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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 18755 | . . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 )) |
5 | 4 | simprbi 498 | . 2 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ) |
6 | oveq2 7366 | . . . . 5 ⊢ (𝑥 = 𝑋 → (𝑦 + 𝑥) = (𝑦 + 𝑋)) | |
7 | 6 | eqeq1d 2739 | . . . 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 3581 | . 2 ⊢ ((∀𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐵 (𝑦 + 𝑥) = 0 ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
10 | 5, 9 | sylan 581 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵) → ∃𝑦 ∈ 𝐵 (𝑦 + 𝑋) = 0 ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∃wrex 3074 ‘cfv 6497 (class class class)co 7358 Basecbs 17084 +gcplusg 17134 0gc0g 17322 Mndcmnd 18557 Grpcgrp 18749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-iota 6449 df-fv 6505 df-ov 7361 df-grp 18752 |
This theorem is referenced by: dfgrp2 18776 grprcan 18785 grpinveu 18786 grprinv 18802 |
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