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Mirrors > Home > MPE Home > Th. List > grpid | Structured version Visualization version GIF version |
Description: Two ways of saying that an element of a group is the identity element. Provides a convenient way to compute the value of the identity element. (Contributed by NM, 24-Aug-2011.) |
Ref | Expression |
---|---|
grpinveu.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinveu.p | ⊢ + = (+g‘𝐺) |
grpinveu.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpid | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqcom 2801 | . 2 ⊢ ( 0 = 𝑋 ↔ 𝑋 = 0 ) | |
2 | grpinveu.b | . . . . . . 7 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpinveu.o | . . . . . . 7 ⊢ 0 = (0g‘𝐺) | |
4 | 2, 3 | grpidcl 17889 | . . . . . 6 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
5 | grpinveu.p | . . . . . . . 8 ⊢ + = (+g‘𝐺) | |
6 | 2, 5 | grprcan 17894 | . . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑋 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) |
7 | 6 | 3exp2 1347 | . . . . . 6 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ( 0 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))))) |
8 | 4, 7 | mpid 44 | . . . . 5 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )))) |
9 | 8 | pm2.43d 53 | . . . 4 ⊢ (𝐺 ∈ Grp → (𝑋 ∈ 𝐵 → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 ))) |
10 | 9 | imp 407 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ 𝑋 = 0 )) |
11 | 2, 5, 3 | grplid 17891 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
12 | 11 | eqeq2d 2804 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = ( 0 + 𝑋) ↔ (𝑋 + 𝑋) = 𝑋)) |
13 | 10, 12 | bitr3d 282 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → (𝑋 = 0 ↔ (𝑋 + 𝑋) = 𝑋)) |
14 | 1, 13 | syl5rbb 285 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ((𝑋 + 𝑋) = 𝑋 ↔ 0 = 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1522 ∈ wcel 2080 ‘cfv 6228 (class class class)co 7019 Basecbs 16312 +gcplusg 16394 0gc0g 16542 Grpcgrp 17861 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-reu 3111 df-rmo 3112 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-nul 4214 df-if 4384 df-sn 4475 df-pr 4477 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-mpt 5044 df-id 5351 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-iota 6192 df-fun 6230 df-fv 6236 df-riota 6980 df-ov 7022 df-0g 16544 df-mgm 17681 df-sgrp 17723 df-mnd 17734 df-grp 17864 |
This theorem is referenced by: isgrpid2 17897 grpidd2 17898 subg0 18039 qus0 18091 ghmid 18105 symgid 18260 isdrng2 19202 lmod0vid 19356 psr0 19867 cnfld0 20251 ldual0v 35830 erng0g 37674 |
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