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Mirrors > Home > MPE Home > Th. List > grpinvex | Structured version Visualization version GIF version |
Description: Every member of a group has a left inverse. (Contributed by NM, 16-Aug-2011.) (Revised by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
grpcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpcl.p | ⊢ + = (+g‘𝐺) |
grpinvex.p | ⊢ 0 = (0g‘𝐺) |
Ref | Expression |
---|---|
grpinvex | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpcl.b | . . . 4 ⊢ 𝐵 = (Base‘𝐺) | |
2 | grpcl.p | . . . 4 ⊢ + = (+g‘𝐺) | |
3 | grpinvex.p | . . . 4 ⊢ 0 = (0g‘𝐺) | |
4 | 1, 2, 3 | isgrp 18910 | . . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
5 | 4 | simprbi 495 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
6 | oveq2 7434 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋)) | |
7 | 6 | eqeq1d 2730 | . . . 4 ⊢ (𝑥 = 𝑋 → ((𝑦 + 𝑥) = 0 ↔ (𝑦 + 𝑋) = 0 )) |
8 | 7 | rexbidv 3176 | . . 3 ⊢ (𝑥 = 𝑋 → (∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ↔ ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 )) |
9 | 8 | rspccva 3610 | . 2 ⊢ ((∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
10 | 5, 9 | sylan 578 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ‘cfv 6553 (class class class)co 7426 Basecbs 17189 +gcplusg 17242 0gc0g 17430 Mndcmnd 18703 Grpcgrp 18904 |
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-ext 2699 |
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-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-grp 18907 |
This theorem is referenced by: dfgrp2 18933 grprcan 18944 grpinveu 18945 grprinv 18961 |
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