 | 3adant2 1128 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘((invg‘𝐺)‘𝑌)) = 𝑌) |
18 | 15, 17 | oveq12d 7444 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 + ((invg‘𝐺)‘𝑌)) + ((invg‘𝐺)‘((invg‘𝐺)‘𝑌))) = ((𝑋 − 𝑌) + 𝑌)) |
19 | 10, 12, 18 | 3eqtr3rd 2777 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 − 𝑌) + 𝑌) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 Grpcgrp 18904 invgcminusg 18905 -gcsg 18906 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-0g 17432 df-mgm 18609 df-sgrp 18688 df-mnd 18704 df-grp 18907 df-minusg 18908 df-sbg 18909 |
This theorem is referenced by: grpsubsub4 19003 grpnpncan 19005 grpnnncan2 19007 dfgrp3 19009 xpsgrpsub 19031 nsgconj 19128 conjghm 19217 conjnmz 19220 sylow2blem1 19589 ablpncan3 19785 lmodvnpcan 20813 ipsubdir 21588 ipsubdi 21589 coe1subfv 22204 mdetunilem9 22550 subgntr 24039 ghmcnp 24047 tgpt0 24051 r1pid 26124 archiabllem1a 32928 archiabllem2a 32931 ornglmulle 33052 orngrmulle 33053 kercvrlsm 42556 hbtlem5 42601 |
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