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Mirrors > Home > MPE Home > Th. List > grpidlcan | Structured version Visualization version GIF version |
Description: If left adding an element of a group to an arbitrary element of the group results in this element, the added element is the identity element and vice versa. (Contributed by AV, 15-Mar-2019.) |
Ref | Expression |
---|---|
grpidrcan.b | ⊢ 𝐵 = (Base‘𝐺) |
grpidrcan.p | ⊢ + = (+g‘𝐺) |
grpidrcan.o | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpidlcan | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 )) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpidrcan.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpidrcan.p | . . . . 5 ⊢ + = (+g‘𝐺) | |
3 | grpidrcan.o | . . . . 5 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | grplid 18888 | . . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
5 | 4 | 3adant3 1132 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ( 0 + 𝑋) = 𝑋) |
6 | 5 | eqeq2d 2743 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ (𝑍 + 𝑋) = 𝑋)) |
7 | simp1 1136 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝐺 ∈ Grp) | |
8 | simp3 1138 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑍 ∈ 𝐵) | |
9 | 1, 3 | grpidcl 18886 | . . . 4 ⊢ (𝐺 ∈ Grp → 0 ∈ 𝐵) |
10 | 9 | 3ad2ant1 1133 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 0 ∈ 𝐵) |
11 | simp2 1137 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → 𝑋 ∈ 𝐵) | |
12 | 1, 2 | grprcan 18894 | . . 3 ⊢ ((𝐺 ∈ Grp ∧ (𝑍 ∈ 𝐵 ∧ 0 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵)) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 )) |
13 | 7, 8, 10, 11, 12 | syl13anc 1372 | . 2 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = ( 0 + 𝑋) ↔ 𝑍 = 0 )) |
14 | 6, 13 | bitr3d 280 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑍 + 𝑋) = 𝑋 ↔ 𝑍 = 0 )) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6543 (class class class)co 7411 Basecbs 17148 +gcplusg 17201 0gc0g 17389 Grpcgrp 18855 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-iota 6495 df-fun 6545 df-fv 6551 df-riota 7367 df-ov 7414 df-0g 17391 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-grp 18858 |
This theorem is referenced by: grpidssd 18935 |
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