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| Mirrors > Home > MPE Home > Th. List > grpidd | Structured version Visualization version GIF version | ||
| Description: Deduce the identity element of a magma from its properties. (Contributed by Mario Carneiro, 6-Jan-2015.) |
| Ref | Expression |
|---|---|
| grpidd.b | ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) |
| grpidd.p | ⊢ (𝜑 → + = (+g‘𝐺)) |
| grpidd.z | ⊢ (𝜑 → 0 ∈ 𝐵) |
| grpidd.i | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) |
| grpidd.j | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) |
| Ref | Expression |
|---|---|
| grpidd | ⊢ (𝜑 → 0 = (0g‘𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqid 2737 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
| 3 | eqid 2737 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 4 | grpidd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
| 5 | grpidd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
| 6 | 4, 5 | eleqtrd 2843 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
| 7 | 5 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺))) |
| 8 | 7 | biimpar 477 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → 𝑥 ∈ 𝐵) |
| 9 | grpidd.p | . . . . . 6 ⊢ (𝜑 → + = (+g‘𝐺)) | |
| 10 | 9 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → + = (+g‘𝐺)) |
| 11 | 10 | oveqd 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥)) |
| 12 | grpidd.i | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
| 13 | 11, 12 | eqtr3d 2779 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
| 14 | 8, 13 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
| 15 | 10 | oveqd 7448 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 )) |
| 16 | grpidd.j | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) | |
| 17 | 15, 16 | eqtr3d 2779 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
| 18 | 8, 17 | syldan 591 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
| 19 | 1, 2, 3, 6, 14, 18 | ismgmid2 18681 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 0gc0g 17484 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-riota 7388 df-ov 7434 df-0g 17486 |
| This theorem is referenced by: ress0g 18775 imasmnd2 18787 smndex1id 18924 isgrpde 18975 xrs0 33008 |
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