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Mirrors > Home > MPE Home > Th. List > Mathboxes > sn-grpridd | Structured version Visualization version GIF version |
Description: The identity element of a group is a right identity. Deduction associated with grprid 18740. (Contributed by SN, 29-Jan-2025.) |
Ref | Expression |
---|---|
grplidd.b | ⊢ 𝐵 = (Base‘𝐺) |
grplidd.p | ⊢ + = (+g‘𝐺) |
grplidd.u | ⊢ 0 = (0g‘𝐺) |
grplidd.g | ⊢ (𝜑 → 𝐺 ∈ Grp) |
grplidd.1 | ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
Ref | Expression |
---|---|
sn-grpridd | ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grplidd.g | . 2 ⊢ (𝜑 → 𝐺 ∈ Grp) | |
2 | grplidd.1 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | |
3 | grplidd.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
4 | grplidd.p | . . 3 ⊢ + = (+g‘𝐺) | |
5 | grplidd.u | . . 3 ⊢ 0 = (0g‘𝐺) | |
6 | 3, 4, 5 | grprid 18740 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 + 0 ) = 𝑋) |
7 | 1, 2, 6 | syl2anc 584 | 1 ⊢ (𝜑 → (𝑋 + 0 ) = 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 Basecbs 17042 +gcplusg 17092 0gc0g 17280 Grpcgrp 18707 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pr 5382 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-iota 6445 df-fun 6495 df-fv 6501 df-riota 7307 df-ov 7354 df-0g 17282 df-mgm 18456 df-sgrp 18505 df-mnd 18516 df-grp 18710 |
This theorem is referenced by: (None) |
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