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Mirrors > Home > MPE Home > Th. List > grpridd | Structured version Visualization version GIF version |
Description: The identity element of a group is a right identity. Deduction associated with grprid 18949. (Contributed by SN, 29-Jan-2025.) |
Ref | Expression |
---|---|
grpbn0.b | ⊢ 𝐵 = (Base‘𝐺) |
grplid.p | ⊢ + = (+g‘𝐺) |
grplid.o | ⊢ 0 = (0g‘𝐺) |
grplidd.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grplidd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
grpridd | ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplidd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | grplidd.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | grpbn0.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | grplid.p | . . 3 ⊢ + = (+g‘𝐺) | |
5 | grplid.o | . . 3 ⊢ 0 = (0g‘𝐺) | |
6 | 3, 4, 5 | grprid 18949 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
7 | 1, 2, 6 | syl2anc 582 | 1 ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 ‘cfv 6549 (class class class)co 7419 Basecbs 17199 +gcplusg 17252 0gc0g 17440 Grpcgrp 18914 |
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 5300 ax-nul 5307 ax-pr 5429 |
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 2930 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-dif 3947 df-un 3949 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-iota 6501 df-fun 6551 df-fv 6557 df-riota 7375 df-ov 7422 df-0g 17442 df-mgm 18619 df-sgrp 18698 df-mnd 18714 df-grp 18917 |
This theorem is referenced by: grprcan 18954 ghmqusnsglem1 19260 ablsubaddsub 19798 rnglidlmcl 21141 rloccring 33081 evl1deg1 33405 evl1deg3 33406 irredminply 33534 primrootscoprmpow 41721 primrootscoprbij 41724 |
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