![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > grpsubadd0sub | Structured version Visualization version GIF version |
Description: Subtraction expressed as addition of the difference of the identity element and the subtrahend. (Contributed by AV, 9-Nov-2019.) |
Ref | Expression |
---|---|
grpsubid.b | ⊢ 𝐵 = (Base‘𝐺) |
grpsubid.o | ⊢ 0 = (0g‘𝐺) |
grpsubid.m | ⊢ − = (-g‘𝐺) |
grpsubadd0sub.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
grpsubadd0sub | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpsubid.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpsubadd0sub.p | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | eqid 2725 | . . . 4 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
4 | grpsubid.m | . . . 4 ⊢ − = (-g‘𝐺) | |
5 | 1, 2, 3, 4 | grpsubval 18941 | . . 3 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
6 | 5 | 3adant1 1127 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ((invg‘𝐺)‘𝑌))) |
7 | grpsubid.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
8 | 1, 4, 3, 7 | grpinvval2 18978 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
9 | 8 | 3adant2 1128 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((invg‘𝐺)‘𝑌) = ( 0 − 𝑌)) |
10 | 9 | oveq2d 7429 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + ((invg‘𝐺)‘𝑌)) = (𝑋 + ( 0 − 𝑌))) |
11 | 6, 10 | eqtrd 2765 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 − 𝑌) = (𝑋 + ( 0 − 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6543 (class class class)co 7413 Basecbs 17174 +gcplusg 17227 0gc0g 17415 Grpcgrp 18889 invgcminusg 18890 -gcsg 18891 |
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 2696 ax-sep 5295 ax-nul 5302 ax-pow 5360 ax-pr 5424 ax-un 7735 |
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 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5145 df-opab 5207 df-mpt 5228 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-riota 7369 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7987 df-2nd 7988 df-0g 17417 df-mgm 18594 df-sgrp 18673 df-mnd 18689 df-grp 18892 df-minusg 18893 df-sbg 18894 |
This theorem is referenced by: chfacfscmulgsum 22775 chfacfpmmulgsum 22779 |
Copyright terms: Public domain | W3C validator |