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Mirrors > Home > ILE Home > Th. List > isgrpid2 | GIF version |
Description: Properties showing that an element 𝑍 is the identity element of a group. (Contributed by NM, 7-Aug-2013.) |
Ref | Expression |
---|---|
grpinveu.b | ⊢ 𝐵 = (Base‘𝐺) |
grpinveu.p | ⊢ + = (+g‘𝐺) |
grpinveu.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
isgrpid2 | ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpinveu.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpinveu.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | grpinveu.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | grpid 12844 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 ↔ 0 = 𝑍)) |
5 | 4 | biimpd 144 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑍) = 𝑍 → 0 = 𝑍)) |
6 | 5 | expimpd 363 | . 2 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) → 0 = 𝑍)) |
7 | 1, 3 | grpidcl 12836 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
8 | 1, 2, 3 | grplid 12838 | . . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 0 ∈ 𝐵) → ( 0 + 0 ) = 0 ) |
9 | 7, 8 | mpdan 421 | . . . 4 ⊢ (𝐺 ∈ Grp → ( 0 + 0 ) = 0 ) |
10 | 7, 9 | jca 306 | . . 3 ⊢ (𝐺 ∈ Grp → ( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 )) |
11 | eleq1 2240 | . . . 4 ⊢ ( 0 = 𝑍 → ( 0 ∈ 𝐵 ↔ 𝑍 ∈ 𝐵)) | |
12 | id 19 | . . . . . 6 ⊢ ( 0 = 𝑍 → 0 = 𝑍) | |
13 | 12, 12 | oveq12d 5890 | . . . . 5 ⊢ ( 0 = 𝑍 → ( 0 + 0 ) = (𝑍 + 𝑍)) |
14 | 13, 12 | eqeq12d 2192 | . . . 4 ⊢ ( 0 = 𝑍 → (( 0 + 0 ) = 0 ↔ (𝑍 + 𝑍) = 𝑍)) |
15 | 11, 14 | anbi12d 473 | . . 3 ⊢ ( 0 = 𝑍 → (( 0 ∈ 𝐵 ∧ ( 0 + 0 ) = 0 ) ↔ (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
16 | 10, 15 | syl5ibcom 155 | . 2 ⊢ (𝐺 ∈ Grp → ( 0 = 𝑍 → (𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍))) |
17 | 6, 16 | impbid 129 | 1 ⊢ (𝐺 ∈ Grp → ((𝑍 ∈ 𝐵 ∧ (𝑍 + 𝑍) = 𝑍) ↔ 0 = 𝑍)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 0gc0g 12693 Grpcgrp 12809 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-reu 2462 df-rmo 2463 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-riota 5828 df-ov 5875 df-inn 8916 df-2 8974 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-0g 12695 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 |
This theorem is referenced by: (None) |
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