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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 2610 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2610 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
3 | eqid 2610 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | grpidd.z | . . 3 ⊢ (𝜑 → 0 ∈ 𝐵) | |
5 | grpidd.b | . . 3 ⊢ (𝜑 → 𝐵 = (Base‘𝐺)) | |
6 | 4, 5 | eleqtrd 2690 | . 2 ⊢ (𝜑 → 0 ∈ (Base‘𝐺)) |
7 | 5 | eleq2d 2673 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐵 ↔ 𝑥 ∈ (Base‘𝐺))) |
8 | 7 | biimpar 501 | . . 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 6566 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = ( 0 (+g‘𝐺)𝑥)) |
12 | grpidd.i | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 + 𝑥) = 𝑥) | |
13 | 11, 12 | eqtr3d 2646 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
14 | 8, 13 | syldan 486 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → ( 0 (+g‘𝐺)𝑥) = 𝑥) |
15 | 10 | oveqd 6566 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = (𝑥(+g‘𝐺) 0 )) |
16 | grpidd.j | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥 + 0 ) = 𝑥) | |
17 | 15, 16 | eqtr3d 2646 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
18 | 8, 17 | syldan 486 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (Base‘𝐺)) → (𝑥(+g‘𝐺) 0 ) = 𝑥) |
19 | 1, 2, 3, 6, 14, 18 | ismgmid2 17090 | 1 ⊢ (𝜑 → 0 = (0g‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 0gc0g 15923 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-riota 6511 df-ov 6552 df-0g 15925 |
This theorem is referenced by: ress0g 17142 imasmnd2 17150 isgrpde 17266 xrs0 29006 |
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